analysis, geometry, and modeling in finance advanced methods in option pricing

Analysis, Geometry, and Modeling in Finance

Advanced Methods in Option Pricing

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CHAPMAN & HALL/CRC Financial Mathematics Series

Aims and scope: The eld of nancial mathematics forms an ever-expanding slice of the nancial sector. This series aims to capture new developments and summarize what is known over the whole spectrum of this eld. It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners. The inclusion of numerical code and concrete real-world examples is highly encouraged.

Series EditorsM.A.H. Dempster Centre for Financial Research Judge Business School University of Cambridge

Dilip B. Madan Robert H. Smith School of Business University of Maryland

Rama Cont Center for Financial EngineeringColumbia UniversityNew York

Published TitlesAmerican-Style Derivatives; Valuation and Computation, Jerome Detemple

Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,

Pierre Henry-Labordère

Credit Risk: Models, Derivatives, and Management, Niklas Wagner

Engineering BGM, Alan Brace

Financial Modelling with Jump Processes, Rama Cont and Peter Tankov

An Introduction to Credit Risk Modeling, Christian Bluhm, Ludger Overbeck, and

Christoph Wagner

Introduction to Stochastic Calculus Applied to Finance, Second Edition,

Damien Lamberton and Bernard Lapeyre

Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and

John J. H. Miller

Portfolio Optimization and Performance Analysis, Jean-Luc Prigent

Quantitative Fund Management, M. A. H. Dempster, Georg Pug, and Gautam Mitra

Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers

Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and

Ludger Overbeck

Understanding Risk: The Theory and Practice of Financial Risk Management,

David Murphy

Proposals for the series should be submitted to one of the series editors above or directly to:CRC Press, Taylor & Francis Group4th, Floor, Albert House1-4 Singer StreetLondon EC2A 4BQUK

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Analysis, Geometry, and Modeling in Finance

Advanced Methods in Option Pricing

Pierre Henry-Labordère

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Henry-Labordère, Pierre.Analysis, geometry, and modeling in finance: advanced methods in option

pricing / Pierre Henry-Labordère.p. cm. -- (Chapman & Hall/CRC financial mathematics series ; 13)

Includes bibliographical references and index.ISBN 978-1-4200-8699-7 (alk. paper)1. Options (Finance)--Mathematical models. I. Title. II. Series.

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To Emma and Veronique

List of Tables

2.1 Example of one-factor short-rate models. . . . . . . . . . . . . 47

5.1 Example of separable LV models satisfying C(0) = 0. . . . . . 1245.2 Feller criteria for the CEV model. . . . . . . . . . . . . . . . . 130

6.1 Example of SVMs. . . . . . . . . . . . . . . . . . . . . . . . . 1516.2 Example of metrics for SVMs. . . . . . . . . . . . . . . . . . . 155

8.1 Example of stochastic (or local) volatility Libor market models. 2088.2 Libor volatility triangle. . . . . . . . . . . . . . . . . . . . . . 214

9.1 Feller boundary classification for one-dimensional Ito processes. 2679.2 Condition at z = 0. . . . . . . . . . . . . . . . . . . . . . . . . 2769.3 Condition at z = 1. . . . . . . . . . . . . . . . . . . . . . . . . 2769.4 Condition at z =∞ . . . . . . . . . . . . . . . . . . . . . . . 2789.5 Example of solvable superpotentials . . . . . . . . . . . . . . 2789.6 Example of solvable one-factor short-rate models. . . . . . . . 2799.7 Example of Gauge free stochastic volatility models. . . . . . . 2819.8 Stochastic volatility models and potential J(s). . . . . . . . . 284

10.1 Example of potentials associated to LV models. . . . . . . . . 295

11.1 A dictionary from Malliavin calculus to QFT. . . . . . . . . . 310

B.1 Associativity diagram. . . . . . . . . . . . . . . . . . . . . . . 360B.2 Co-associativity diagram. . . . . . . . . . . . . . . . . . . . . 360

List of Figures

1.1 Implied volatility (multiplied by ×100) for EuroStoxx50 (03-09-2007). The two axes represent the strikes and the maturitydates. Spot S0 = 4296. . . . . . . . . . . . . . . . . . . . . . . 2

4.1 Manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 2-sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3 Line bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1 Comparison of the asymptotic solution at the first-order (resp.second-order) against the exact solution (5.23). f0 = 1, σ =0.3, β = 0.33, τ = 10 years. . . . . . . . . . . . . . . . . . . . 133

5.2 Comparison of the asymptotic solution at the first-order (resp.second-order) against the exact solution (5.23). f0 = 1, σ =0.3, β = 0.6, τ = 10 years. . . . . . . . . . . . . . . . . . . . . 134

5.3 Market implied volatility (SP500, 3-March-2008) versus Dupirelocal volatility (multiplied by ×100). T = 1 year. Note thatthe local skew is twice the implied volatility skew. . . . . . . 137

5.4 Comparison of the asymptotic implied volatility (5.41) at thezero-order (resp. first-order) against the exact solution (5.42).f0 = 1, σ = 0.3, τ = 10 years, β = 0.33. . . . . . . . . . . . . 142

5.5 Comparison of the asymptotic implied volatility (5.41) at thezero-order (resp. first-order) against the exact solution (5.42).f0 = 1, σ = 0.3%, τ = 10 years, β = 0.6. . . . . . . . . . . . . 143

6.1 Poincare disk D and upper half-plane H2 with some geodesics.In the upper half-plane, the geodesics correspond to verticallines and to semi-circles centered on the horizon =(z) = 0 andin D the geodesics are circles orthogonal to D. . . . . . . . . 169

6.2 Probability density p(K,T |f0) = ∂2C(T,K)∂2K . Asymptotic solu-

tion vs numerical solution (PDE solver). The Hagan-al formulahas been plotted to see the impact of the mean-reverting term.Here f0 is a swap spot and α has been fixed such that the Blackvolatility αfβ−1

0 = 30%. . . . . . . . . . . . . . . . . . . . . . 1726.3 Implied volatility for the SABR model τ = 1Y. α = 0.2, ρ =

−0.7, ν2

2 τ = 0.5 and β = 1. . . . . . . . . . . . . . . . . . . . 174

6.4 Implied volatility for the SABR model τ = 5Y. α = 0.2, ρ =−0.7, ν2

2 τ = 2.5 and β = 1. . . . . . . . . . . . . . . . . . . . 1756.5 Implied volatility for the SABR model τ = 10Y. α = 0.2, ρ =

−0.7, ν2

2 τ = 5 and β = 1. . . . . . . . . . . . . . . . . . . . . 1766.6 Exact conditional probability for the normal SABR model ver-

sus numerical PDE. . . . . . . . . . . . . . . . . . . . . . . . 178

7.1 Basket implied volatility with constant volatilities. ρij = e−0.3|i−j|,σi = 0.1 + 0.1× i, i, j = 1, 2, 3, T = 5 years. . . . . . . . . . . 197

7.2 Basket implied volatility with CEV volatilities. ρij = e−0.3|i−j|,σi = 0.1 + 0.1× i, i, j = 1, 2, 3, βCEV = 0.5, T = 5 years. . . . 198

7.3 Basket implied volatility with LV volatilities (NIKKEI-SP500-EUROSTOCK, 24-04-2008). ρij = e−0.4|i−j|, T = 3 years. . . 199

7.4 CCO. Zero-correlation case. . . . . . . . . . . . . . . . . . . . 2027.5 Exact versus asymptotic prices. 3 Assets, 1 year. . . . . . . . 2047.6 Exact versus asymptotic prices. 3 Assets, 5 years. . . . . . . . 204

8.1 Instantaneous correlation between Libors in a 2-factor LMM.k1 = 0.25, k2 = 0.04, θ1 = 100%, θ2 = 50% and ρ = −20%. . . 210

8.2 Comparison between HW2 and BGM models. . . . . . . . . . 2168.3 Calibration of a swaption with the SABR model. . . . . . . . 2258.4 The figure shows implied volatility smiles for swaption 10 ×

0.5, 5 × 15 and 10 × 10 using our asymptotic formula and theAndersen-Andreasen expression. Here ν = 0 and β = 0.2. . . 236

8.5 The figure shows implied volatility smiles for swaption 10×0.5,5 × 15 and 10 × 10 using our asymptotic formula and a MCsimulation. Here ν = 20% and ρia = 0%. . . . . . . . . . . . . 237

8.6 CEV parameters βk calibrated to caplet smiles for EUR-JPY-USD curves. The x coordinate refers to the Libor index. . . . 238

8.7 The figure shows implied volatility smiles for swaption 5 × 15(EUR, JPY, USD curves-February 17th 2007) using our asymp-totic formula and a MC simulation. Here ν = 20% and ρia =−30%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

8.8 The figure shows implied volatility smiles for swaption 10× 10(EUR, JPY-February 17th 2007) using our asymptotic formulaand a MC simulation. Here ν = 20% and ρia = −30%. . . . . 240

8.9 The figure shows implied volatility smiles for swaption 10 ×10 (EUR-February 17th 2007) using different value of (ν, ρ).The LMM has been calibrated to caplet smiles and the ATMswaptions x× 2, x× 10. . . . . . . . . . . . . . . . . . . . . . 241

Symbol Description

P Probability measure, usu-ally the risk-neutral mea-sure.

PT Forward measure.Pαβ Forward swap measure.Ps Spot Libor measure.EP Expectation with respect to

the measure P.dPdQ Radon-Nikodyn derivative

of the measure P with re-spect to Q.

F Filtration.Wt Brownian motion.t1 ∧ t2 min(t1, t2).Γ(E) Smooth sections on a vector

bundle E.d Exterior derivative.gij Metric.Ai Abelian connection.P Parallel gauge transport.Q Potential.p(t, x|x0)Conditional probability at

time t between x and x0.∆(x, y) VanVleck-Morette determi-

nant.PtT Bond between the dates t

and T .DtT Discount factor between

the dates t and T .

sαβ Swap between the dates Tαand Tβ .

Li(t) Libor at time t between thedates Ti−1 and Ti.

rt Instantaneous interest rate.ftT Forward interest curve.1(x) Heaviside function: 1 if x ≥

0 and 0 otherwise.δ(x) Dirac function.R+ x ∈ R , x ≥ 0R∗+ x ∈ R , x > 0δji Kronecker symbol: δji = 1

if i = j, zero otherwise.< ·, · > scalar product on L2([0, 1]).csc csc(x) = 1

sin(x)

csch csch(x) = 1sinh(x)

sec sec(x) = 1cos(x)

sech sech(x) = 1cosh(x)

GBM Geometric Brownian Mo-tion.

LV Local Volatility.LVM Local Volatility Model.LMM Libor Market Model.PDE Partial Differential Equa-

tion.SDE Stochastic Differential

Equation.SV Stochastic Volatility.SVM Stochastic Volatility Model.

Contents

1 Introduction 1

2 A Brief Course in Financial Mathematics 72.1 Derivative products . . . . . . . . . . . . . . . . . . . . . . . 72.2 Back to basics . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Sigma-algebra . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Probability measure . . . . . . . . . . . . . . . . . . . 102.2.3 Random variables . . . . . . . . . . . . . . . . . . . . 102.2.4 Conditional probability . . . . . . . . . . . . . . . . . 122.2.5 Radon-Nikodym derivative . . . . . . . . . . . . . . . 13

2.3 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 132.4 Ito process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Stochastic integral . . . . . . . . . . . . . . . . . . . . 152.4.2 Ito’s lemma . . . . . . . . . . . . . . . . . . . . . . . . 192.4.3 Stochastic differential equations . . . . . . . . . . . . . 21

2.5 Market models . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Pricing and no-arbitrage . . . . . . . . . . . . . . . . . . . . 25

2.6.1 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.2 Self-financing portfolio . . . . . . . . . . . . . . . . . . 26

2.7 Feynman-Kac’s theorem . . . . . . . . . . . . . . . . . . . . . 322.8 Change of numeraire . . . . . . . . . . . . . . . . . . . . . . 342.9 Hedging portfolio . . . . . . . . . . . . . . . . . . . . . . . . 412.10 Building market models in practice . . . . . . . . . . . . . . 43

2.10.1 Equity asset case . . . . . . . . . . . . . . . . . . . . . 432.10.2 Foreign exchange rate case . . . . . . . . . . . . . . . 452.10.3 Fixed income rate case . . . . . . . . . . . . . . . . . . 462.10.4 Commodity asset case . . . . . . . . . . . . . . . . . . 49

2.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Smile Dynamics and Pricing of Exotic Options 553.1 Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Static replication and pricing of European option . . . . . . 573.3 Forward starting options and dynamics of the implied volatility 62

3.3.1 Sticky rules . . . . . . . . . . . . . . . . . . . . . . . . 623.3.2 Forward-start options . . . . . . . . . . . . . . . . . . 633.3.3 Cliquet options . . . . . . . . . . . . . . . . . . . . . . 633.3.4 Napoleon options . . . . . . . . . . . . . . . . . . . . . 64

3.4 Interest rate instruments . . . . . . . . . . . . . . . . . . . . 643.4.1 Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.2 Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4.3 Swaption . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.4 Convexity adjustment and CMS option . . . . . . . . 68

3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Differential Geometry and Heat Kernel Expansion 754.1 Multi-dimensional Kolmogorov equation . . . . . . . . . . . . 75

4.1.1 Forward Kolmogorov equation . . . . . . . . . . . . . 764.1.2 Backward Kolmogorov’s equation . . . . . . . . . . . . 78

4.2 Notions in differential geometry . . . . . . . . . . . . . . . . 804.2.1 Manifold . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.2 Maps between manifolds . . . . . . . . . . . . . . . . . 814.2.3 Tangent space . . . . . . . . . . . . . . . . . . . . . . 824.2.4 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2.5 Cotangent space . . . . . . . . . . . . . . . . . . . . . 844.2.6 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.7 Vector bundles . . . . . . . . . . . . . . . . . . . . . . 884.2.8 Connection on a vector bundle . . . . . . . . . . . . . 904.2.9 Parallel gauge transport . . . . . . . . . . . . . . . . . 934.2.10 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 944.2.11 Curvature of a connection . . . . . . . . . . . . . . . . 1014.2.12 Integration on a Riemannian manifold . . . . . . . . . 102

4.3 Heat kernel on a Riemannian manifold . . . . . . . . . . . . 1034.4 Abelian connection and Stratonovich’s calculus . . . . . . . . 1074.5 Gauge transformation . . . . . . . . . . . . . . . . . . . . . . 1084.6 Heat kernel expansion . . . . . . . . . . . . . . . . . . . . . . 1104.7 Hypo-elliptic operator and Hormander’s theorem . . . . . . . 116

4.7.1 Hypo-elliptic operator . . . . . . . . . . . . . . . . . . 1164.7.2 Hormander’s theorem . . . . . . . . . . . . . . . . . . 117

4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Local Volatility Models and Geometry of Real Curves 1235.1 Separable local volatility model . . . . . . . . . . . . . . . . 123

5.1.1 Weak solution . . . . . . . . . . . . . . . . . . . . . . . 1245.1.2 Non-explosion and martingality . . . . . . . . . . . . . 1265.1.3 Real curve . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2 Local volatility model . . . . . . . . . . . . . . . . . . . . . . 1345.2.1 Dupire’s formula . . . . . . . . . . . . . . . . . . . . . 1345.2.2 Local volatility and asymptotic implied volatility . . . 136

5.3 Implied volatility from local volatility . . . . . . . . . . . . . 145

6 Stochastic Volatility Models and Geometry ofComplex Curves 1496.1 Stochastic volatility models and Riemann surfaces . . . . . . 149

6.1.1 Stochastic volatility models . . . . . . . . . . . . . . . 1496.1.2 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . 1536.1.3 Associated local volatility model . . . . . . . . . . . . 1576.1.4 First-order asymptotics of implied volatility . . . . . . 159

6.2 Put-Call duality . . . . . . . . . . . . . . . . . . . . . . . . . 1626.3 λ-SABR model and hyperbolic geometry . . . . . . . . . . . 164

6.3.1 λ-SABR model . . . . . . . . . . . . . . . . . . . . . . 1656.3.2 Asymptotic implied volatility for the λ-SABR . . . . . 1656.3.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.4 Analytical solution for the normal and log-normal SABR model 1766.4.1 Normal SABR model and Laplacian on H2 . . . . . . 1766.4.2 Log-normal SABR model and Laplacian on H3 . . . . 178

6.5 Heston model: a toy black hole . . . . . . . . . . . . . . . . . 1816.5.1 Analytical call option . . . . . . . . . . . . . . . . . . 1816.5.2 Asymptotic implied volatility . . . . . . . . . . . . . . 183

6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7 Multi-Asset European Option and Flat Geometry 1877.1 Local volatility models and flat geometry . . . . . . . . . . . 1877.2 Basket option . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.2.1 Basket local volatility . . . . . . . . . . . . . . . . . . 1917.2.2 Second moment matching approximation . . . . . . . 195

7.3 Collaterized Commodity Obligation . . . . . . . . . . . . . . 1967.3.1 Zero correlation . . . . . . . . . . . . . . . . . . . . . . 2007.3.2 Non-zero correlation . . . . . . . . . . . . . . . . . . . 2017.3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . 203

8 Stochastic Volatility Libor Market Models and HyperbolicGeometry 2058.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.2 Libor market models . . . . . . . . . . . . . . . . . . . . . . 207

8.2.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . 2088.2.2 Pricing with a Libor market model . . . . . . . . . . . 216

8.3 Markovian realization and Frobenius theorem . . . . . . . . . 2208.4 A generic SABR-LMM model . . . . . . . . . . . . . . . . . 2248.5 Asymptotic swaption smile . . . . . . . . . . . . . . . . . . . 226

8.5.1 First step: deriving the ELV . . . . . . . . . . . . . . 2268.5.2 Connection . . . . . . . . . . . . . . . . . . . . . . . . 2308.5.3 Second step: deriving an implied volatility smile . . . 2338.5.4 Numerical tests and comments . . . . . . . . . . . . . 234

8.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2378.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9 Solvable Local and Stochastic Volatility Models 2479.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.2 Reduction method . . . . . . . . . . . . . . . . . . . . . . . . 2499.3 Crash course in functional analysis . . . . . . . . . . . . . . . 251

9.3.1 Linear operator on Hilbert space . . . . . . . . . . . . 2529.3.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 2559.3.3 Spectral decomposition . . . . . . . . . . . . . . . . . 256

9.4 1D time-homogeneous diffusion models . . . . . . . . . . . . 2629.4.1 Reduction method . . . . . . . . . . . . . . . . . . . . 2629.4.2 Solvable (super)potentials . . . . . . . . . . . . . . . . 2699.4.3 Hierarchy of solvable diffusion processes . . . . . . . . 2739.4.4 Natanzon (super)potentials . . . . . . . . . . . . . . . 274

9.5 Gauge-free stochastic volatility models . . . . . . . . . . . . 2799.6 Laplacian heat kernel and Schrodinger equations . . . . . . . 2849.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

10 Schrodinger Semigroups Estimates and Implied VolatilityWings 28910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 28910.2 Wings asymptotics . . . . . . . . . . . . . . . . . . . . . . . . 29010.3 Local volatility model and Schrodinger equation . . . . . . . 293

10.3.1 Separable local volatility model . . . . . . . . . . . . . 29310.3.2 General local volatility model . . . . . . . . . . . . . . 295

10.4 Gaussian estimates of Schrodinger semigroups . . . . . . . . 29610.4.1 Time-homogenous scalar potential . . . . . . . . . . . 29610.4.2 Time-dependent scalar potential . . . . . . . . . . . . 298

10.5 Implied volatility at extreme strikes . . . . . . . . . . . . . . 30010.5.1 Separable local volatility model . . . . . . . . . . . . . 30010.5.2 Local volatility model . . . . . . . . . . . . . . . . . . 302

10.6 Gauge-free stochastic volatility models . . . . . . . . . . . . 30310.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

11 Analysis on Wiener Space with Applications 30911.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 30911.2 Functional integration . . . . . . . . . . . . . . . . . . . . . . 310

11.2.1 Functional space . . . . . . . . . . . . . . . . . . . . . 31011.2.2 Cylindrical functions . . . . . . . . . . . . . . . . . . . 31011.2.3 Feynman path integral . . . . . . . . . . . . . . . . . . 311

11.3 Functional-Malliavin derivative . . . . . . . . . . . . . . . . . 31311.4 Skorohod integral and Wick product . . . . . . . . . . . . . . 317

11.4.1 Skorohod integral . . . . . . . . . . . . . . . . . . . . . 31711.4.2 Wick product . . . . . . . . . . . . . . . . . . . . . . . 319

11.5 Fock space and Wiener chaos expansion . . . . . . . . . . . . 32211.5.1 Ornstein-Uhlenbeck operator . . . . . . . . . . . . . . 324

11.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

11.6.1 Convexity adjustment . . . . . . . . . . . . . . . . . . 32511.6.2 Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . 32711.6.3 Local volatility of stochastic volatility models . . . . . 332

11.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

12 Portfolio Optimization and Bellman-Hamilton-Jacobi Equa-tion 33912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33912.2 Hedging in an incomplete market . . . . . . . . . . . . . . . 34012.3 The feedback effect of hedging on price . . . . . . . . . . . . 34312.4 Non-linear Black-Scholes PDE . . . . . . . . . . . . . . . . . 34512.5 Optimized portfolio of a large trader . . . . . . . . . . . . . . 345

A Saddle-Point Method 351

B Monte-Carlo Methods and Hopf Algebra 353B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

B.1.1 Monte Carlo and Quasi Monte Carlo . . . . . . . . . . 354B.1.2 Discretization schemes . . . . . . . . . . . . . . . . . . 354B.1.3 Taylor-Stratonovich expansion . . . . . . . . . . . . . 356

B.2 Algebraic Setting . . . . . . . . . . . . . . . . . . . . . . . . 358B.2.1 Hopf algebra . . . . . . . . . . . . . . . . . . . . . . . 359B.2.2 Chen series . . . . . . . . . . . . . . . . . . . . . . . . 363

B.3 Yamato’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 365

References 369

Index 379

Chapter 1

Introduction

Faire des mathematiques, c’est donner le meme nom a des chosesdifferentes.

— Henri Poincare

With the numerous books on mathematical finance published each year, theusefulness of a new one may be questioned. This is the first book settingout the applications of advanced analytical and geometrical methods usedin recent physics and mathematics to the financial field. This means thatnew results are obtained when only approximate and partial solutions werepreviously available.We present powerful tools and methods (such as differential geometry, spectraldecomposition, supersymmetry) that can be applied to practical problemsin mathematical finance. Although encountered across different domains intheoretical physics and mathematics (for example differential geometry ingeneral relativity, spectral decomposition in quantum mechanics), they remainquite unheard of when applied to finance and allow to obtain new resultsreadily. We introduce these methods through the problem of option pricing.An option is a financial contract that gives the holder the right but not theobligation to enter into a contract at a fixed price in the future. The simplestexample is a European call option that gives the right but not the obligationto buy an asset at a fixed price, called strike, at a fixed future date, calledmaturity date. Since the work by Black, Scholes [65] and Merton [32] in 1973,a general probabilistic framework has been established to price these options.In this framework, the financial variables involved in the definition of an op-tion are random variables and their dynamics follow stochastic differentialequations (SDEs). For example, in the original Black-Scholes-Merton model,the traded assets are assumed to follow log-normal diffusion processes withconstant volatilities. The volatility is the standard deviation of a probabil-ity density in mathematical finance. The option price satisfies a (parabolic)partial differential equation (PDE), called the Kolmogorov-Black-Scholes pric-ing equation, depending on the stochastic differential equations introduced tomodel the market.The market model depends on unobservable or observable parameters suchas the volatility of each asset. They are chosen, we say calibrated, in orderto reproduce the price of liquid options quoted on the market such as Euro-

1

2 Analysis, Geometry, and Modeling in Finance

pean call options. If liquid options are not available, historical data for themovement of the assets can be used.

21-S

ep-0

720

-Mar

-08

20-M

ar-0

9

19-M

ar-1

0

16-D

ec-1

1

20-D

ec-1

3

03-S

ep-2

064

4.4

3436

.8

4296

4940

.4

7088

.405101520253035404550

FIGURE 1.1: Implied volatility (multiplied by ×100) for EuroStoxx50 (03-09-2007). The two axes represent the strikes and the maturity dates. SpotS0 = 4296.

This book mainly focuses on the study of the calibration and the dynamics ofthe implied volatility (commonly called smile). The implied volatility is thevalue of the volatility that, when put in the Black-Scholes formula, reproducesthe market price for a European call option. In the original log-normal Black-Scholes model, regarding the hypothesis of constant volatility, the impliedvolatility is flat as a function of the maturity date and the strike. However,the implied volatility observed on the market is not flat and presents a U-shape (see Fig. 1.1). This is one of the indications that the Black-Scholesmodel is based on several unrealistic assumptions which are not satisfied un-der real market conditions. For example, assets do not follow log-normal

Introduction 3

processes with constant volatilities as they possess “fat tail” probability den-sities. These unrealistic hypotheses, present in the Black-Scholes model, canbe relaxed by introducing more elaborate models called local volatility modelsand stochastic volatility models that, in most cases, can calibrate the form ofthe initially observed market smile and give dynamics for the smile consistentwith the fair value of exotic options. To be thorough, we should mentionjump-diffusion models. Although interesting and mathematically appealing,we will not discuss these models in this book as they are extensively discussedin [9].On the one hand, local volatility models (LVMs) assume that the volatilityσloc(t, f) of an asset only depends on its price f and on the time t. As shownby Dupire [81], there is a unique diffusion term σloc(t, f), depending on thederivatives of European call options with respect to the time to maturity andthe strike, which can exactly reproduce the current market of European calloption prices. Dupire then shows how to get the calibration solution. However,the dynamics is not consistent with what is observed on the market. In thiscontext, a better alternative is to introduce (time-homogeneous) stochasticvolatility models.On the other hand, stochastic volatility models (SVMs) assume that thevolatility itself follows a stochastic process. LVMs can be seen as a degen-erate example of SVMs as it can be shown that the local volatility functionrepresents some kind of average over all possible instantaneous volatilities ina SVM.For LVMs or SVMs, the implied volatility can be obtained by various methods.For LVMs (resp. SVMs) European call options satisfy the Kolmogorov andBlack-Scholes equation which is a one-dimensional (resp. two-dimensional ormore) parabolic PDE. This PDE can be traditionally solved by a finite differ-ence scheme (for example Crank-Nicholson) or by a Monte-Carlo simulationvia the Feynman-Kac theorem. These pricing methods turn out to be fairlytime-consuming on a personal computer and they are generally not appropri-ate when one tries to calibrate the model to a large number of (European)options. In this context, it is much better to use analytical or asymptoticalmethods. This is what the book is intended for.For both local and stochastic volatility models, the resulting Black-ScholesPDE is complicated and only a few analytical solutions are known. Whenexact solutions are not available, singular perturbative techniques have beenused to obtain asymptotic expressions for the price of European-style options.We will explain how to obtain these asymptotic expressions, in a unified man-ner, with any kind of stochastic volatility models using the heat kernel expan-sion technique on a Riemannian manifold.In order to guide the reader, here is the general description of the book’schapters:• A quick introduction to the theory of option pricing: The purpose is toallow the reader to acquaint with the main notions and tools useful for pricingoptions. In this context, we review the construction of Ito diffusion processes,


martingales and change of measures. Problems have been added at the endso that the reader can check his understanding.• A recall of a few definitions regarding the dynamics of the implied volatil-ity: In particular, we review the different forward starting options (cliquet,Napoleon...) and options on volatility that give a strong hint on the dynamicsof the smile (in equity markets). A similar presentation is given for the studyof the dynamics of swaption implied volatilities using specific interest rateinstruments.• A review of the heat kernel expansion on a Riemannian manifold endowedwith an Abelian connection: thanks to the rewriting of the Kolmogorov-Black-Scholes equation as a heat kernel equation, we give a general asymptotic so-lution in the short-time limit to the Kolmogorov equation associated to ageneral multi-dimensional Ito diffusion process. The main notions of differen-tial geometry, useful to grasp the heat kernel expansion, are reviewed carefully.Therefore no prerequisite in geometry is needed.• A focus on the local and stochastic volatility models: In the geometricalframework introduced in the previous chapter, the stochastic (resp. local)volatility model corresponds to the geometry of complex (resp. real) curves(i.e., Riemann surfaces). For example, the SABR stochastic volatility model,particularly used in the fixed-income market, can be associated to the geom-etry of the (hyperbolic) Poincare plane. By using the heat kernel expansion,we obtain a general asymptotic implied volatility for any SVM, in particularthe SABR model with a mean-reverting drift.• Applications to the pricing of multi-asset European options such as equitybaskets, Collateralized Commodity Obligations (CCO) and swaptions: In par-ticular, we find an asymptotic implied volatility formula for a European basketwhich is valid for a general multi-dimensional LVM and an asymptotic swap-tion implied volatility valid for a stochastic volatility Libor market model. Inthis chapter, we review the main issues on the construction and calibration ofa Libor market model.• A classification of solvable LVMs and SVMs: Solvable means that the priceof a European call option can be written in terms of hypergeometric func-tions. Recasting the Black-Scholes-Kolmogorov PDE in a geometrical settingas described in chapter 4, we show how to reduce the complexity of thisequation. The three main ingredients are the group of diffeomorphisms, thegroup of gauge transformations and the supersymmetry. For LVMs, thesethree transformations allow to reduce the backward Kolmogorov equation toa (Euclidean) Schrodinger type equation with a scalar potential for which aclassification of solvable potentials is already known. This chapter illustratesthe power of the differential geometry approach as it reproduces and enlargesthe classification of solvable LVMs and SVMs. In addition, a new useful toolis introduced: the spectral decomposition of unbounded linear operators. Areview of the theory of unbounded operators on a Hilbert space will be givenin this context.• Chapter 10 studies the large-strike behavior of the implied volatility for

Introduction 5

LVMs and SVMs. We use two-sided Gaussian estimates of Schrodinger equa-tions with scalar potentials belonging to the Kato class.• In chapter 11, we give a brief overview of the Malliavin calculus. We focuson two applications: Firstly, we obtain probabilistic representations of sensi-tivities of derivatives products according to model parameters. Secondly, weshow how to compute by Monte-Carlo simulation the local volatility functionassociated to SVMs.• In the last chapter, our asymptotic methods are applied to non-linear PDEs,mainly the Bellman-Hamilton-Jacobi equation. We focus on the problem ofpricing options when the market is not complete or is illiquid. In the lattercase, the hedging strategy of a large trader has an impact on the market. Theresulting Black-Scholes equation becomes a non-linear PDE.• Two appendices are included. The first one summarizes the saddle-pointmethod and the second one briefly explains Monte-Carlo methods. In thispart, we highlight the Hopf algebra structure of Taylor-Stratonovich expan-sions of SDEs. This allows to prove the Yamato theorem giving a necessaryand sufficient condition to represent asset prices as functionals of Brownianmotions.Throughout this book, we have tried to present not only a list of theoreticalresults but also as many as possible numerical ones. The numerical imple-mentations have been done with Mathematica c© and C++. Also, problemshave been added at the end of most chapters. They are based on recentlypublished research papers and allow the reader to check his understandingand identify the main issues arising about the financial industry.

Book audience

In the derivatives finance field, one can mainly identify four types of profes-sionals.

• The structurer who designs the derivatives products.

• The trader who is directly in contact with the market and daily cali-brates the computer models. He determines the current prices of theproducts and the necessary hedging.

• The salespeople who are in charge of selling the products at the condi-tions decided by the trader.

• The quantitative analyst who is someone with a strong background inmathematics and is in charge of developing the mathematical models& methods and computer programs aiming at the optimum pricing andhedging of the financial risks by the traders.


Among the above mentioned professionals, this book will be useful to quanti-tative analysts and highly motivated traders to better understand the modelsthey are using.Also, many Ph.D. students in mathematics and theoretical physics move tofinance as quantitative analysts. A purpose of this book is to explain the newapplications of some advanced mathematics to better solve practical problemsin finance.It is however not intended to be a full monograph on stochastic differentialgeometry. Detailed proofs are not included and are replaced by relevant ref-erences.

Acknowledgments

I would like to thank my father, Prof. A. Henry-Labordere, for advice in thewriting of this book. I would like to thank my colleagues in the Equity Deriva-tives Quantitative research team at Societe Generale for useful discussions andfeedback on the contents of this book.Finally, this book could not have been written without the support from mywife Veronique and my daughter Emma.

About the author

Dr. Pierre Henry-Labordere works in the Equity Derivatives Quantitativeresearch team at Societe Generale as a quantitative analyst. After receivinghis Ph.D. at Ecole Normale Superieure (Paris) in the Theory of Superstrings,he worked in the theoretical physics department at Imperial College (London)before moving to finance in 2004. He also graduated from Ecole Centrale Parisand holds DEAs in theoretical physics and mathematics.

Chapter 2

A Brief Course in FinancialMathematics

The enormous usefulness of Mathematics in the natural sciencesis something bordering on the mysterious and that there is norational explanation for it.1

—E. Wigner

Abstract Finance is dominated by stochastic methods. In particular,traded assets are assumed to follow time-continuous Ito diffusion processes.The notions of (local) martingale, Ito diffusion process, equivalent measureappear naturally. In this chapter, we review the main ideas in mathematicalfinance and motivate the mathematical concepts.

2.1 Derivative products

Instead of giving a general definition of a derivative product, let us presenta few classical examples. The first one is a European call option on a singleasset which is a contract which gives to the holder the opportunity but notthe obligation to buy an asset at a given price (called strike) at a fixed futuredate (called maturity). Let us suppose a call option is bought with a 100 $strike and a five year maturity T = 5. The initial price of the asset, calledthe spot, is 100 $. If the stock price is 150 $ in five years, the option can beexercised and the buyer is free to purchase the 100 $ stock as the contractallows. The counterpart which has sold the contract must then buy the 150 $stock on the market (if he does not have it) and sell it at 100 $. When thebuyer has received the stock for 100 $, he can sell it directly on the marketat 150 $ (we assume that there is no transaction cost). So his net earning is50− C$ and the bank has lost −50 + C$, C being the price of the contract.In practice, we don’t have a two-player zero sum game as the bank hedges itsrisk by investing in a portfolio with traded assets and options. The earning

1“The Unreasonable Effectiveness of Mathematics in the Natural Sciences” in Communica-tions in Pure and Applied Mathematics, vol. 13, No. I (February 1960).

7


of the bank at the maturity date T is therefore −50 + C + πT where πT is theportfolio value at T . As a result, both the option buyer and the seller canhave a positive earning at T .We define the payoff representing the potential gain at maturity date T as

f(ST ) = max(ST −K, 0)

with K the strike and ST the stock price at maturity T . The net earning ofthe buyer will be

f(ST )− C

whether he calls the option or not because he purchased the contract C. Abank, after defining the characteristics of the option, needs to compute thefair value C and to build a hedging strategy to cover its own risk.More generally, each derivative product is characterized by a payoff dependingon financial (random) variables such as equity stocks, commodity assets, fixed-income rates and foreign exchange rates between various currencies.In practice, the derivative products are more complex than a European calloption. Several characteristic features can however be outlined. A derivativecan be

• European: meaning that the option can be exercised by the holder ata specified maturity date. The simplest example is the European calloption that we have presented above. Another example most straightfor-ward would be given by a European put option which gives the holderthe right but not the obligation to sell an asset at a fixed price (i.e.,strike) at the maturity date T . The payoff at T is

max(K − ST , 0)

• American: meaning that the option can be exercised either within aspecific time frame or at set dates (in the latter case the option is calledBermudan). The simplest example is an American put option whichgives to the holder the right but not the obligation to sell an asset at afixed price (i.e., strike) at any time up to the maturity date.

• Asian: meaning that the option depends on the path of some assets.For example, a European Asian call option on a single stock with strikeK is defined by the following payoff at the maturity date T

max

(1

T − t

∫ T

t

Sudu−K, 0

)with Su the stock price at a time u. In practice, the integral is under-stood as a Riemann sum over the business days from t1 = t to tN = T ,1N

∑Ni=1 Sti . The value of the payoff depends therefore on the arithmetic

average of the stock price from t up to the maturity date T .

A Brief Course in Financial Mathematics 9

• Barrier: meaning that a European option is activated if an asset hasnot or has reached a barrier level up to the maturity date.

Moreover, an option does not necessarily depend on a single asset but candepend on many financial products. In this case, one says that we have amulti-asset option. For example, a classical multi-asset option is a Europeanbasket option whose payoff at maturity T with strike K depends on the valueof an (equity) basket

max

(∑i=1

ωiSiT −K, 0

)

with ωi the weight normalized by∑i=1 ωi = 1 and SiT the price of the asset i

at maturity T . Increasing the complexity level, one can also consider optionsdepending on equity assets, fixed income rates and foreign exchange rates(FX). One says in this case that the option is hybrid. For example, let usconsider an option that pays a foreign exchange rate FXTi (resp. a fixedcoupon ci) at some specified date Ti if a stock price STi at Ti is greater (resp.lower) than a barrier U . The payoff is therefore2

FXTi1(STi − U) + ci1(U − STi)

In the following sections, we model the financial random variables that enterin the definition of a derivative product with Ito diffusion processes. For thereader with no familiarity with probability theory, we have included in thenext section a reminder of definitions and results in probability. Details andproofs can be found in [26].

2.2 Back to basics

2.2.1 Sigma-algebra

In probability, the space of events that can appear are formalized by the notionof a σ-algebra. If Ω is a given set, then a σ-algebra for Ω is a family F ofsubsets of Ω with the following properties:

1. ∅ ∈ F

2. F ∈ F ⇒ F c ∈ F where F c = Ω \ F is the complement of F in Ω

3. F1, F2, · · · ∈ F =⇒ F = ∪∞i=1Fi ∈ F

21(x) is the Heaviside function meaning that 1(x) = 1 (resp. 0) if x > 0 (resp. x ≤ 0).


The pair (Ω,F) is called a measurable space. In the following, an element of Ωis noted ω. If we replace the third condition by finite union (and intersection),we have an algebra.

Example 2.1 Borel σ-algebraWhen we have a topological space X, the smallest σ-algebra generated by theopen sets is called the Borel σ-algebra that we note below B(X).

We can endow a measurable space with a probability measure.

2.2.2 Probability measure

A probability measure P on a measurable space (Ω,F) is a function

P : F → [0, 1]

such that

1. P(∅) = 0, P(Ω) = 1

2. If F1, F2, · · · ∈ F and (Fi)∞i=1 are disjoint sets then

P(∪∞i=1Fi) =∞∑i=1

P(Fi)

The triple (Ω,F ,P) is called a probability space. It is called a complete prob-ability space if A ⊂ B , B ∈ F and P(B) = 0 then A ∈ F . The sets B withzero measure are called negligible sets. We say that a property holds almostsurely (a.s.) if it holds outside a negligible set.In the following, all probability spaces will be assumed to be complete. Itis often simpler to construct a measure on an algebra which generates theσ-algebra. The issue is then to extend this measure to the σ-algebra itself.This extension exists and is unique as stated by the Caratheodory extensiontheorem.

THEOREM 2.1 Caratheodory extension theoremLet A be an algebra and P : A → [0, 1] be a probability measure on A. There

exists a unique probability measure P′ : σ(A) → [0, 1] on the σ-algebra σ(A)generated by A such that P′|A = P.

2.2.3 Random variables

A function X : Ω→ Rn is called F-measurable if

X−1(U) ≡ ω ∈ Ω;X(ω) ∈ U ∈ F


for all open sets U ∈ Rn.

Example 2.2 Continuous functionEvery continuous function f from (Rn,B(Rn)) endowed with its Borel σ-algebra B(Rn) to (Rm,B(Rm)) is measurable. Indeed, according to the def-inition of a continuous function, f−1(U) with U ∈ B(Rm) an open set is anopen set and therefore belongs to B(Rn).

By definition, a random variable (r.v.) X : Ω → Rn is a F-measurablefunction. Note that the terminology is not suitable as a r.v. is actually not avariable but a function.Every r.v. X can induce a probability measure, noted mX , on Rn defined by

mX(B) = P(X−1(B)) (2.1)

with B an open set of Rn. Note that this relation is well defined as X−1(B) ∈F by definition of the measurability of X.At this stage we can introduce the expectation of a r.v. First, we restrainthe definition to a particular class of r.v., the simple r.v.s (also called stepfunctions) which take only a finite number of values and hence can be writtenas3

X =m∑i=1

xi1Ai (2.2)

where Ai ∈ F . For this class of (measurable) functions, the expectation EP[X]is the number

EP[X] =m∑i=1

xiP(Ai)

From this definition, it is clear that EP[·] is a linear operator acting on thevector space of simple r.v. The operator EP[·] can be extended to non-negativer.v. by

EP[X] = sup EP[Y ] , Y simple r.v. , 0 ≤ Y ≤ X

Note that the expectation above can be ∞. Finally, for an arbitrary r.v, X,we decompose X into a difference of two non-negative r.v. X+ = max(X, 0)and X− = −min(X, 0)

X = X+ −X−

31Ai (x) = 0 if x ∈ Ai, zero otherwise.


and we set

EP[X] = EP[X+]− EP[X−] (2.3)

This expectation is not always defined. When EP[X+] and EP[X−] are finite,the r.v. is called integrable. This is equivalent to EP[|X|] < ∞. L1(Ω,F ,P)denotes the set of all integrable r.v. Moreover, the set of all k times integrabler.v. is noted Lk(Ω,F ,P). The expectation (2.3) of a r.v. is called the firstmoment and it is also possible to define higher moments:

mk ≡ EP[|X|k] <∞ , X ∈ Lk(Ω,F ,P)

For example, the variance of a r.v., defined for X ∈ L2(Ω,F ,P), is

Var[X] = EP[X2]− EP[X]2

Note that when the random variable X has a density p(x) on R, it is integrableif and only if

∫R |x|p(x)dx <∞ and the expectation value is

EP[X] =∫

Rxp(x)dx

2.2.4 Conditional probability

In probability theory, it may be useful to compute expectations of r.v. condi-tional to some information that we have. This is formalized by the notion ofconditional expectation.

DEFINITION 2.1 Conditional expectation Let X ∈ L1(Ω,F ,P) andlet G be a sub σ-algebra of F . Then the conditional expectation of X given G,denoted EP[X|G], is defined as follows:

1. EP[X|G] is G-measurable (⇒ it is a r.v.)

2. EP[XY ] = EP[EP[X|G]Y ] holds for any bounded G-measurable r.v. Y .

It can be shown that the map X → EP[X|G] is linear.If we have two r.v. X and Y admitting a probability density, the conditionalexpectation of X ∈ L1 conditional to Y = y can be computed as follows:The probability to have X ∈ [x, x + dx] and Y ∈ [y, y + dy] is by definitionp(x, y)dxdy. Then, we can prove that

EP[X|Y = y] =∫

Rxp(x, y)dx (2.4)

Indeed, following the definition, one needs to show that

EP[f(Y )EP[X|Y ]] = EP[f(Y )X]

for all bounded measurable function f . Injecting our guess (2.4) into thisequation, we obtain a trivial equality.


2.2.5 Radon-Nikodym derivative

This brief review of probability theory will conclude with the Radon-Nikodymderivative which will be used when we discuss the Girsanov theorem in section2.8.Let P be a probability space on (Ω,F) and let Q be a finite measure on (Ω,F)(i.e., Q takes its values in R+ and Q(A) <∞ ∀A ∈ F).We say that Q is equivalent to P (denoted Q ∼ P) if Q(A) = 0 if and only ifP(A) = 0 for every A ∈ F . The Radon-Nikodym theorem states that Q ∼ Pif and only if there exists a non-negative r.v. X such that

Q(A) = EP[1AX] , ∀A ∈ F

Moreover X is unique P-almost surely and we note

X =dQdP

X is called the Radon-Nikodym derivative of Q with respect to P.At this stage, we can define a stochastic process. Complements can be foundin [34] and [27].

2.3 Stochastic processes

DEFINITION 2.2 Stochastic process A n-dimensional stochasticprocess is a family of random variables Xtt≥0 defined on a probability space(Ω,F ,P) and taking values in Rn.

In chapter 4, we will consider stochastic processes that take their values in aRiemannian manifold.In finance, an asset price is modeled by a stochastic process. The past valuesof the price are completely known (historical data). The information thatwe have about a stochastic process up to a certain time (usually today) isformalized by the notion of filtration.

DEFINITION 2.3 Filtration A filtration on (Ω,F) is a family Ftt≥0

of σ sub-algebras Ft ⊂ F such that for 0 ≤ s ≤ t, Fs ⊂ Ft.

The σ-algebra Ft represents the information we have up to the time t. As Ftis a sub σ-algebra of F , we can define the conditional expectation of the r.v.Xs according to the filtration Ftt, with s > t

EP[Xs|Ft]


By definition, this r.v. is Ft-measurable. This is not necessarily the case forXt which should only be F-measurable. When the r.v. Xt is Ft-measurablefor every t, we will say that the process X is adapted to the filtration Ftt≥0.

The most important example of stochastic processes is the Brownian motionwhich will be our building block for elaborating more complex stochastic pro-cesses. From this Brownian motion, we can generate a natural filtration.

Example 2.3 Brownian motionA Brownian motion (or Wiener process) Wt is a continuous adapted processdefined on a probability space (Ω,F ,P) with the properties

• W0 = 0 (with probability one).

• All increments on non-overlapping time intervals are independent: thatis Wtn −Wtn−1 , ..., Wt2 −Wt1 are independent for all 0 ≤ t1 ≤ · · · ≤ tn.

• Each increment Wti − Wti−1 is normally distributed with zero meanEP[Wti −Wti−1 ] = 0 and variance

EP[(Wti −Wti−1)2] = (ti − ti−1) (2.5)

So it means that

EP[(Wti+1 −Wti)(Wtj+1 −Wtj )] = δji (ti+1 − ti)

The conditional probability of Wti such as Wti−1 ≡ y is

p(Wti ≡ x|Wti−1 ≡ y) =1√

2π(ti − ti−1)e− (x−y)2

2(ti−ti−1) (2.6)

from which we can reproduce the variance (2.5).Note that EP[WtWs] = min(t, s).

DEFINITION 2.4 Brownian filtration Let Wt be a Brownian motion.Then, we denote FWt the increasing family of σ-algebras generated by (Ws)s≤t,the information on the Brownian motion up to time t. In other words, FWtis the smallest σ-algebra containing all sets of the form

ω;Wt1(ω) ∈ F1, · · · ,Wtk(ω) ∈ Fk

where tj ≤ t and Fj ⊂ R are Borel sets, j ≤ k = 1, 2, · · ·

Example 2.4By definition of the filtration FWt , the process Wt is FWt -adapted. This is notthe case for the process W2t.


In the following, without any specification, we denote (Ω,F ,P) theprobability space where our Brownian motion is defined and theBrownian filtration FWt will be noted Ft.

2.4 Ito process

The price of an asset on a financial market has a deterministic (we say a drift)and a fluctuating (we say a diffusion) part. We will represent the variation ofthe asset price ∆St ≡ St+∆t − St between t ≥ 0 and t+ ∆t > t by

∆St = b(t, St)∆t︸︷︷︸ + σ(t, St)(Wt+∆t −Wt)︸︷︷︸ (2.7)

drift diffusion

with Wt a Brownian motion. The term S−1σ is called the (log-normal) volatil-ity.Here b(·, ·) and σ(·, ·) are two measurable functions on R+ × R. The Itodiffusion process is formally defined by considering the limit ∆t→ 0 in (2.7).We now give a precise meaning to this limit under the integral form

St = S0 +∫ t

0

b(s, Ss)ds+∫ t

0

σ(s, Ss)dWs (2.8)

The first term is a classical Lebesgue integral that should be finite P-almostsurely, i.e.,

P[∫ t

0

|b(s, Ss(ω))|ds <∞] = 1

whereas the second term is different as it involves a r.v. Wt+∆t −Wt.

2.4.1 Stochastic integral

As usual in the theory of integration, we will define the integral∫ t

0σ(s, Ss)dWs

according to a class of simple functions and then extend the definition to alarger class of functions that can be approximated by these simple functions.In this context, we introduce the class z of simple (Ft-adapted) functionsdefined on the interval [0, t] by

f(s, ω) =n−1∑j=0

f(tj , ω)1tj≤s<tj+1 , ω ∈ Ω (2.9)

where we have partitioned the interval [0, t] into n subintervals by means ofpartitioning points

t0 = 0 ≤ t1 ≤ · · · ≤ tn−1 ≤ tn = t


The Ito integral is thus defined as∫ t

0

f(s, ω)dWs ≡n−1∑j=0

f(tj , ω)(Wtj+1(ω)−Wtj (ω)

)(2.10)

Note that we could have introduced simple functions instead

f 12(s, ω) =

n−1∑j=0

f(tj , ω) + f(tj+1, ω)2

1tj≤s<tj+1

and defined the Stratonovich integral as∫ t

0

f(s, ω) dWs ≡n−1∑j=0

f(tj , ω) + f(tj+1, ω)2

(Wtj+1(ω)−Wtj (ω)

)Now, we introduce Υ to be the class of functions

f(t, ω) : [0,∞)× Ω→ R

such that

1. (t, ω)→ f(t, ω) is a B(R+)×F-measurable function.

2. f(t, ω) is Ft-adapted.

3. P[∫ TSf(t, ω)2dt <∞] = 1 ∀ 0 ≤ S < T <∞.

One can show that each f ∈ Υ can be approximated as a limit of simplefunctions φn ∈ z in L2([S, T ] × P) ∀S < T meaning that for each f ∈ Υ,there exists a sequence φn ∈ z such that∫ T

S

EP[(φn(t, ·)− f(t, ·))2]dt→n→∞ 0

∫ t0f(s, ω)dWs is then defined as the limit in L2(P) of

∫ t0φn(s, ω)dWs given by

(2.10).

DEFINITION 2.5 Let f ∈ Υ. Then the Ito integral of f (from S to T )is defined by ∫ T

S

f(t, ω)dWt(ω) = limn→∞

∫ T

S

φn(t, ω)dWt(ω)

where lim is the limit in L2(P) and the φn is a sequence of simple functionssuch that

limn→∞

EP[∫ T

S

(f(t, ω)− φn(t, ω))2dt] = 0 (2.11)


The limit does not depend on the actual choice φn as long as (2.11) holds.

Finally, we have given a meaning to the equation (2.8) which is usually writtenformally as

dSt = b(t, St)dt+ σ(t, St)dWt (2.12)

Following a similar path, it is possible to define a n-dimensional Ito process

xit = xi0 +∫ t

0

bi(s, xs)ds+∫ t

0

m∑j=1

σij(s, xs)dWjs , i = 1, · · · , n

that we formally write as

dxit = bi(t, xt)dt+m∑j=1

σij(t, xt)dWjt

Here Wt is an uncorrelated m-dimensional Brownian motion with zero meanEP[W j

t ] = 0 and variance: EP[W jtW

it ] = δijt.

DEFINITION 2.6 Ito process-SDE Let Wt(ω) = (W 1t (ω), · · · ,Wm

t (ω))denote an m-dimensional Brownian motion. If each process σij(t, x) belongsto the class Υ and each process bi(t, x) is Ft-adapted and4

P[∫ t

0

|b(s, xs(ω))|ds <∞ ∀ t ≥ 0] = 1

then the process


σij(t, xt)dWjt

is an Ito process also called a stochastic differential equation (SDE).

Example 2.5As an example of a computation of an Ito integral, we calculate

∫ t0WsdWs.

Let us introduce the simple function

φn(s, ω) =n−1∑j=0

Wtj (ω)1tj≤s<tj+1

4| · | is the Euclidean norm on Rn: |x| =pPn

i=1(xi)2.


with t0 = 0 and tn = t. This simple function satisfies condition (2.11):

EP[∫ t

0

(Ws(ω)− φn(s, ω))2ds] =

n−1∑j=0

∫ tj+1

tj

EP[(Ws −Wtj )2]dt

By using EP[(Ws−Wtj )2] = |s− tj | according to the definition of a Brownian

motion, we have

EP[∫ t

0

(Ws − φn(s, ·))2ds] =

n−1∑j=0

∫ tj+1

tj

(s− tj)ds

=n−1∑j=0

12

(tj+1 − tj)2 → 0 as ∆tj ≡ tj+1 − tj → 0

Therefore,

∫ t

0

WsdWs = lim∆tj→0

n−1∑j=0

Wtj (Wtj+1 −Wtj )

As∑n−1j=0 Wtj (Wtj+1 −Wtj ) = 1

2W2t − 1

2

∑n−1j=0 (Wtj+1 −Wtj )

2 and since

n−1∑j=0

(Wtj+1 −Wtj )2 → in L2(P)

we obtain ∫ t

0

WsdWs =12

(W 2t − t)

Similarly, we can compute the same integral in the Stratonovich conventionand we obtain ∫ t

0

Ws dWs =12W 2t

Note that the Stratonovich integral looks like a conventional Riemann integral.In the Ito integral, there is an extra term − 1

2 t.

In the same way the computation of a Riemann integral is done without usingthe Riemann sums, the Ito integral is usually computed without using thedefinition 2.5. A handier approach relies on the use of Ito’s lemma that weexplain in the next section.


2.4.2 Ito’s lemma

As we will see in the following section, the fair value C of a European calloption on a single asset at time t depends implicitly on the time t and theasset price St. This leads to the question of the infinitesimal variation of thefair price of the option between t and t + ∆t. The asset price St changes toSt + ∆St (2.7), during the time ∆t, and the variation of C is

∆Ct ≡ C(t+ ∆t, St + ∆St)− C(t, St)

By using a Taylor expansion assuming that C is C1,2([0,∞)× R) (i.e., (resp.twice) continuously differentiable on [0,∞) (resp. R)) then

∆Ct = ∂tC∆t+ ∂SC∆St +12∂2SC(∆St)2 +R

with R the rest of the Taylor expansion. The arguments of the function Chave not been written so as not to burden the notations. Replacing ∆St byits variation (2.7), we obtain

∆Ct ≡ ∂tC∆t+ ∂SC(b(t, St)∆t+ σ(t, St)∆Wt)

+12∂2SC (b(t, St)∆t+ σ(t, St)∆Wt)

2 +R

where we have noted ∆Wt ≡Wt+∆t −Wt. The total variation between t = 0up to t is obtained as the sum of the infinitesimal variations ∆Ct

C(t, St)− C(0, S0) =∑i

(∂tCi + b(ti, Sti)∂SCi)∆ti +∑i

∂SCσ(ti, Sti)∆Wti

+∑i

12∂2SCi (b(ti, Sti)∆ti + σ(ti, Sti)∆Wti)

2 +∑i

Ri

where we have set Ci ≡ C(ti, Sti). As ∆ti → 0, we have∑i

(∂tCi + b(ti, Sti)∂SCi)∆ti →∫ t

0

(∂sC(s, Ss) + b(s, Ss)∂SC(s, Ss))ds

and ∑i

∂SCiσ(ti, Sti)∆Wti →∫ t

0

∂SCσ(s, Ss)dWs

where the last term should be understood as an Ito integral. Moreover, onecan show that the sums of the terms in (∆ti)2 and (∆ti)∆Wi go to zero as∆ti → 0. Finally, we have in L2(P)∑

i

σ(ti, Sti)2(∆Wti)

2 →∫ t

0

σ(t, St)2dt


To prove this, we set vi = σ(ti, Sti)2 and consider

EP[

(∑i

vi(∆Wti)2 −

∑i

vi∆ti

)2

] =∑ij

EP[vivj(

(∆Wti)2 −∆ti

)((

∆Wtj

)2 −∆tj)

]

For i < j (similarly j > i), the r.v. (with zero mean) vivj((∆Wti)

2 −∆ti)

and (∆Wtj )2−∆tj are independent and therefore the sum cancels. For i = j,

the sum reduces to

EP[v2i ((∆Wti)

2 −∆ti)2] = EP[v2i ]EP[(∆Wti)

4 − 2∆ti(∆Wti)2 + (∆ti)2]

According to the definition of a Brownian motion, we have EP[(∆Wti)2] = ∆ti,

EP[(∆Wti)4] = 3(∆ti)2 (see exercise 2) and we obtain

EP[((∆Wti)

2 −∆ti)2

] = 2(∆ti)2

Finally

EP[

(∑i

vi(∆Wti)2 −

∑i

vi∆ti

)2

] = 2∑ij

EP[v2i ](∆ti)2 → 0 as ∆ti → 0

The argument above also proves that∑iRi → 0 as ∆ti → 0. Therefore,

putting all our results together, we obtain the Ito lemma

C(t, St) = C(0, S0) +∫ t

0

(∂sC +

σ(s, Ss)2

2∂2SC + b(s, Ss)∂SC

)ds

+∫ t

0

σ(s, Ss)∂SCdWs

This equation can be formally written in a differential form as

dCt =(∂tC +

σ(t, St)2

2∂2SC + b(t, St)∂SC

)dt+ σ(t, St)∂SCdWt

or in the equivalent form

dCt = ∂tCdt+ ∂SCdSt +12∂2SCdS2

t

with dSt given by (2.12) and dS2t computed using the formal rules

dtdWt = dt.dt = 0dWtdWt = dt


dCt is therefore an Ito diffusion process with a drift (∂tC + σ(t,St)2

2 ∂2SC +

b(t, St)∂SC) and a diffusion term σ(t, St)∂SC. Similarly for a function of n-dimensional Ito diffusion processes xit (4.4), we can generalize Ito’s lemma andwe obtain

THEOREM 2.2 The general Ito formulaLet xiti=1,··· ,n be an n-dimensional Ito process


σij(t, xt)dWjt (2.13)

Let C(·, ·) be a C1,2([0,∞)×Rn) real function. Then the process Ct ≡ C(t, xt)is again an Ito process given by

dCt =n∑i=1

m∑j=1

σij(t, x)∂C(t, x)∂xi

dW jt +∂tC(t, x) +

n∑i=1

bi(t, x)∂C(t, x)∂xi

+12

n∑i,j=1

m∑k=1

σik(t, x)σjk(t, x)∂2C(t, x)∂xi∂xj

dt

This can be also written as

dCt = ∂tC(t, x)dt+n∑i=1

∂C(t, x)∂xi

dxi +12

n∑i,j=1

∂2C(t, x)∂xi∂xj

dxidxj (2.14)

where dxitdxjt is computed from (2.13) using the formal rules dt.dW i

t = dt.dt =0 and dW i

t .dWjt = δji dt.

This formula is one of the key tools in financial mathematics. In the nextparagraph, we present a few examples of resolution of stochastic differentialequations (SDE) using Ito’s lemma and then explicit under which conditionsa SDE admits a unique (strong or weak) solution.

2.4.3 Stochastic differential equations

Example 2.6 Geometric Brownian motionA Geometric Brownian motion (GBM) is the core of the Black-Scholes marketmodel. A GBM is given by the following SDE

dXt = µ(t)Xtdt+ σ(t)XtdWt

with the initial condition Xt=0 = X0 ∈ R and µ(t), σ(t) two time-dependentdeterministic functions. We observe that Xt is a positive Ito process. By


using Ito’s lemma on ln(Xt), we have

d ln(Xt) =(µ(t)− 1

2σ(t)2

)dt+ σ(t)dWt

and by integration, we deduce the solution

Xt = X0eR t0 µ(s)ds− 1

2

R t0 σ

2(s)ds+R t0 σ(s)dWs (2.15)

From Ito isometry (see exercise 2.5), lnXt is a Gaussian r.v. N (mt, Vt) witha mean mt and a variance Vt equal to

mt =∫ t

0

(µ(s)− 1

2σ(s)2

)ds+ lnX0

Vt =∫ t

0

σ(s)2ds

Example 2.7 Ornstein-Uhlenbeck processAn Ornstein-Uhlenbeck process is given by the following SDE

dXt = γXtdt+ σdWt

Xt=0 = X0 ∈ R

where γ and σ are two real constants. If σ = 0, we know that the solution isXt = X0e

γt. Let us try the ansatz Xt = eγtYt. Applying Ito’s lemma on Xt,we obtain the GBM on Yt

dYt = σe−γtdWt

and finally we have

Xt = eγt(X0 + σ

∫ t

0

e−γsdWs

)By using the Ito isometry (see exercise 2.5), the process Xt is a Gaussian r.v.N (mt, Vt) with a mean mt and a variance Vt equal to

mt = eγtX0

Vt =σ2

2γ(e2γt − 1)


Strong solution

After these examples, let us present the conditions under which a SDE admitsa unique solution. From a classical theorem on ordinary differential equations,x = f(x) admits a unique solution if f is a Lipschitz function, meaning that

|f(x)− f(y)| ≤ K|x− y| , ∀x, y

with K a constant. There is a similar condition for the existence and unique-ness of a SDE:

THEOREM 2.3 Existence and Uniqueness of SDE (Strong solution)

Let T > 0 and b(·, ·) : [0, T ] × Rn → Rn, σ(·, ·) : [0, T ] × Rn → Rn×m bemeasurable functions satisfying

|b(t, x)|+ |σ(t, x)| ≤ C(1 + |x|) , x ∈ Rn, t ∈ [0, T ] (2.16)

for some constant C (where |b| =√∑n

i=1 |bi|2 and |σ| =√∑n

i=1

∑mj=1 |σij |2)

and such that ∀ x, y ∈ Rn , t ∈ [0, T ]5

|b(t, x)− b(t, y)|+ |σ(t, x)− σ(t, y)| ≤ D|x− y| (2.17)

for some constant D. Then the SDE

dXit = bi(t,Xt)dt+

m∑j=1

σij(t,Xt)dWjt (2.18)

Xt=0 = X0 ∈ R (2.19)

has a unique time-continuous solution Xt for ∀ 0 ≤ t ≤ T with the propertythat Xt is adapted to the filtration Ft generated by W i

ss≤t,i=1,···m. Moreover,

EP[∫ T

0

|Xt|2dt] <∞

The Brownian motion is given as an input and Xt is adapted to the filtrationgenerated by the Brownian motion. The solution Xt is called a strong solution.

REMARK 2.1 Note that for time-homogeneous SDEs (i.e., b ≡ b(x) andσ ≡ σ(x)), the condition (2.16) is implied by (2.17) and therefore is irrelevant.

5This condition means that the coefficients b and σ are globally Lipschitz. We recall thatany function with a bounded first derivative is globally Lipschitz.


Weak solution

The existence and uniqueness of SDEs holds under rather restrictive analyticalconditions on the SDE coefficients which are not satisfied in many commonlyused market models. In order to circumvent this difficulty, we need to intro-duce weak solutions.In comparison with strong solutions where the Brownian motion is given asan input, in a weak solution, the Brownian motion Wt is given exogenously:one finds simultaneously Wt and Xt. The only inputs are the drifts andthe diffusion parameters. In the pricing of derivative products, one needs toconsider weak solutions only as we are solely interested in the law of Xt.

DEFINITION 2.7 Weak solution [34] A process Xt on a completeprobability space (Ω,F ,P) is said to be a weak solution to the SDE (2.18) ifand only if

• Xt is a continuous Ft-adapted process for some complete filtration Ftwhich satisfies (2.18).

• Wt is a Ft-Brownian motion on (Ω,F ,P).

Equation (2.18) is said to be unique in law if any two weak solutions X1 andX2 which are defined on two probability spaces (Ω1,F1,P1) and (Ω2,F2,P2)with the filtrations F1

t and F2t have the same law.

When we examine local and stochastic volatility models in chapters 5 and 6,we will give some weaker conditions on the coefficients of SDEs which implyexistence and uniqueness in law of weak solutions.

Example 2.8The GBM and the Ornstein-Uhlenbeck processes satisfy the condition (2.17)and therefore have a unique (strong) solution.

2.5 Market models

The tools to model a market have now been presented. A market model willbe characterized by a n-dimensional Ito diffusion process


σij(t, xt)dWjt , i = 1 · · ·n (2.20)

and by another positive stochastic process Bt satisfying

dBt = rtBtdt ; B0 = 1 (2.21)


The stochastic process xi can model equity assets, fixed-income rates, foreignexchange currencies or other unobservable r.v. (such as a stochastic volatilitythat we introduce in chapter 6). Bt is the money market account which whenyou invest 1 at time t gives 1+rtdt at t+dt with rt the instantaneous interestrate at t. We define the discount factor from 0 to t by

D0t ≡1Bt

= e−R t0 rsds (2.22)

and set the discount factor between two dates t and T > t by

DtT ≡D0T

D0t(2.23)

The fact that the equation for dBt does not have a Brownian term means thatBt is a non-risky asset with a fixed return rt. However rt is not necessarily adeterministic function and may depend on the risky asset xit or other unob-servable Ito processes. The instantaneous rate rt can therefore be a stochasticprocess.

2.6 Pricing and no-arbitrage

Arising from previous sections, the natural follow-up question is to find a pricefor these contracts. It is called the pricing problem. The pricing at time t isan operator which associates a number (i.e., price) to a payoff

Πt : P → R∗+

Different rules must be imposed on the pricing operator Πt:If a trader holds a book of (non-American) options, each one being character-ized by a payoff fi, the total value of the book should be

Πt

(∑i

fi

)=∑i

Πt (fi)

Therefore each payoff fi can be priced independently to get the value of thewhole portfolio. This implies that Πt is a linear form on the space of payoffP which is the space of measurable functions on a measurable space (Ω,F).If we impose that Πt is continuous on P, the Riesz representation theoremstates that there exists a density µt such that

Πt(f) ≡∫fdµt


Below, we show that this can be written as a conditional expectation

EP[f |Ft] (2.24)

with P a measure on (Ω,F) and Ft a filtration of F . In order to characterizethe measure P, we need to introduce one of the most important notions infinance: the no-arbitrage condition.

2.6.1 Arbitrage

Before getting into the mathematical details, let us present this notion throughan example. Let us suppose that the real estate return is greater than thefixed income rate. The trader will borrow money and invest in the real estate.If the real estate return remains the same, the trader earn money. We say thatthere is an arbitrage situation as money can be earned without any risk. Letus suppose now that as we apply this winning strategy and start earning a lotof money, others who observe our successful strategy will start doing the samething. According to the offer-demand law, as more and more people will investin the real estate by borrowing money, the fixed income rate will increase andthe real estate return will decrease. The equilibrium will be reached when thereal estate return µ will converge to the fixed income return r. So in orderto have no-arbitrage, we should impose that µ = r. More generally, we willsee in the following that the no-arbitrage condition imposes that the drift oftraded assets in our market model is fixed to the instantaneous rate in a well-chosen measure P (not necessarily unique) called the risk-neutral measure. Todefine precisely the meaning of no-arbitrage, we introduce a class of strategiesthat could generate arbitrage. In this context, we introduce the concept of aself-financing portfolio.

2.6.2 Self-financing portfolio

Let us assume that we have a portfolio consisting of an asset and a moneymarket account Bt. We will try to generate an arbitrage at a maturity date Tby selling, buying the asset (we assume that there is no transaction cost) andinvesting the profit in the market-money account Bt with a return rt. Theportfolio at a time ti is composed of ∆i assets (with asset price xi = xti) anda money market account Bi = Bti . The portfolio value πi is

πi = ∆ixi +Bi (2.25)

During our strategy, no cash is injected in the portfolio. This way, we intro-duce a self-financing portfolio by assuming that the variation of the portfoliobetween dates ti and ti+1 is

πi+1 − πi = ∆i (xi+1 − xi) + ri∆tiBi (2.26)


with ∆ti = ti+1− ti. At time ti, we hold ∆i assets with a price xi and at timeti+1, the new value of the portfolio comes from the variation of the asset pricebetween time ti and ti+1 and the increase of the value of the money marketaccount due to the fixed income return ri∆ti. By using the condition above(2.26), we derive that the money market account at time ti+1 is

Bi+1 = Bi(1 + ri∆ti)− xi+1(∆i+1 −∆i)

Therefore, by recurrence, we have

Bn =n−1∏k=0

(1 + rk∆tk)B0 −n∑k=1

xk(∆k −∆k−1) (2.27)

with xk = xk∏n−1i=k (1 + ri∆ti) , k = 1, · · ·n − 1 and xn = xn. By plugging

(2.27) into (2.25), the variation of the portfolio between t0 = t and tn = T is

πT −n−1∏k=0

(1 + rk∆tk)πt =n−1∑k=0

∆k(xk+1 − xk) (2.28)

In the continuous-time limit assuming that the initial value of the portfolioat t is zero, πt = 0, the discrete sum (2.28) converges to the Ito integral

πT =∫ T

t

∆(s, ω)dxs (2.29)

with πT = D0T πT and

xt = xtD0t (2.30)

From this expression, we formalize the notion of arbitrage:

DEFINITION 2.8 Arbitrage A self-financing portfolio is called anarbitrage if the corresponding value process πt satisfies π0 = 0 and

πT ≥ 0 Phist −almost surely and Phist[πT > 0] > 0

with Phist the historical (or real) probability measure under which we modelour market.

It means that at the maturity date T , the value of the portfolio is non-negativeand there is a non-zero probability that the return is positive: there is norisk to lose money and a positive probability to win money. Under whichconditions for a specific market model can we build a self-financing portfoliogenerating arbitrage? To answer this question, we need to define the notionof (local) martingale.


Martingale

A n-dimensional stochastic process Mtt≥0 on (Ω,F ,P) is called a martingalewith respect to a filtration Ft≥0 and with respect to P if

DEFINITION 2.9 Martingale

• (a) Mt is Ft-adapted for all t.

• (b) EP[|Mt|] <∞ ∀ t ≥ 0.

• (c) EP[Mt|Fs] = Ms , t ≥ s.

Note that condition (c) implies that EP[Ms] = M0. However, this conditionis much weaker than (c).

Example 2.9

Wt is a martingale.

Example 2.10

xt = W 2t − t is a martingale with respect to the Brownian filtration FWt

because

EP[W 2t − t|FWs ] = EP[(Wt −Ws)2 + 2(Wt −Ws)Ws +W 2

s − t|FWs ]= (t− s) +W 2

s − t = W 2s − s

where we have used that Wt −Ws and Ws are independent r.v.

Local martingale

If we apply Ito’s lemma to the martingale xt = W 2t −t, we find that xt satisfies

a driftless Ito diffusion process

dxt = 2WtdWt

More generally, a driftless Ito process such as∫ t

0asdWs is called a local mar-

tingale.6 In particular, a martingale is a local martingale. In chapter 5, weexplain the Feller criterion which gives a necessary and sufficient conditionfor which a positive local martingale is a martingale in one dimension. Astandard criterion is if

∫ T0

EP[a2s]ds < ∞ then the local martingale

∫ t0asdWs

is a martingale for t ∈ [0, T ].

6A more technical definition can be found in [27].


Fundamental theorem of Asset pricing

All the tools to formulate the fundamental theorem of asset pricing charac-terizing that a market model does not generate any arbitrage opportunitieshave now been introduced.Let us call Ftt≥0 the natural filtration generated by the Brownianmotions W i

t i=1,··· ,m that appear in our market model.

DEFINITION 2.10 Asset

An asset means a positive Ito process that can be traded (i.e., sold or bought)on the market.

For example, an equity stock is an asset. The instantaneous interest rate rtis not a financial instrument that can be sold or bought on the market andtherefore is not an asset.

LEMMA 2.1

Let us suppose that there exists a measure P on FT such that Phist ∼ P andsuch that the discounted asset process xtt∈[0,T ] (2.30) is a local martingalewith respect to P. Then the market xt, Btt∈[0,T ] has no arbitrage.

PROOF (Sketch) Let us suppose that we have an arbitrage (see definition2.8). From (2.29), we have that πt is a local martingale under P. Assumingthat πt is in fact a martingale, we obtain that

EP[πT ] = π0 = 0

As πT ≥ 0, we have πT = 0 P-almost surely, which contradicts the fact thatP[πT > 0] > 0 (and therefore Phist[πT > 0] > 0).Note that the martingale condition can be relaxed by introducing the class ofadmissible portfolio [34].

A measure P ∼ Phist such that the normalized process xtt∈[0,T ] is a localmartingale with respect to P is called an equivalent local martingale measure.Conversely, one can show that the no-arbitrage condition implies the existenceof an equivalent local martingale measure [11].

THEOREM 2.4 Fundamental theorem of Asset pricing

The market model defined by (Ω,F ,Phist) and asset prices xt∈[0,T ] is arbitrage-free if and only if there exists a probability measure P ∼ Phist such that thediscounted assets xtt∈[0,T ] (defined by (2.30)) are local martingales with re-spect to P.


In practice, arbitrage situation may exist. Statistical arbitrage traders try todetect them to generate their profits. On a long scale period, the market canbe considered at equilibrium and thus there is no arbitrage.In the following, we assume that our market model is arbitrage free, meaningthat the assets xit = D0tx

it are local martingales under P and therefore driftless

processes. Therefore, there exists a diffusion function σij(t, ω) such that theprocesses xit satisfy the SDE under P

dxit =m∑j=1

σij(t, ω)dW jt

As xit = D−10t x

it, we have under P

dxit = rtxitdt+

m∑j=1

σij(t, ω)dW jt

with σij(t, ω) = D−10t σ

ij(t, ω) and Wt a Brownian motion under P. A measure

P for which the drift of the traded assets is fixed to the instantaneous rate rtis called a risk-neutral measure.Note that a derivative product Ct can be considered as an asset as it can bebought and sold on the market. Therefore according to the theorem above,D0tCt should be a local martingale in an arbitrage-free market model under arisk-neutral measure P. Assuming the integrability condition to ensure thatD0tCt is not only a local martingale but also a martingale, we obtain thepricing formula

D0tCt = EP[D0TCT |Ft]

and using (2.22)

Ct = EP[e−R TtrsdsCT |Ft] (2.31)

Using the Markov property, it can be shown that Ct = C(t, xt) [34].A fair price is given by the mean value of the discounted payoff under a risk-neutral measure P. This is the main formula in option pricing theory. It is anequilibrium price fair to both the buyer and the seller. The seller will add apremium to the option price.A pricing operator Πt (2.24) is therefore the mean value according to a risk-neutral measure for which traded assets have a drift fixed to the instanta-neous fixed income rate. Note that in some market models such as Libormarket models (see section 2.10.3.3), the instantaneous rate is not defined.We therefore require that the discounted traded assets be local martingalesunder another measure, the forward measure, that we explain in example(2.13). Below, we introduce the classical Black-Scholes market model andprice a European call option using pricing formula (2.31).


Example 2.11 Black-Scholes market model and Call option priceThe market model consists of one asset St and a deterministic money marketaccount with constant interest rate r. Therefore, under a risk-neutral measureP, the process fTt ≡ St

DtT= Ste

r(T−t), called the forward of maturity T anddenoted xt above, is a local martingale and therefore driftless. We model itby a GBM

dfTt = σfTt dWt

with the constant volatility σ and the initial condition fT0 = S0erT . The

solution is (see example 2.6)

fTt = fT0 e−σ2

2 t+σWt (2.32)

We want to find in this framework the fair price of a European call optionwith payoff max(ST − K, 0). First, we remark that as ST = fTT , the payoff(at maturity) can be written as

max(fTT −K, 0)

Moreover as r is constant, the discount factor is DtT = e−r(T−t) and thepricing formula (2.31) gives at t = 0

C(0, S0) = e−rTEP[max(fTT −K, 0)]

By plugging the value for fTT in (2.32), we have

C(0, S0) = e−rTEP[(fT0 e−σ2

2 T+σWT −K)1

(WT −

ln KfT0

σ+σT

2

)]

By using the probability density (2.6) for WT , we obtain

C(0, S0) = e−rT∫ ∞

ln KfT0σ −σT2

(fT0 e−σ2

2 T+σx −K)e−

x22T

√2πT

dx

The integration over x does not present any difficulty and we obtain

C(0, S0) = S0N(d+)−Ke−rTN(d−) (2.33)

with

d± = −ln(Ke

−rT

S0)

σ√T

± σ√T

2

and with N(x) =∫ x−∞ e−

y2

2dy√2π

the cumulative normal distribution. This is

the Black-Scholes formula for a European call option.


2.7 Feynman-Kac’s theorem

According to the previous example (2.11), the fair price can be obtained byintegrating the payoff according to the probability density. Another approachto get the same result shows that Ct satisfies a parabolic partial differentialequation (PDE). This is the Feynman-Kac theorem.First, we assume that under a risk-neutral measure P our market model isdescribed by (2.20, 2.21). The fair value C(t, x) depends on the n-dimensionalIto diffusion processes xit characterizing our market model plus the moneymarket account. Using the general Ito formula (2.14) we obtain that the driftof D0tC is

D(t, xt) ≡ D−10t Drift[d(D0tC)] = ∂tC(t, xt)− rtC(t, xt) (2.34)

+n∑i=1

bi(t, xt)∂C(t, xt)∂xi

+12

n∑i,j=1

m∑k=1

σik(t, xt)σjk(t, xt)

∂2C(t, xt)∂xi∂xj

As C is a traded asset under a risk-neutral measure P, D0tC is a local martin-gale and its drift should cancel. Then one can show under restrictive smooth-ness assumption on C, D(t, xt) = 0 implies that D(t, x) = 0 for all x in thesupport of the diffusion. We obtain the PDE

∂tC(t, x) +n∑i=1


+12

n∑i,j=1

m∑k=1


−rtC(t, x) = 0

More precisely, the Feynman-Kac theorem stated as below is valid under thefollowing hypothesis [12]Hypothesis A: The functions b(·, ·) and σ(·, ·) satisfy the Lipschitz condition(2.17). The function C also satisfies the Lipschitz condition in [0, T ] × Rn;moreover, the functions b, σ, C, ∂xb, ∂xσ, ∂xC, ∂xxb, ∂xxσ, ∂xxC exist, arecontinuous and satisfy the growth condition (2.16).

THEOREM 2.5 Feynman-KacLet f ∈ C2(Rn), r ∈ C(Rn) and r be lower bounded.

Given C(t, x) = EP[e−R Ttrsdsf(xT )|Ft], we have under hypothesis A

−∂tC(t, x) = DC(t, x)− rtC(t, x) , t > 0, x ∈ Rn

with the terminal condition C(T, x) = f(x) and with the second-order differ-ential operator D defined as

DC(t, x) =n∑i=1


+12

n∑i,j=1

m∑k=1



This theorem is fundamental as we can convert the mean-value (2.31) intoa PDE. We will apply this theorem to the case of the Black-Scholes marketmodel and reproduce the fair value of a European call option (2.33) by solvingexplicitly this PDE.

Example 2.12 Black-Scholes PDE and call option price rederivedStarting from the hypothesis that the forward fTt = Ste

r(T−t) follows a log-normal diffusion process in the Black-Scholes model

dfTt = σfTt dWt

the resulting Black-Scholes PDE for a European call option, derived from theFeynman-Kac theorem, is

−∂tC(t, f) =12σ2f2∂2

fC(t, f)− rC(t, f)

with the terminal condition C(T, f) = max(f −K, 0). The solution is givenby

C(t, f) = e−r(T−t)∫ ∞

0

max(f ′ −K, 0)p(T, f ′|t, f)df ′ (2.35)

p(T, f ′|t, f) is the fundamental solution of the PDE

−∂tp(T, f ′|t, f) =12σ2f2∂2

fp(T, f′|t, f)

with the initial condition limt→T p(T, f ′|t, f) = δ(f ′ − f). Doing the changeof variable s =

√2σ ln( ff ′ ) and setting τ = T − t, one can easily show that the

new function p′(τ, s) defined by

p(T, f ′|t, f) = e12 ln( f

f′ )−σ28 τp′(τ, s)

√2

σf ′(2.36)

satisfies the heat kernel equation on R

∂τp′(τ, s) = ∂2

sp′(τ, s)

with the initial condition limτ→0 p′(τ, s) = δ(s). The solution is the Gaussian

heat kernel

p′(τ, s) =1√4πτ

e−s24τ

From (2.36) we obtain the conditional probability for a log-normal process

p(T, f ′|t, f) =1

f ′√

2πσ2(T − t)e−

(ln( f′f

)+σ2(T−t)2 )2

2σ2(T−t)


Doing the integration over f ′ in (2.35), we obtain our previous result (2.33).

The Feynman-Kac theorem allows to price European options using a PDE ap-proach. For Barrier options we need the following extension of the Feynman-Kac theorem.

THEOREM 2.6 Localized Feynman-KacLet τ be the stopping time the process leaves the domain D:

τ = inft : Xt /∈ D

Given C(t, x) = EP[e−R Ttrsdsf(xT )1τ>T |Ft], we have

−∂tC(t, x) = DC(t, x)− rtC(t, x) , t > 0, x ∈ D (2.37)

with the terminal condition C(T, x) = f(x) and the Dirichlet boundary condi-tion

C(x) = 0 , x /∈ D , ∀ t ∈ [0, T ]

REMARK 2.2 This theorem can be formally derived from the originalFK theorem by taking

r(x) = 0 , x ∈ D= ∞ , x /∈ D

With this choice, the mean value is localized to the domain of D.

2.8 Change of numeraire

The Girsanov theorem is a key tool in finance as it permits to considerablysimplify the analytical or numerical computation of (2.31) by choosing anappropriate numeraire. By definition,

DEFINITION 2.11 A numeraire is any positive continuous asset.

Under the no-arbitrage condition, the drift of a numeraire, being an asset,is constrained to be the instantaneous interest rate rt under a risk-neutralmeasure P.


Before stating the Girsanov theorem, we motivate it with a simple problemconsisting in the pricing of a European option depending on two assets S1

and S2 whose payoff at time T is the spread option

max(S2T − S1

T , 0)

From (2.31), we know that the fair price at time t ≤ T is

C = EP[D0T

D0tmax(S2

T − S1T , 0)|Ft]

For our market model, we assume that the two assets S1 and S2 follow thediffusion processes under a risk-neutral measure P

dS1t

S1t

= rtdt+ σ1(t, S1).dWt

dS2t

S2t

= rtdt+ σ2(t, S2).dWt

where Wt is a m-dimensional Brownian motion.7

Let us suppose that the two assets are valued in euros. As an alternative,the second asset could be valued according to the first one. In this case,the numeraire is the first asset and we can consider the dimensionless assetXt = S2

t

S1t. By using Ito formula, the dynamics of Xt is then

dXt

Xt= (σ2(t, S2)− σ1(t, S1)).(dWt − σ1(t, S1)dt)

Note that to get this result easily, it is better to apply the Ito formula onlnXt beforehand. We observe that Xt has a non-trivial drift under P.Let us define

dWt = dWt − σ1(t, S1)dt

and the dynamics of Xt can be written as

dXt

Xt= (σ2(t, S2)− σ1(t, S1)).dWt

This rewriting is completely formal at this stage as Wt is not a Brownianmotion according to P. However, according to the Girsanov theorem, we candefine a new measure P equivalent to P such that Wt is a Brownian motionunder P. Therefore, Xt is a local martingale under P. More precisely, we have

7σ1(t, S1).dWt meansPmj=1 σ

1j (t, S1)dW j

t with W jt j = 1, · · · ,m a m-dimensional uncor-

related Brownian motion.


THEOREM 2.7 GirsanovLet Xt ∈ Rm be an Ito diffusion process of the form

dXt = λtdt+ dWt , t ≤ T , X0 = 0

where T ≤ ∞ a given constant, λt a Ft-adapted process and (Wt,Ft,P) is am-dimensional Brownian motion. Then Mt given by

Mt = e−R t0 λsdWs− 1

2

R t0 λ

2sds (2.38)

is a (positive) local martingale with respect to Ft.Let us define the measure P on FT by the Radon-Nikodym derivative

dPdP|FT = MT

If Mt is a Ft-martingale (see remark 2.3 below) then P is a probability measureon FT and Xt is a m-dimensional Brownian motion according to P for 0 ≤t ≤ T .

REMARK 2.3 Under the Novikov condition

EP[e12

R T0 λ2

sds] <∞ (2.39)

the positive local martingale Mt given by (2.38) is a martingale. This condi-tion can be weakened by

EP[e12

R T0 λsdWs ] <∞ (2.40)

This theorem is standard and the proof can be found in [34] for example (seealso exercise 2.7). Note that the fact that Mt is a local martingale can beeasily proved by observing that

dMt = −λtMtdt

as an application of Ito’s lemma.When dealing with conditional expectation, we have that for a FT -measurabler.v. X satisfying EP[|X|] <∞,

EP[X|Ft] = EP[XMT

Mt|Ft] (2.41)

PROOF By using in the following order the definition of EP, the definitionof the conditional expectation and the martingale property of Mt, we have


∀ A ∈ Ft<T

EP[1A1Mt

EP[XMT |Ft]] = EP[1AEP[XMT |Ft]]

= EP[1AXMT ]

= EP[1AX]

Coming back to our example, applying the Girsanov theorem, the change ofmeasure from P to P is (here λt = −σ1(t, S1

t ))

MT ≡dPdP|FT = e

R T0 σ1dWs− 1

2

R T0 (σ1)2ds

=S1T

S10BT

=S1T

S10

D0T (2.42)

Therefore the fair value is

C = EP[D0T

D0tmax(S2

T − S1T , 0)|Ft] = EP[

D0T

D0tS1T max(XT − 1, 0)|Ft]

= EP[D0TMt

D0tMTS1T max(XT − 1, 0)|Ft]

= S1t EP[max(XT − 1, 0)|Ft]

In the second line, we have used (2.41) and in the last equation (2.42).Before closing this section, let us present the general formula enabling to trans-form an Ito diffusion process under a measure P1 (associated to a numeraireN1) into a new Ito process under the measure P2 (associated to a numeraireN2).

Change of numeraire: General formula

Let us consider a n-dimensional Ito process Xt given under P1 by

dXt

Xt= µN1

X dt+ σX .dW1t

and under the numeraire P2 by

dXt

Xt= µN2

X dt+ σX .dW2t

Note that the diffusion terms are the same in the two equivalent measures P1

and P2 as the change of measure only affects the drift terms. Moreover, the


writing of the dynamics for Xt using a “log-normal” form dXtXt

is solely forconvenience. Under the Girsanov theorem, the two measures differ by

dW 2t = dW 1

t + λtdt

with

µN1X = µN2

X + σX .λt (2.43)

Here σX .λt is a n-dimensional vector with components∑mj=1 σ

iX jλ

jt . The

change of measure is

dP2

dP1|FT ≡MT = e−

R T0 λsdW

1s− 1

2

R T0 λ2

sds

If we price a European contract (with maturity T ) and payoff fT , the fairvalue can be written in two different manners using the measure P1 and P2

C = EP1[N1t

N1T

fT |Ft] = EP2[N2t

N2T

fT |Ft]

= EP2[Mt

MT

N1t

N1T

fT |Ft]

Therefore,

Mt

MT=N2t N

1T

N2TN

1t

Modulo an irrelevant multiplicative factor, we deduce that

Mt =N2t

N1t

Moreover, we assume that the two numeraires under the measure P1 followthe Ito diffusion processes

dN1

N1= µN1dt+ σ1.dW 1

t

dN2

N2= µN2dt+ σ2.dW 1

t

As we have dMt = −λtMtdW1t , we obtain the identification

λt = σ1 − σ2

and therefore from (2.43), we have

µN2X = µN1

X + σX .(σ2 − σ1)


The result is summarized in the following proposition

PROPOSITION 2.1Consider two numeraires N1 and N2 with Ito diffusion processes under P1

(associated to the numeraire N1)

dN1

N1= µ1dt+ σ1.dW 1

t

dN2

N2= µ2dt+ σ2.dW 1

t

Then if we consider an Ito diffusion process given under P1 by

dXt

Xt= µN1

X dt+ σX .dW1t

then the process Xt under the measure P2 (associated to the numeraire N2) is

dXt

Xt= (µN1

X + σX .(σ2 − σ1))dt+ σX .dW2t

In our previous example, the two processes S1t and S2

t are written under themeasure P1 associated to the numeraire S1

t

dS1

S1= (rt + (σ1)2)dt+ σ1.dW 1

t

dS2

S2= (rt + σ2.σ1)dt+ σ2.dW 1

t

as the volatility associated to S1 is σ1 and the volatility associated to themoney market account Bt is zero. From Ito’s lemma, we can check that S2

S1is

a local martingale under P1 as expected

d

(S2

S1

)=(S2

S1

)(σ2 − σ1).dW 1

t

In order to illustrate the power of the change of measure technique, we presenta new measure, the forward measure, particularly useful to compute the fairvalue of an option (2.31) when the discount factor D0t is a stochastic process.

Example 2.13 Forward and Forward measureWe want to price a European call option (with a maturity T and a strike

K) and we assume that the discount factor D0t is a stochastic process. Thefair price is under a risk-neutral measure P

C = EP[D0T

D0tmax(ST −K, 0)|Ft] (2.44)


with D0t = e−R t0 rsds. We set

PtT ≡ EP[D0T

D0t|Ft]

= EP[e−R Ttrsds|Ft]

the fair value of a contract paying 1 at the maturity T . It is called a bond.Since PtT is a traded asset, its dynamics under the measure P is

dPtTPtT

= rtdt+ σP .dWt

with σP the volatility of the bond PtT (left unspecified). Moreover, we assumethat under P

dStSt

= rtdt+ σS .dWt

Doing a change of measure from P to PT associated to the numeraire PtT(called the forward measure), the dynamics of PtT and St under PT are

dPtTPtT

= (rt + σP .σP )dt+ σP .dWTt

dStSt

= (rt + σS .σP )dt+ σS .dWTt

The Radon-Nikodym derivative is

MT ≡dPT

dP|FT =

PTTBT

= D0T (2.45)

Moreover, we define the forward fTt = StPtT

(fTT = ST as PTT = 1). As beingthe ratio of a traded asset St and a bond PtT , fTt is a local martingale underthe forward measure PT . Indeed, it can be checked using Ito’s lemma that fTtis a local martingale

dfTt = fTt(σS .dWT

t − σP .dWTt

)The European call option can be written as

C = EPT [D0T

D0t

dPdPT

max(fTT −K, 0)] = EPT [D0T

D0t

Mt

MTmax(fTT −K, 0)]

and finally using (2.45)

C = PtTEPT [max(fTT −K, 0)] (2.46)

Using the forward measure, the (stochastic) discount factor has disappearedin the pricing formula (2.44).

We conclude this section with a PDE interpretation of the Girsanov transformfor Ito diffusion processes.


PDE interpretation of the Girsanov transform

We restrict our discussion to one-dimensional Ito diffusion processes althougheverything we discuss below can be trivially extended to higher-dimensionalprocesses.We assume that under a measure P (not necessarily being a risk-neutral mea-sure), the spot process St satisfies the SDE

dSt = µ(St)dt+ σ(St)dWt

and under a new measure P

dSt = r(St)dt+ σ(St)dWt

Let us introduce the process Mt defined by the SDE

dMt =(µ(St)− r(St)

σ(St)

)MtdWt

According to the Feynman-Kac theorem, C(t, S,M) ≡ EP[MTΦ(ST )|Ft] sat-isfies the PDE

∂tC(t, S,M) + r(S)∂S C(t, S,M) +12σ(S)2∂2

S C(t, S,M)

+12M2

(µ(S)− r(S)

σ(S)

)2

∂2M C(t, S,M) +M (µ(S)− r(S)) ∂MS C(t, S,M) = 0

with the terminal condition C(T, S,M) = MΦ(S). Let us try the ansatzC(t, S,M) = MC(t, S). We obtain that the function C(t, S) satisfies the PDE

(∂t + µ(S)∂S +12σ(S)2∂2

S)C(t, S) = 0

with the terminal condition C(T, S) = Φ(S). By the Feynman-Kac theorem,this is equivalent to

MtEP[Φ(ST )] = EP[MTΦ(ST )]

This is the Girsanov transform.

2.9 Hedging portfolio

Up to this stage, the pricing problem has been our main focus. However, inorder to reduce the risk, a specific strategy called a hedging strategy shouldalso be used. For example, should we sell a European option characterized


by an unbounded payoff ΦT at the maturity T , the loss could potentially beunlimited if we did nothing.Let us assume that our market model is described by a money market accountBt, n traded assets xiti=1···n and m unobservable Ito processes xαt withα = n+ 1, · · · , n+m. The (hedging) strategy consists in buying and sellingthe assets xit from t = 0 to t = T and investing the profit in the money marketaccount. The amount of asset i that we hold at time t is noted ∆i

t. Fromsection 2.6, the value of a self-financing portfolio at the maturity T is

πT = EP[DtTΦT |Ft]− ΦT +∫ T

t

n∑i=1

∆i(t, x)dxit (2.47)

with xit = xitD0t. The first term is the fair value of the contract with payoffΦT as given by the pricing formula (2.31). The second term is the payoff ΦTexercised at maturity T . The last term is the value of a self-financing portfolio(2.29). From (2.47), we introduce the notion of a complete market.

DEFINITION 2.12 Complete market The payoff ΦT is attainable(in the market (x,B)t∈[0,T ]) if there exists a self-financing portfolio π suchthat πT = 0 P-almost surely. If each payoff ΦT is attainable, then the marketis called complete. Otherwise, the market is called incomplete.

Let us see under which conditions a market is complete. As the fair pricef(t, xt) of an option with payoff ΦT is a local martingale, we have that undera risk-neutral measure P

df(t, xt) =n∑i=1

∂f(t, xt)∂xi

dxit +n+m∑α=n+1

∂f(t, xt)∂xα

[dxαt ]

with [dxαt ] the local martingale part of dxαt (i.e., we disregard the drift part).Integrating this equation between t and T and using that f(t, xt) = EP[DtTΦT |Ft],we obtain

ΦT ≡ f(T, xT ) = EP[DtTΦT |Ft] +∫ T

t

n∑i=1

∂f

∂xidxit +

∫ T

t

n+m∑α=n+1

∂f

∂xα[dxαt ]

By plugging this expression in (2.47), we obtain

πT =∫ T

t

n∑i=1

(∆i(t, xt)−

∂f(t, xt)∂xi

)dxit −

∫ T

t

n+m∑α=n+1

∂f(t, xt)∂xα

[dxαt ]

Therefore if we choose

∆i(t, x) =∂f(t, x)∂xi


and the market model is composed of traded assets only (plus a money market-account), then ΦT is attainable and the market is complete. We say thatwe have a dynamic Delta hedging strategy which consists in holding ∆i(t, x)asset i at time t. The resulting risk at maturity T cancels as the option ΦTis attainable.On the contrary, if we have unobservable Ito processes xα such as a stochas-tic volatility (i.e., [dxαt ] 6= 0), the model is incomplete. More generally, wecan prove that

THEOREM 2.8 Second theorem of asset pricing

A market defined by the assets (S0t , S

1t , · · · , Sdt )t∈[0,T ] (plus a money market

account), described as Ito processes on (Ω,F ,P), is complete if and only ifthere is a unique locale martingale measure Q equivalent to P.

2.10 Building market models in practice

The building of a market model is to be a two-step process: The first stepconsists in introducing the right financial traded assets relevant to our pricing.One could model equity assets, foreign exchange (Fx) rates, commodity assets.... In the second step, we model the fixed income rate which enter the defini-tion of the money market account. Once the dynamics of the interest rate hasbeen fixed, we should specify an Ito diffusion process for traded assets. In thiscontext, a choice of measure should be done. Usually, we use the risk-neutralor the forward measure. Let us now describe the modeling in different cases.

2.10.1 Equity asset case

For an equity asset, we know that under a risk-neutral measure (associatedto the money market account as a numeraire) the drift is constrained to bethe instantaneous interest rate rt.The forward fTt = St

PtTis a strictly positive local martingale in the forward

measure PT (associated to the bond PtT as numeraire). We can use the fol-lowing proposition 2.2 which characterizes a strictly positive local martingale.Previous to that, we recall the definition of a quadratic variation process:

DEFINITION 2.13 Quadratic variation For Xt a continuous stochas-tic process, the quadratic variation is defined by

< X,X >t (ω) = lim sup∆ti→0

n−1∑i=1

|Xti+1(ω)−Xti(ω)|2


where 0 = t1 < t2 < · · · < tn = t and ∆ti = ti+1 − ti. If Xt and Yt are twocontinuous stochastic processes, we define

< X,Y >t=12

(< X + Y,X + Y >t − < X,X >t − < Y, Y >t)

Note that for Xt =∫ t

0σsdWs with σt an Ft-adapted process, we have

< X,X >t=∫ t

0

σ2sds

PROPOSITION 2.2

If ft is a strictly positive local martingale then there exists a local martingaleXt such that

ft = f0e(Xt− 1

2<X,X>t)

Moreover if < X,X >t is absolutely continuous meaning that there exists aL2(dP× [0, t]) r.v., σ·, such that

< X,X >t=∫ t

0

σ2sds

then

Xt =∫ t

0

σs.dWs

with Wt a Brownian motion. σs is called the (stochastic) volatility.

In particular, if we assume that the forward and the bond are Ito processes,the dynamics of fTt under PT is

dfTt = (σt − σPt ).dWTt

with σPt the volatility of the bond PtT . In the risk-neutral measure P, we have

dStSt

= rtdt+ σt.dWt

At this stage, the form of σ and σP and the number of Brownians we use todrive the dynamics of fTt are unspecified. We will come back to this pointwhen we discuss fixed income rate modeling.


2.10.2 Foreign exchange rate case

For Fx rates, the specification of the dynamics is slightly more evolved. Letthe exchange rate between two currencies (domestic and foreign) be S

d/ft .

By convention, one unit in the foreign currency corresponds to Sd/ft in the

domestic currency. The domestic (resp. foreign) money market account isnoted Dd

t (resp. Dft ) and the domestic (resp. foreign) bond between t and T

is P dtT (resp. P ftT ). We also denote Pd (resp. Pf ) the domestic (resp. foreign)risk-neutral measure.Let us consider two different strategies:The first strategy consists in investing one unit in foreign currency at t up tothe maturity T and then convert the value in the domestic currency using theexchange rate Sd/fT at T . The undiscounted portfolio valued in the domestic

currency at T is DftDfT

Sd/fT .

In the second strategy, we directly convert our initial unit investment in thedomestic currency at t and then invest in the domestic money market ac-count. The value of this new portfolio, still valued in the domestic currency,is Ddt

DdTSd/ft .

In order to avoid arbitrage, we should impose that the expectations of thesetwo different discounted portfolios are equal

EPd [DdTD

ft

DdtD

fT

Sd/fT |Ft] = S

d/ft

equivalent to

EPd [DdT

DfT

Sd/fT |Ft] =

Ddt

Dft

Sd/ft

Therefore, DdtDftSd/ft should be a (local) martingale under the domestic risk-

neutral measure Pd. As DdtDft

= e−R t0 (rd−rf )ds, the drift of the exchange rate

Sd/ft is constrained to be the difference between the domestic and foreign

rates:

dSd/ft

Sd/ft

= (rd − rf )dt+ σd/f .dWdt (2.48)

The product of the Fx forward (defined as fd/ft = Sd/ft P ftTPdtT

) with the domestic

bond P dtT is a contract paying 1 in foreign currency valued in the domesticcurrency. Therefore fd/ft is a local martingale under the domestic forwardmeasure PTd . Its dynamics under PTd is

dfd/ft

fd/ft

= σd/f .dWdt − σPd .dW d

t + σP f .dWdt


with σS the volatility of the Fx rate and σPd (resp. σP f ) the volatility of thedomestic (resp. foreign) bond.To conclude this section, we consider the dynamics of an asset, valued in theforeign currency, under the domestic risk-neutral measure Pd. In Pf , we have

dSft

Sft= rfdt+ σS .dW

ft

By definition, the process Sd/ft Sft is the foreign asset valued in the domesticcurrency and therefore should be driven under Pd by

dSd/ft Sft

Sd/ft Sft

= rddt+ σS .dWdt + σd/f .dW

dt

for which we deduce that

d ln(Sd/ft Sft

)=(rd −

12σS .σS −

12σd/f .σd/f − σS .σd/f

)dt

+ σS .dWdt + σd/f .dW

dt

Then, by using (2.48), we deduce that the dynamics of Sft under Pd is

dSft

Sft=(rf − σd/f .σS

)dt+ σS .dW

dt

2.10.3 Fixed income rate case

As previously discussed, the modeling of equity assets, Fx rates or marketmodels in general depends on the modeling of the money market account(resp. bond) if we work in a risk-neutral measure (resp. forward measure).The modeling of a fixed-income rate is slightly more complex as there is nostandard way of doing this. In the following section, we quickly review themain models that have been introduced in this purpose: short-rate models,HJM model and Libor Market Models. Details and extensive references canbe found in [7].

2.10.3.1 Short rate models

In the framework of short-rate models, one decides to impose the dynamicsof the instantaneous interest rate rt. As rt is not a traded financial contract,there is no restriction under the no-arbitrage condition on its dynamics. Forexample, we can impose the time-independent one-dimensional Ito diffusionprocess

drt = a(rt)dt+ b(rt)dWt


TABLE 2.1: Example of one-factor short-rate models.one-factor short-rate model SDE

Vasicek-Hull-White drt = k(θ − rt)dt+ σdWt

CIR drt = k(θ − rt)dt+ σ√rtdWt

Dooleans drt = krtdt+ σrtdWEV-BK drt = rt(η − α ln(rt))dt+ σrtdWt

See table (2.1) for a list of functions a and b that are commonly used. Bydefinition, the discount factor is

Dt = e−R t0 rsds

and the fair pricing formula for a payoff ΦT in a risk-neutral measure P is

C = EP[e−R TtrsdsΦT ]

Moreover, the value of a bond quoted at t maturing at T is

PtT = EP[e−R Ttrsds]

Note that using the Feynman-Kac theorem, one gets under PdPtTPtT

= rtdt+ σP .dWt (2.49)

2.10.3.2 HJM model

The main drawback of short-rate models is that it is slightly suspicious toassume that the forward curve defined by

ftT = −∂ ln(PtT )∂T

is driven by the short rate rt = limT→t ftT . This situation can be improvedby directly modeling the forward curve ftT . This is the purpose of the HJMframework. A priori, ftT is not a financial contract and therefore there shouldbe no restriction under the no-arbitrage condition on its dynamics. However,we should have (2.49) and we will see that this constraint imposes the driftof the forward as a function of its volatility. Without loss of generality, theforward ftT follows the SDE

dftT = µ(t, T )dt+ σ(t, T )dWt

We will now derive the SDE followed by the bond PtT . Starting from thedefinition, we have

ln(PtT ) = −∫ T

t

ftsds

= −∫ T

t

ds(f0s +∫ t

0

µ(u, s)du+∫ t

0

σ(u, s)dWu)


Ito’s lemma gives

d ln(PtT ) =(f0t +

∫ t

0

µ(u, t)du+∫ t

0

σ(u, t)dWu

)−∫ T

t

ds(µ(t, s)dt+ σ(t, s)dWt

)Recognizing the first bracket as rt ≡ ftt, we obtain

d ln(PtT ) =

(rt −

∫ T

t

µ(t, s)ds

)dt−

∫ T

t

σ(t, s)dsdWt

and

dPtTPtT

=

(rtdt−

∫ T

t

µ(t, s)ds+12

∫ T

t

∫ T

t

σ(t, u)σ(t, s)dsdu

)dt

−∫ T

t

σ(t, s)dsdWt

Therefore, there is no-arbitrage if there exists a Ft-adapted function λt suchthat ∫ T

t

µ(t, s)ds =12

∫ T

t

∫ T

t

σ(t, u)σ(t, s)dsdu− λt∫ T

t

σ(t, s)ds

Differentiating with respect to T , we obtain

µ(t, T ) = σ(t, T )∫ T

t

σ(t, u)du− λtσ(t, T )

Finally, under the risk-neutral measure P, the forward curve follows the SDE

dftT = σ(t, T )∫ T

t

σ(t, u)du+ σ(t, T )dWt

where we have set dWt = dWt − λtdt.Note that from a HJM model, we can derive a short rate model rt = limt→T ftTgiven by

rt = f0t +∫ t

0

σ(s, t)ds∫ t

s

σ(s, u)du+∫ t

0

σ(s, t)dWs

We should note that this process is not necessarily Markovian.


2.10.3.3 Libor market models

The main drawback of the HJM model is that the (continuous) forward curveis an idealization and is not directly observable on the market. The forwardrates observed on the market are the Libor rates corresponding to the ratesbetween two future dates, usually spaced out three or six months apart. Wenote Li(t) the value of the Libor forward rate at time t between two futuredates Ti−1 and Ti (we say the tenor structure). The Libor Market Model(LMM) corresponds to the modeling of the forward rate curve via the Liborvariables. The forward bond at t between Ti−1 and Ti is by definition

P (t, Ti−1, Ti) ≡1

1 + τLi(t)

Besides the no-arbitrage condition requires that

P (t, Ti−1, Ti) =PtTiPtTi−1

which gives

Li(t) =1τi

(PtTi−1

PtTi− 1)

As the product of the bond PtTi with the forward rates Li(t) is a difference oftwo bonds with maturities Ti−1 and Ti, 1

τi(PtTi−1 − PtTi), therefore a traded

asset, Li(t) is a (local) martingale under Pi, the (forward) measure associatedwith the numeraire PtTi . Therefore we assume the following driftless dynamicsunder Pi

dLi(t) = Liσi(t)dW it

σi(t) can be a deterministic function; in this case we get the BGM model,or a stochastic process (which can depend on the Libors). In chapter 8, wewill discuss in details the BGM model and its extension including stochasticvolatility processes. In particular, we will explain how the asymptotic methodspresented in this book can be used to calibrate such a model.

2.10.4 Commodity asset case

Commodity assets are particular as their dynamics does not depend on themodeling of interest rates as it is the case for equity assets and Fx ratesbecause it is impossible to build a self-financing portfolio with commodityassets (which can not be stored). On the commodity market, one can tradecommodity future ftT which is the contract paying the spot ST ≡ fTT at thematurity T . Being a traded asset, ftT should be a local martingale under theforward measure PT

dftT = σ.dWt


where σ is an unspecified volatility.Note that the spot defined as St = limt→T ftT does not necessarily follow aMarkovian dynamics. This phenomenon is analog to what is observed in theHJM model for the instantaneous interest rate.

2.11 Problems

Exercises 2.1 Central limit theoremWe have N independent r.v. xi = +1,−1 such that p(+1) = p(−1) = 1

2 .We define the sum X =

∑Ni=1 xi

1. Compute the mean-value E[X].

2. Compute the variance E[X2]− E[X]2.

3. Compute the probability P (M) such that X = M .

4. Take the limit N →∞ of√N2 P (

√Nx) with x ∈ R.

5. Deduce the result above using the central limit theorem [26] (Hint: usethe Stirling formula).

Exercises 2.2Prove that (2.1) satisfies the definition of a measure.

Exercises 2.3 Wick’s identityLet Wt be a Brownian motion.

1. Prove that

EP[eiuWt ] = e−u2t2

2. Deduce the following formula, called the Wick identity, ∀n ∈ N

EP[(Wt)2n] =(2n− 1)!

2n−1(n− 1)!tn

EP[(Wt)2n+1] = 0


Exercises 2.4 Exponential martingaleLet us define Mt = exp(Wt − t

2 ) with Wt a Brownian motion. Prove thatMt is a martingale using the definition 2.9. Then prove that Mt is a (local)martingale using Ito’s lemma.

Exercises 2.5 Ito’s isometryLet us consider a deterministic function f : R→ R.

1. Prove that

EP[∫ t

0

f(s)dWs] = 0

EP[(∫ t

0

f(s)dWs

)2

] =∫ t

0

f(s)2ds

Hint: Assume that f(s) can be approximated by simple functions χ(s) =χi1si<s<si+1 (for the L2 norm).

These expressions can be extended to the case where f(s, ω) is anadapted process. We have the Ito isometry formula:

EP[∫ t

0

f(s, ω)dWs] = 0

EP[(∫ t

0

f(s, ω)dWs

)2

] =∫ t

0

EP[f(s, ω)2]ds

2. Let Z ∈ N(0, σ2) be a normal r.v. with zero mean and variance equalto σ2. Prove that

E[eZ ] = eσ22

3. Deduce that

EP[eR t0 f(s)dWs ] = e

12

R t0 f(s)2ds

Exercises 2.6 Ornstein-Uhlenbeck SDEThe Ornstein-Uhlenbeck process is defined by the following SDE

dXt = −αXtdt+ σdWt

with the initial condition X0 = x0 ∈ R.

1. Is there a unique solution?

Hint: Verify the Lipschitz condition.


2. Compute the mean-value EP[Xt].

3. Compute the variance EP[X2t ]− EP[Xt]2.

Hint: You should need the result 1 in problem (2.5).

4. Deduce the distribution for Xt.

Exercises 2.7 Simplified Girsanov’s theoremLet us suppose that Wt satisfies the SDE

dWt = dWt + λ(t)dt

with λ(·) a deterministic function and Wt a Brownian with respect to a mea-sure P. We want to prove that Wt is a Brownian motion with respect to themeasure P such that

dPdP|Ft = e−

R t0 λ(s)dWs− 1

2

R t0 λ(s)2ds

1. Prove that

EP[eiuWt ] = EP[dPdPeiuWt ]

2. By completing the square, prove that

EP[eiuWt ] = EP[e−R t0 λ(s)dWs− 1

2

R t0 λ(s)2dseiuWt ] = e−

u2t2

3. Deduce that Wt is a Brownian motion according to P.

Exercises 2.8 Barrier option in Bachelier’s model

In the Bachelier model, a forward ft follows a normal process

dft = dWt

Here we have taken a unit volatility. Note that this model has a severe draw-back as the forward can become negative.In this exercise, we want to price a barrier option (i.e., down-and-out call)which pays a call with strike K at the maturity T if the forward has notreached the barrier level B. We take K > B.

1. By using the localized Feynman-Kac theorem, prove that the fair valueC of the barrier option satisfies the PDE

∂tC(t, f) +12∂2fC(t, f) = 0 ,∀ f > B

C(T, f) = max(f −K, 0)1f>BC(t, B) = 0 ,∀ t ∈ [0, T ]


2. Show that the solution can be written as

C(t, f) = u(t, f)− u(t, 2B − f)

with u a call option with strike K and maturity T . This is called theimage’s method.

Exercises 2.9 P&L Theta-GammaWe study the variation of a trader’s self-financing portfolio composed of anasset with price St at time t and a European option with payoff φ(ST ) atthe maturity date T . The fair value of the option is C(t, St) at time t. Theportfolio’s value is at time t:

πt = C(t, St)−∆tSt

We take zero interest rate.

1. We assume that on the market the asset follows a Black-Scholes log-normal process with a volatility σreal. By using Ito’s formula, provethat the infinitesimal variation of the self-financing delta-hedge portfoliobetween t and t+ dt is

dπt = ∂tCdt+12S2∂2

SCσ2realdt

The term ∂tC (resp. ∂2SC) is called the Theta (resp. the Gamma) of the

option.

2. We assume that the option was priced using a Black-Scholes model witha volatility σmodel. Prove that

dπt =12S2∂2

SC(σ2

real − σ2model

)dt

3. Deduce that the expectation value EP[∆πT ] of the variation of the port-folio between t = 0 to t = T is

EP[∆πT ] =12

∫ T

0

EP[S2∂2SC]

(σ2

real − σ2model

)dt

This expression is called the profit and loss (P&L) Theta-Gamma. Fora European call option, the Gamma is positive and therefore, if σreal >σmodel (resp. σreal < σmodel) , the P&L is positive (resp. negative).When the market realized volatility σreal equals the model volatilityσmodel, the P&L vanishes. In this context, σmodel is called the break-even volatility.

Chapter 3

Smile Dynamics and Pricing ofExotic Options

Abstract We review the definition of the implied volatility in equity mar-kets and present examples of exotic options which give an insight into thedynamics of the implied volatility. Similar definitions and examples are alsopresented for fixed-income markets.

3.1 Implied volatility

The Black-Scholes implied volatility σBS t(K,T ), starting at t, also calledsmile, is defined as the “wrong number” which, when put in the Black-Scholesformula CBS for a European call option with strike K and maturity T quotedat t, reproduces the fair price on the market Cmkt. More precisely, we havethe following definition

DEFINITION 3.1 The implied volatility σBS t(K,T ) is such that

Cmkt(K,T |t) = CBS(K,T, σBSt(K,T )|S, t)

where

CBS(K,T, σ|S, t) = SN(d+)−KPtTN(d−) (3.1)

with

d± = −ln(KPtTS )

σBS t(K,T )√T − t

± σBS t(K,T )√T − t

2

Here N(x) =∫ x−∞ e−

x22 dx√

2πis the cumulative normal distribution, PtT the

bond value expiring at T and S is the spot.

The solution is unique because CBS is strictly increasing in σ. In practice, thenumerical solution can be found using a dichotomy algorithm. As a function

55


of the strike K and the maturity T , σBS t(K,T ) describes a surface or amembrane if the time t is added. For a fixed maturity T (and time t), theform of σBS t as a function of the strike can vary according to the marketconsidered. In equity markets (resp. Fx markets), the smile is a decreasing(resp. parabolic) function of the strike. An important characteristic of thesmile is its skew defined as the derivative of the smile according to the log ofthe strike at-the-money (i.e., the strike is fixed to the spot):

DEFINITION 3.2 Volatility Skew:

S(T ) = K∂σBS

∂K|K=S

We can also define the skew at-the-money forward (i.e., the strike is fixed tothe forward fTt ):

SF (T ) = K∂σBS

∂K|K=fTt

We will now describe several useful properties satisfied by the smile whichrestrain the form that the implied volatility surface can take.

• Let us consider a time-homogeneous stochastic volatility model given bythe following SDE under the forward measure PT

dfTtfTt

= σtdWt

with σt an adapted process and fTt the forward. We assume that con-ditional on the filtration generated by the process σt, the forward islog-normal. Then the implied volatility is a function of the moneynessm = K/ft, the time-to-maturity τ = T − t and the volatility σt

σBS t(K,T ) = σtΦ(m,σ2t τ) (3.2)

PROOF This can be derived using the fact that the parameters[σ2t τ ] and m = K/ft are dimensionless. Thus, if we write the implied

volatility

σBS t(K,T ) = σtΦ(τ, ft,K, σt)

the general function Φ(·) must be dimensionless and therefore dependson dimensionless parameters only, i.e., m and σtτ

2

Smile Dynamics and Pricing of Exotic Options 57

• Given the inequality −1 < P−1tT

∂Cmkt

∂K < 0,1 we derive the following(rough) bounds for the skew at-the-money forward

− N(d−)√T − tn(d−)

|K=fTt≤ −SF (T ) ≤ N(d+)√

T − tn(d+)|K=fTt

with n(x) = e−x22√

2π.

PROOF From the Black-Scholes formula (3.1), we have

−P−1tT

∂Cmkt

∂K|K=fTt

= N(d−)|K=fTt− SF (T )n(d−)|K=fTt

√T − t

From the inequality 0 < −P−1tT

∂Cmkt

∂K < 1 and (N(d+) +N(d−)) |K=fTt=

1, we obtain the bounds above.

Example 3.1 Merton modelThe Merton model is given by the SDE on the forward fTt

dfTt = σ(t)fTt dWt

Thanks to a change of local time t′ =∫ t

0σ(s)2ds, we obtain the GBM

dfTt′ = fTt′ dWt′

The implied volatility at the maturity T is therefore

σBS t(T )2(T − t) =∫ T

t

σ(s)2ds

By definition, European call-put options are sensitive to the implied volatilitygiven a specific strike and maturity. In the following section, we will see that ageneral European option quoted at time t with a maturity date T on a singleasset depends on the full implied volatility at T : K → σBS t(K,T ).

3.2 Static replication and pricing of European option

A European option on a single stock or index is characterized by a payoffφ(ST ) depending only on the price ST of a single stock or an index at the

1 ∂Cmkt

∂K= −PtTEPT [1(ST −K)].


maturity date T . This payoff φ(ST ) can be decomposed (we say replicated)over an infinite sum of calls and puts of different strikes as written below.Firstly, we have the following identity for ST > 0

φ(ST ) = φ(St) + φ′(St)(ST − St) +∫ St

0

φ′′(K) max(K − ST , 0)dK

+∫ ∞St

φ′′(K) max(ST −K, 0)dK (3.3)

The prime indicates a derivative with respect to S. The derivation is done ac-cording to the distribution theory where φ(ST ) is considered as a distributionbelonging to the space D′. For example, the first derivative of a call payoffmax(x −K, 0) is the Heaviside function 1(x −K) and the second derivativeis the Dirac function δ(x−K).

PROOF We take x > 0 and start from the representation of the functionf(·) as

f(x) =∫ ∞

0

f(K)δ(K − x)dK

=∫ x∗

0

f(K)∂2

∂K2max(K − x, 0)dK +

∫ ∞x∗

f(K)∂2

∂K2max(x−K, 0)dK

Integrating by parts twice gives (3.3) with x ≡ ST and x∗ ≡ St

As the fair value of an option is given in the forward measure PT by

PtTEPT [φ(ST )|Ft]

multiplying both sides of (3.3) by the bond value PtT and taking the pricingoperator EPT [·|Ft], we obtain the fair price of a European option with payoffφ(ST )

PtTEPT [φ(ST )|Ft] = PtTφ(St) + φ′(St)St(1− PtT ) +∫ St

0

φ′′(K)PBS(t,K, T )

+∫ ∞St

φ′′(K)CBS(t,K, T ) (3.4)

with

CBS(t,K, T ) = PtTEPT [max(ST −K, 0)|Ft]

and

PBS(t,K, T ) = PtTEPT [max(K − ST , 0)|Ft]


the fair price of European call and put options with strike K and maturity Tquoted at t ≤ T . We have used that

PtTEP[ST ] = PtTEP[fTT ] = PtT fTt = St

We observe that the fair value of the payoff φ(ST ) is completely model inde-pendent as it can be decomposed over an infinite sum of European call-putoptions that are quoted on the market. Note that in practice only a finitenumber of strikes are quoted on the market and the pricing of (3.4) requiresthe extrapolation of the implied volatility outside the known strikes (the so-called wings). The payoff φ is therefore sensitive to the wings of the impliedvolatility.A common (European) derivative product sensitive to the wings of the impliedvolatility is a variance swap option.

Example 3.2 Variance swapA variance swap option is a contract which pays at the maturity date T the(daily) realized variance of the stock from t to T minus a strike. The strike iscalled the variance swap fair value, noted VS below. The payoff at T is

1N

N−1∑i=0

ln(Si+1

Si

)2

−VS

where Si is the stock price at ti (with t0 = t and tN = T ). By marketconvention, the strike VS is fixed such that the price of the option quoted att is zero. Therefore VS is

VS = EPT [1N

N−1∑i=0

ln(Si+1

Si

)2

|Ft] (3.5)

where we have used the forward measure PT for convenience sake. Note thatthe discount factor PtT cancels out.Assumption 1: We assume that the stock price St can be described by anIto diffusion process and that St follows the SDE in a risk-neutral measure P

dStSt

= rtdt+ σt.dWt

We make no assumptions on rt and σt, which therefore can be either deter-ministic or stochastic processes.In the large N limit, VS, given by (3.5), converges to the integral

VS =1

T − t

∫ T

t

EPT [σ2s |Ft]ds


Then, thanks to a change of measure, the process St in the forward measurePT is

dStSt

= (rt + σt.σPt )dt+ σt.dW

Tt (3.6)

with σPt the volatility of the bond PtT . By applying Ito’s lemma on ln(St),we have∫ T

t

σ2s

2ds = − ln

(STSt

)+∫ T

t

(rs + σs.σPs )ds+

∫ T

t

σs.dWTs (3.7)

If we apply the pricing operator EPT [·|Ft] on both sides of this equation, wededuce that the variance swap VS is related to a log-contract option with apayoff ln

(STSt

)VS = − 2

T − tEPT [ln

(STSt

)|Ft] +

2T − t

∫ T

t

EPT [(rs + σs.σPs )|Ft]ds

where we have used that EPT [∫ TtσsdW

Ts |Ft] = 0 by Ito’s isometry.

Assumption 2: Let us assume that the instantaneous rate is a deterministicconstant process rs ≡ r(s) and therefore the volatility of the bond σPt cancels.The value of VS simplifies to

VS = − 2T − t

EPT [ln(STSt

)|Ft] +

2T − t

∫ T

t

r(s)ds

From (3.4), we obtain the static replication

VS =2P−1

tT

(T − t)

(∫ St

0

dKPBS(t,K, T )

K2+∫ ∞St

dKCBS(t,K, T )

K2

)

+2

T − t(lnP−1

tT + 1− P−1tT

)(3.8)

In this equation, we have used that lnPtT = −∫ Ttr(s)ds.

Example 3.3 Generalized Variance swapThe construction above can be generalized to any option which pays at the

maturity date T

1N

N−1∑i=0

f(Si)(

ln(Si+1

Si

))2

−K

Here f is a measurable function. For

f(S) = 1(S

St∈ [A,B])


we have a corridor variance swap [71]. As for the variance swap, the strike Kis fixed such that the value of the option quoted at t is zero. Thus K is underthe forward measure PT (for convenience sake) equal to

K =1N

N−1∑i=0

EPT [f(Si)(

ln(Si+1

Si

))2

|Ft]

In the large N limit, K becomes

K =1

(T − t)

∫ T

t

EPT [f(Ss)σ2s |Ft]ds (3.9)

We want to specify the function f(Ss) such that there still exists a staticreplication formula for the generalized variance swap. In this context, wenote that using the Ito lemma, we have the relation

g(ST )− g(St) =∫ T

t

Ss∂Sg(Ss)σs.dWTs +

∫ T

t

S2s∂

2Sg(Ss)

σ2s

2ds

+∫ T

t

Ss∂Sg(Ss)(rs + σs.σPs )ds

for a function g ∈ C2(R) and the Ito diffusion process St following (3.6).Moreover, taking the mean value under PT , we obtain

EPT [g(ST )|Ft]− g(St) =∫ T

t

EPT [S2s∂

2Sg(Ss)

σ2s

2|Ft]ds

+∫ T

t

EPT [Ss∂Sg(Ss)(rs + σs.σPs )|Ft]ds (3.10)

Comparing (3.9) and (3.10), if we impose that the function f(Ss) satisfies

S2

2∂2Sg(S) = f(S) (3.11)

and that the interest rate is a deterministic process r(s) (i.e., the bond’svolatility cancels), then there is a static replication formula

K =1

(T − t)

(EPT [g(ST )|Ft]− g(St)−

∫ T

t

r(s)EPT [Ss∂Sg(Ss)|Ft]ds

)The general solution of (3.11) can be written as a sum of a particular solutiongp(S) and the homogeneous solution aS + b with a and b two constants:

g(S) = gp(S) + aS + b

We obtain

K =1

(T − t)

(EPT [gp(ST )|Ft]− gp(St)−

∫ T

t

r(s)EPT [Ss∂Sgp(Ss)|Ft]ds

)


The term EPT [gp(ST )|Ft] (resp. EPT [Ss∂Sgp(Ss)|Ft]) can be decomposed asa sum of call and put options with maturity T (resp. s) using (3.4). Forf(S) = 1, the relation (3.11) gives gp(S) = −2 ln(S) and we reproduce thevariance swap replication formula (3.8). For a corridor variance swap withf(S) = 1

(SSt∈ [A,B]

), we obtain

gp(S) = 1(S

St∈ [A,B]

)lnS

St+ 1(S > BSt)

(S

BSt+ lnB − 1

)+ 1(S < ASt)

(S

ASt+ lnA− 1

)

3.3 Forward starting options and dynamics of the im-plied volatility

From the fair value of European call-put options, the value of the impliedvolatility can be deduced. However, this surface evolves with time and de-scribes a non-trivial dynamics. For European options, we have seen in thesection on static replication that the implied volatility dynamics has no im-pact on their valuations as their fair values depend only on the initial impliedvolatility. However, path-dependent exotic options will depend strongly onimplied volatility dynamics. It is therefore important to have at our disposalfinancial instruments that can inform on the dynamics followed by the impliedvolatility. Such exotic options are the forward-starting options and in prac-tice the cliquet options. Using these instruments, we will define the forwardimplied volatility and the forward skew. As a foreword we give simple rulesthat have been used to estimate the evolution of the implied volatility.

3.3.1 Sticky rules

The observation that the implied volatility surface evolves with time has ledpractitioners to develop simple rules to estimate its evolution.

• Sticky delta: A commonly used rule is the “sticky moneyness” rule whichstipulates that when viewed in relative coordinates (m = ln(KSt ), τ =T − t), the surface σBS t(τ,m) remains constant from day to day:

∀(τ,m) σBS t+∆t(τ,m) = σBS t(τ,m)

• Sticky strike: Another well-known rule is the so-called “sticky rule” : inthis case one assumes that the level of implied volatilities in absolute


strikes does not change:

∀(τ,K) σBS t+∆t(τ,K) = σBS t(τ,K)

3.3.2 Forward-start options

A forward-start option on a single asset is a contract according to which theholder receives an option at T1 with an expiry date T2 > T1. For a forward-start call option, the payoff at T2 is

max(ST2

ST1

−K, 0)

Its fair price is given at time t by

C(t, St) = PtT1EPT1 [S−1T1

max (ST2 −KST1, 0) |Ft]

with PT1 the forward measure. By using the tower property, we have

C(t, St) = PtT1EPT1 [S−1T1

max (ST2 −KST1 , 0) |Ft]

= PtT1EPT1 [S−1T1

EPT1 [max (ST2 −KST1 , 0) |FT1 ]|Ft]

Assuming that St follows a Black-Scholes log-normal model with a volatilityσ and that we have a deterministic rate, we obtain

C(t, St) = PtT1EPT1 [N(d+)−KPT1T2N(d−)|Ft]= PtT1N(d+)−KPtT2N(d−) (3.12)

with d± = − ln(KPT1T2 )

σBS√T2−T1

± σBS√T2−T12 . From this Black-Scholes-like formula,

we define the forward implied volatility:

DEFINITION 3.3 The forward implied volatility is defined as the “wrongnumber” σ which when put in the formula (3.12) reproduces the fair valuequoted on the market.

We can also define a forward skew as the skew of the forward implied volatil-ity. Usually the forward implied volatility is derived from more complicatedforward-start options called cliquet options.

3.3.3 Cliquet options

A cliquet is a series of forward-start call options, with periodic settlements.The profit can be accumulated until final maturity, or paid out at each resetdate. The payoff is the sum

N∑i=1

max(STi+1

STi−K, 0

)


In the next subsection, we present an example of a new derivative productwhich strongly depends on the dynamics of the (forward) implied volatility.

3.3.4 Napoleon options

Napoleon options are financial instruments that give the traders the oppor-tunity to play with the forward volatility of a market. The main factors ofthe payoff of a Napoleon option are a fixed coupon and the worst return ofan index over specified time periods.Let T0 < T1 < · · · < TN be several settlement period dates. The payoff

will depend on the performance of a single stock or index Rj ≡Stj+1i

Stji

−

1tji∈[Ti,Ti+1] between each period [Ti, Ti+1]. It is given at each Ti+1 by

min(

cap,max(

coupon + minj

[Rj ],floor))

When the market volatility is small, the buyer of a Napoleon option will havea good chance to get a payoff around the fixed coupon. When the marketvolatility is large, the buyer will get at least the minimum return determinedby the floor.Napoleon options vary from one contract to another. Some Napoleon op-tion contracts have multiple coupon calculation/payment periods and eachcoupon consists of performances of the underlying index on multiple timeperiods. Some have a single coupon payment but multiple performance cal-culation periods. Some have only one coupon payment and one performancecalculation period also. Besides, caps and floors on performances and couponsare specified differently across contracts.In the following section, we define the liquid instruments that are used tocalibrate an interest-rate model. As usual, we will compute their fair valuesin a Black-Scholes-like model and this enables us to define an implied volatility.

3.4 Interest rate instruments

3.4.1 Bond

The first liquid interest-rate instrument is a bond which is a contract whichpays 1 at a maturity date T . Its fair value at t < T is noted PtT and it isgiven in the risk-neutral measure P by

PtT = EP[e−R Ttrsds|Ft]

Similarly, we can define a forward bond which is a contract which pays a bond,expiring at T2, at maturity T1 < T2. Its value at t < T1 is noted P (t, T1, T2).


From a no-arbitrage argument, we have that

P (t, T1, T2) =PtT2

PtT1

(3.13)

A Libor L(t, T1, T2) is defined as

P (t, T1, T2) =1

1 + (T2 − T1)L(t, T1, T2)

By using (3.13), we obtain

L(t, T1, T2) =1

(T2 − T1)

(PtT1

PtT2

− 1)

3.4.2 Swap

A swap is a contract composed of two flux (we say two legs). One pays aLibor rate and the other one a fixed rate. At each instant Ti in Tα+1, ..., Tβ ,the fixed legs pay (τi = Ti − Ti−1)

τiK

whereas the floating leg pays

τiL(Ti, Ti−1, Ti)

The discounted payoff at time t for the payer of the fixed leg (receiving thefloating leg) is

β∑i=α+1

DtTiτi (L(Ti, Ti−1, Ti)−K)

We remind the reader that DtT is the discount factor as defined in (2.23). Byusing the forward measure PTi associated to the bond PtTi , we have that

EP[DtTiτi (L(Ti, Ti−1, Ti)−K) |Ft] = PtTiτiEPTi

[(L(Ti, Ti−1, Ti)−K) |Ft]

Therefore, the price of the swap is

β∑i=α+1

τiPtTiEPTi [(L(Ti, Ti−1, Ti)−K) |Ft] =β∑

i=α+1

τiPtTi (L(t, Ti−1, Ti)−K)

where we have used that L(t, Ti−1, Ti) is a martingale in the forward measurePT i as explained in 2.10.3.3. From the relation L(t, Ti−1, Ti) = 1

τi

(PtTi−1PtTi

− 1)

,we obtain that the sum of the two legs is

−PtTβ + PtTα −Kβ∑

i=α+1

τiPtTi


By definition, the swap rate is the strike K = sαβ,t such that the value of theswap contract at time t is zero

sαβ,t =PtTα − PtTβ∑βi=α+1 τiPtTi

(3.14)

The natural next question is how one can reconstruct the yield curve betweentwo dates t and T from the swap data (note that a Libor L(t, Tα−1, α) =sα−1α,t is an example of swap). Let us consider a set of m forward Libor ratesLk that are associated to the tenor structure Tkk = 1, · · · , n. We want toreconstruct at any time the yield curve described by the Lk ≡ L(t, Tk−1, Tk)Libor rates from the swaps characterized by the set of pairs

S = εj = (s(j), e(j)) , j = 1, · · ·m , 1 ≤ s(j) < e(j) ≤ m

In accordance with our previous notation, a swap sαβ has s = α+1 and e = β.We have the following result

PROPOSITION 3.1 [135]

The yield curve can be built from the set S if and only if n = m and s(j) = j.That is why, at each tenor date, we have to specify a unique swap.

3.4.3 Swaption

A swaption gives the right, but not the obligation, to enter into an interestrate swap at a pre-determined rate, noted K below, on an agreed future date.The maturity date for the swaption is noted Tα and coincides with the firsttenor date of the swap contract. Tβ is the expiry date for the swap. The valueof the swap at Tα is

β∑i=α+1

τiPTαTi (L(Ti, Ti−1, Ti)−K)

Then the discounted value of the swaption at t ≤ Tα is under the risk-neutralmeasure P

Cαβ = EP[DtTα max

(β∑

i=α+1

τiPTαTi(L(Ti, Ti−1, Ti)−K), 0

)]

By using the definition (3.14) of the swap sαβ,t, this can be written as

Cαβ = EP[DtTα

β∑i=α+1

τiPTαTi max (sαβ,Tα −K, 0)]


We consider the measure Pαβ associated to the numeraire∑βi=α+1 τiPtTi . The

Radon-Nikodym change of measure from P to Pαβ is

MTα ≡dPαβ

dP|FTα =

∑βi=α+1 τiPTαTiDTαTα

By using (2.41), we obtain

Cαβ =β∑

i=α+1

τiEPαβ [DtTαPTαTiMt

MTα

max (sαβ,Tα −K, 0)]

As

Mt

MTα

=∑βi=α+1 τiPtTi∑β

i=α+1 τiPTαTiDtTα

we get our final result

Cαβ =β∑

i=α+1

τiPtTiEPαβ [max (sαβ,Tα −K, 0) |Ft]

As the product of the swap rate sαβ,t with the numeraire∑βi=α+1 τiPtTi is a

difference of two bonds PtTα − PtTβ and therefore a traded asset, the swap isa local martingale in the measure Pαβ . Adopting a Black-Scholes-like model,we assume that the swap is log-normal (in the measure Pαβ)

dsαβ,t = σαβsαβ,tdWαβ (3.15)

with the constant volatility σαβ . By noting the similarity with the computa-tion of the fair value of a European call option in the Black-Scholes model,we obtain the fair value of a swaption as

Cαβ =β∑

i=α+1

τiPtTiCBS(K,Tα, σαβ |sαβ,t, t) (3.16)

with sαβ,t the swap spot at t.We can then define the swaption implied volatility as

DEFINITION 3.4 Swaption implied volatility The swaption impliedvolatility is the volatility that when put in the formula (3.16) reproduces themarket price of a swaption.

The at-the-money swaption implied volatilities (i.e., K = sαβ,0) are coded ina matrix, usually called swaption matrix. The columns represent the tenor of


the swap (i.e., Tβ − Tα) and the rows represent the maturity of the swaption(i.e., Tα). Two particular lines are useful for calibration purposes (more detailsare given in chapter 8). The first one is the first column which correspondsto caplets. The second one is the anti-diagonal, called co-terminal swaptions.An extra dimension can be added with the strike K and we have a swaptioncube.

3.4.4 Convexity adjustment and CMS option

Convexity correction arises when an interest rate is paid out at the “wrongtime.” As an example, we have a Constant Maturity Swap (CMS) which paysthe swap rate sαβ,Tα at the maturity date Tα. Therefore the fair value is inthe forward measure PTα

CMSαβ(t) = PtTαEPTα [sαβ,Tα |Ft] (3.17)

As discussed previously, the swap rate is not a local martingale in the forwardmeasure PTα . This is the case under the swap measure Pαβ associated to thenumeraire Cαβ(t) =

∑βi=α+1 τiPtTi . By using this natural measure Pαβ , we

can rewrite CMSαβ(t) as ( Eαβ ≡ EPαβ )

CMSαβ(t) = Cαβ(t)Eαβ [1∑β

i=α+1 τiPTαTisαβ,Tα |Ft]

= Cαβ(t)Eαβ [

(1∑β

i=α+1 τiPTαTi− 1

)sαβ,Tα |Ft]︸︷︷︸+Cαβ(t)s0

αβ

Convexity correction (3.18)

As sαβ is a local martingale under Pαβ (we assume that sαβ,t is also a mar-tingale), the second term in (3.18) is given by the spot swap rate s0

αβ ≡ sαβ,ttimes Cαβ(t). The first term, usually small, is called the convexity correc-tion. To compute this quantity, we assume that the numeraire ratio Rαβ =

PtTαPβi=α+1 τiPtTi

depends on the time t and the swap rate sαβ,t

PtTα∑βi=α+1 τiPtTi

− 1 ≡ G(t, sαβ,t) (3.19)

The convexity correction can be written as

Cαβ(t)Eαβ [G(Tα, sαβ,Tα)sαβ,Tα |Ft]

The simple assumption (3.19) implies a strong constraint on the functionG(·, ·). By definition Rαβ is a local martingale under Pαβ and the functionG(t, sαβ) must be a local martingale under Pαβ meaning that its drift cancels.


As sαβ is a local martingale under Pαβ , for example as given in (3.15), wehave the PDE satisfied by G

∂tG(t, sαβ) +12σαβ(sαβ)2∂2

sαβG(t, sαβ) = 0 (3.20)

with the initial condition

G(0, s0αβ) =

P0Tα

Cαβ(0)− 1 (3.21)

For example, assuming that the function G does not depend on the time,G = G(sαβ), then the solution of (3.20) is a linear function of sαβ

G(sαβ) = a+ bsαβ

The initial condition (3.21) implies that a+ bs0αβ = P0Tα

Cαβ(0) − 1.Then, to compute the convexity correction, we use a static replication for thefunction f(x) ≡ G(Tα, x)x similar to (3.4)

Eαβ [f(sαβ)|Ft] = f(s0αβ) + f ′(s0

αβ)Eαβ [sαβ − s0αβ |Ft]

+∫ s0αβ

0

f′′(x)Eαβ [max(x− sαβ , 0)|Ft]dx

+∫ ∞s0αβ

f′′(x)Eαβ [max(sαβ − x, 0)|Ft]dx

and finally the fair value of the CMS is

CMSαβ(t) = Cαβ(t)s0αβG(Tα, s0

αβ)

+∫ s0αβ

0

G′′(Tα, x)xPαβ(x)dx+∫ ∞s0αβ

G′′(Tα, x)xCαβ(x)dx

+ 2∫ s0αβ

0

G′(Tα, x)Pαβ(x)dx+ 2∫ ∞s0αβ

G′(Tα, x)Cαβ(x)dx

where we have set Cαβ(x) ≡ Cαβ(t)Eαβ [max(sαβ,Tα − x, 0)|Ft]2 the value of acall swaption with strike x. In the particular case when G is linear, we have

CMSαβ(t) = Cαβ(t)s0αβ(a+ bs0

αβ) + 2b

(∫ s0αβ

0

Pαβ(x)dx+∫ ∞s0αβ

Cαβ(x)dx

)(3.22)

2resp. Pαβ(x) ≡ C0αβ(t)Eαβ [max(x− sαβ,Tα , 0)|Ft].


Therefore, if we want to match the CMS (3.22) and the initial condition (3.21),we should impose that the convexity adjustment function G is

G(sαβ) =PtTαCαβ(t)

− 1 +CMSαβ(t)− s0

αβ (PtTα − Cαβ(t))

2(∫ s0αβ

0 Pαβ(x)dx+∫∞s0αβCαβ(x)dx)

(sαβ − s0αβ)

Modulo the fundamental assumption (3.19) on the dependence of the yieldcurve according to the unique swap rate sαβ , the value that we have foundfor the CMS (3.22) is model-independent.Before closing this section, we present an other convexity adjustment whichhas the inconvenience of being model dependent. We assume that G is afunction of the swap rate sαβ only. The dynamics of sαβ follows a log-normalprocess (3.15) in Pαβ . Then, the dynamics of sαβ in PTα is

dsαβsαβ

= σ2αβsαβ

∂sαβG(sαβ)G(sαβ)

dt+ σαβdWα

By approximating the drift by its value at the spot swap rate s0αβ , the process

sαβ becomes log-normal and the value of the CMS (3.17) is given by

CMSαβ(t) = PtTαs0αβe

(Tα−t)σ2αβs

0αβ

∂sαβG(s0αβ)

G(s0αβ

)

3.5 Problems

Exercises 3.1 Volatility swapA Volatility swap is a contract that pays at a maturity date T the realized

volatility σrealized (which is the square root of the realized variance) of a stockor index minus a strike K. By definition, the strike K is set such that thevalue of the volatility swap at t (i.e., today) is zero. Therefore, we have

K = EP[√Vrealized|Ft]

with

Vrealized ≡1N

N−1∑i=0

ln(Si+1

Si

)2

We recall that a variance swap K is

K = EP[Vrealized|Ft]


Assuming a diffusion model for the stock price St, when we take the limitN →∞, the realized variance between t and T is

Vrealized ≡1

T − t

∫ T

t

σ2sds

Here σs is the instantaneous volatility defined by

dSt = σtStdWt

We assume zero interest rate. We have seen in this chapter that the value of avariance swap (i.e., K) is completely model-independent. Indeed, there existsa static replication where the value of the variance swap can be decomposedas an infinite sum of calls and puts which are priced on the market. In thisproblem, we propose to show that a similar model-independent solution existsfor the volatility swap if we assume that the stock St and the volatility σt areindependent processes. This problem is based on the article [72]. Below, wenote VT = Vrealized.

1. As the processes σt and Wt are independent by assumption, prove thatconditional on the filtration FσT generated by the volatility σtt≤T ,XT = ln(STSt ) is a normal process with mean − 1

2 (T − t)VT and variance(T − t)VT .

2. As an application of the tower property, we have ∀p ∈ C,

EP[epXT ] = EP[EP[epXT |FσT ]]

By using question 1. and the equality above, prove that

EP[epXT ] = EP[eλ(p)(T−t)VT ] (3.23)

with λ(p) ≡ p2

2 −p2 . Equivalently, we have p(λ) = 1

2 ±√

14 + 2λ.

3. Let f(t) be a real function, locally summable on R+. The Laplacetransform of f , noted L(z), z = x+ iy ∈ C, is defined as

L(z) =∫ ∞

0

f(t)e−ztdt

If L(z) is defined for all x ≡ Re[z] > x0 and if L(z) is a summablefunction ∀x > x0, then the inversion formula (called the Bromvitchformula) giving f(t) is

1t≥0f(t) =1

2πi

∫ a+i∞

a−i∞L(λ)eλxdλ (3.24)

where a > x0. By using the inversion formula (3.24), prove that theLaplace transform of the square root function is

√t =

12√π

∫ ∞0

1− e−λt

λ32

dλ (3.25)


4. By using (3.25) and (3.23), deduce that

EP[√

(T − t)VT ] =12√π

∫ ∞0

1− EP[e( 12−√

14−2λ)XT |Ft]

λ32

dλ

5. By using the static replication formula (3.4), decompose the Europeanoption EP[e( 1

2−√

14−2λ)XT |Ft] on an infinite sum of calls and puts.

6. Deduce the model-independent formula for the volatility swap.

Exercises 3.2 Ho-Lee equity hybrid modelThe purpose of this exercise is to price exotic options depending on a single

stock and characterized by a large maturity date. As a consequence, theassumption of deterministic rates should be relaxed. In this problem, wepresent a simple toy model which achieves this goal and we focus on calibrationissues. It is assumed that the stock follows a Merton model

dS

S= rtdt+ σ(t)dWt

where σ(t) is a time-dependent volatility. The instantaneous short rate rt isassumed to be driven by

drt = θ(t)dt+ ψdZt

with a constant volatility ψ, a time-dependent drift θ(t) and dZt a Brownianmotion correlated to dWt: dWtdZt = ρdt. This is the short-rate Ho-Lee model[7]. The next questions deal with how to calibrate these model parametersσ(·), ψ and θ(·).

1. Prove that the rate can be written as rt = φ(t) + xt with φ(t) a deter-ministic function that you will specify and xt a process following

dxt = ψdZt

2. Prove that xt is a Brownian process and compute its mean and itsvariance at the time t.

3. By using the results of the question above, prove that the value of abond PtT expiring at T and quoted at t < T is

PtT = A(t, T )e−B(t,T )xt

with A(t, T ) and B(t, T ) two functions that you will specify. A modelsuch that the bond value can be written in this way is called an affinemodel.


4. Deduce that we can choose the function θ(t) in order to calibrate theinitial yield curve P0T , ∀T .

5. From the previous questions, deduce that the bond PtT follows the SDE

dPtTPtT

= rtdt+ σP (t)dZt

with σP (t) = −ψ(T − t).

6. Prove that the (equity) forward fTt = StPtT

in the forward measure PTfollows the SDE

dfTtfTt

= σ(t)dWt − σP (t)dZt

Why is it obvious that the forward is a local martingale?

7. As the forward obeys a log-normal process, prove that the impliedvolatility with maturity T is

σ2BS,t(T − t) =

∫ T

t

(σ(s)2 + σP (s)2 − 2ρσ(s)σP (s)

)ds

8. Deduce the Merton volatility σ(s) in order to calibrate the at-the-moneymarket implied volatility.

Chapter 4

Differential Geometry and HeatKernel Expansion

AΓEΩMETPHTOΣ MH∆EIΣ EIΣITΩ1

—Plato

Abstract In this chapter, we present the key tool of this book: the heatkernel expansion on a Riemannian manifold. In the first section, in orderto introduce this technique naturally, we remind the reader of the link be-tween the multi-dimensional Kolmogorov equation and the value of a Euro-pean option. In particular, an asymptotic implied volatility in the short-timelimit will be obtained if we can find an asymptotic expansion for the multi-dimensional Kolmogorov equation. This is the purpose of the heat kernelexpansion. Rewriting the Kolmogorov equation as a heat kernel equation ona Riemannian manifold endowed with an Abelian connection, we can applyHadamard-DeWitt’s theorem giving the short-time asymptotic solution to theKolmogorov equation. An extension to the time-dependent heat kernel willalso be presented as this case is particularly important in finance in order toinclude term structures. In the next chapters, we will present several appli-cations of this technique, for example the calibration of local and stochasticvolatility models.

4.1 Multi-dimensional Kolmogorov equation

In this part, we recall the link between the valuation of a multi-dimensionalEuropean option and the backward (and forward) Kolmogorov equation. Forthe sake of simplicity, we assume a zero interest rate.

1“Let no one inapt to geometry come in.” Inscribed over the entrance to the Plato Academyin Athens.

75


4.1.1 Forward Kolmogorov equation

We assume that our market model depends on n Ito processes which can betraded assets or unobservable Markov processes such as a stochastic volatility.Let us denote the stochastic processes x ≡ (xi)i=1,··· ,n and α ≡ (αi)i=1,··· ,n.These processes xi satisfy the following SDEs in a risk-neutral measure P

dxit = bi(t, xt)dt+ σi(t, xt)dWi (4.1)dWidWj = ρij(t)dt

with the initial condition x0 = α.Here [ρij ]i,j=1,··· ,n is a correlation matrix (i.e., a symmetric non-degeneratematrix).

REMARK 4.1 In chapter 2, the SDEs were driven by independent Brow-nian motions. The SDEs (4.1) can be framed in this setting by applying aCholesky decomposition: we write the correlation matrix as

ρ = LL†

or in components

ρij =n∑k=1

LikLjk

Then the correlated Brownians Wii=1,··· ,n can be decomposed over a basisof n independent Brownians Zii=1,··· ,n:

dWi =n∑k=1

LikdZk

In order to ensure the existence and uniqueness of SDE (4.1), we imposethe space-variable Lipschitz condition (2.17) and the growth condition (2.16).The no-arbitrage condition implies that the discounted traded assets are (lo-cal) martingales under this equivalent measure P. For P, the drifts bi areconsequently zero for the traded assets according to the theorem 2.4. Notethat the measure P is not unique as the market is not necessarily completeaccording to the theorem 2.8. Finally, the fair value of a European option att with payoff f(xT ) at maturity T is given by the mean value of the payoffconditional on the filtration Ft generated by the processes xis≤t

C(α, t, T ) = EP[f(xT )|Ft] (4.2)

Differential Geometry and Heat Kernel Expansion 77

No discount factors have been added as we assume a zero interest rate. Bydefinition of the conditional mean value, the fair value C depends on theconditional probability density p(T, x|t, α) by

C(α, t, T ) =∫ n∏

i=1

dxif(x)p(T, x|t, α) (4.3)

REMARK 4.2 It is not obvious that the SDE (4.1) admits a (smooth)conditional probability density. We will come back to this point when wediscuss the Hormander theorem in the last section.

Independently, from Ito’s formula, we have

f(xT ) = f(α) +∫ T

t

Df(xs)ds+∫ T

t

n∑i=1

∂f(x)∂xi

σi(t, x)dWi

(4.4)

with D a second-order differential operator

D = bi(t, x)∂

∂xi+

12σi(t, x)σj(t, x)ρij(t)

∂2

∂xi∂xj(4.5)

In (4.5), we have used the Einstein summation convention, meaning that twoidentical indices are summed. For example, bi(t, x) ∂

∂xi =∑ni=1 b

i(t, x) ∂∂xi .

We will adopt this Einstein convention throughout this book.

Taking the conditional mean-value EP[·|Ft] on both sides of (4.4), we obtain

C(α, t, T ) = f(α) + EP[∫ T

t

Df(xs)ds|Ft] (4.6)

where we have used that EP[∫ Tt

∂f(x)∂xi σ

i(t, x)dWi|Ft] = 0 by Ito isometry.Finally, differentiating (4.6) according to the variable T , we obtain

∂C(α, t, T )∂T

= EP[Df(xT )|Ft]

≡∫ n∏

i=1

dxiDf(x)p(T, x|t, α)

Integrating by parts and discarding the surface terms, we obtain

∂C(α, t, T )∂T

=∫ n∏

i=1

dxif(x)D†p(T, x|t, α) (4.7)


with

D† = − ∂

∂xibi(x, t) +

12ρij(t)

∂2

∂xi∂xjσi(x, t)σj(x, t)

Moreover, differentiating (4.3) according to T , we have

∂C(α, t, T )∂T

=∫ n∏

i=1

dxif(x)∂p(T, x|t, α)

∂T(4.8)

Since f(x) is an arbitrary function, we obtain by identifying (4.7) and (4.8)that the conditional probability density satisfies the forward Kolmogorov equa-tion (or Fokker-Planck equation) given by

∂p(T, x|t, α)∂T

= D†p(T, x|t, α)


limT→t

p(T, x|t, α) = δ(x− α)

The initial condition is to be understood in the weak sense, i.e.,

limT→t+

∫ n∏i=1

dxip(T, x|t, α)f(x) = f(α)

for any compactly supported function f on Rn.Coming back to the surface terms in (4.7), we have the boundary conditions(

bi − 12ρij∂i (σiσj)

)f +

12ρijσiσj∂jf = 0 , ∀x ∈ ∂M

4.1.2 Backward Kolmogorov’s equation

From the Feynman-Kac theorem, we have that the option fair value satisfiesthe PDE

−∂tC(α, t, T ) = bi(t, α)∂C(α, t, T )

∂αi+

12ρij(t)σi(t, α)σj(t, α)

∂2C(α, t, T )∂αi∂αj

From (4.3), we obtain that p(T, x|t, α) satisfies the backward Kolmogorovequation

−∂p∂t

= bi(t, α)∂p(T, x|t, α)

∂αi+

12ρij(t)σi(t, α)σj(t, α)

∂2p(T, x|t, α)∂αi∂αj

p(t = T, x|t, α) = δ(α− x)

We assume in the following that bi, σi and ρij are time-independent (foran extension to the time-dependent case, see section (4.6)). Let us define


τ = T − t. p(T, x|t, α) ≡ p(τ, x|α) only depends on the combination T − tand not on t or T separately (i.e., solutions are time-homogeneous). Thenp(τ, x|α) satisfies the backward Kolmogorov PDE

∂p(τ, x|α)∂τ

= Dp(τ, x|α) (4.9)


p(τ = 0, x|α) = δ(x− α)

and with D defined by (with ∂i = ∂∂αi

)

D = bi(α)∂i +12ρijσ

i(α)σj(α)∂2ij

The following section is dedicated to short-time asymptotic solutions for themulti-dimensional Kolmogorov equation (4.9).Note that the coefficients in the Kolmogorov equation do not behave nicelyunder a change of variables. Indeed if we do a change of variables fromx = xii=1,··· ,n to xi

′= f i

′(x) with f a C∞(Rn,Rn)-function, via Ito’s

formula one shows that the xi′

variables satisfy the SDE

dxi′

= bi′(x′)dt+ σi

′(x′)dWi′

with

bi′(x′) =

∂f i′

∂xibi(x) +

12ρijσ

i(x)σj(x)∂2f i

′(x)

∂xi∂xj(4.10)

σi′(x′) =

∂f i′(x)

∂xiσi(x) (4.11)

The volatility σi(x) transforms covariantly (4.11). This is not the case forthe drift (4.10). The non-covariant term is due to the Ito additional term.We will see in the following how to transform the non-covariant Kolmogorovequation into an equivalent covariant equation, the heat kernel equation. Forthis purpose, we will introduce a metric and an Abelian connection, dependingon the drift and the volatility terms, that transform covariantly under a changeof variables.In the next section, we introduce the reader to a few basic notions in differ-ential geometry, useful to formulate a heat kernel equation on a Riemannianmanifold and to state the short-time asymptotic solution to the Kolmogorovequation. Detailed proofs and complements can be found in [14] and [22].


4.2 Notions in differential geometry

The beginner should not be discouraged if he finds that he doesnot have the prerequisite for reading the prerequisites.— P. Halmos

4.2.1 Manifold

A real n-dimensional manifold M is a space which looks like Rn around eachpoint. More precisely, M is covered by open sets Ui (i.e., M is a topologicalspace) which are homeomorphic to Rn meaning that there is a continuousapplication φi (and its inverse) from Ui to Rn for each i. (Ui, φi) is called achart and φi a map. Via a map φi : Ui → Rn, we can endow the open set Uiwith a system of coordinates xkk=1,··· ,n by x = φ(p) with p ∈ Ui.

U_1 U_2

f_1 f_2

f_2of_1^-1

R^N R^N

MANIFOLD

FIGURE 4.1: Manifold


The figure (4.1) shows that a point p belonging to the intersection of the twocharts U1 and U2 corresponds to two systems of coordinates x1 (resp. x2)via the map φ1 (resp. φ2). We should give a rule indicating how to passfrom a system of coordinates to the other. We impose that the applicationsφi,j ≡ φi φ−1

j : Rn → Rn from Rn to Rn are C∞(Rn). We say that we have aC∞-manifold. In conclusion, we have a manifold structure if we define a chartand if the composition of two maps is an infinitely differentiable function. Letus see a simple example of a manifold: the 2-sphere.

Example 4.1 2-SphereThe two-sphere

S2 = (x, y, z) ∈ R3 | x2 + y2 + z2 = 1

can be covered with two patches: UN and US , defined respectively as S2 minusthe north pole and the south pole. The map φN (resp. φS) is obtained by astereographic projection on UN (resp. US). This projection consists in takingthe intersection of the equatorial plane with a line passing through the North(resp. South) pole and a point p on S2 (see Fig. 4.2). The resulting maps are

φN (x, y, z) =(

x

1− z,

y

1− z

)φS(x, y, z) =

(x

1 + z,

y

1 + z

)The change of coordinates defined on UN ∩ US is

φN φ−1S (x, y) =

(x

x2 + y2,

y

x2 + y2

)and is C∞. So S2 is a C∞-manifold.

4.2.2 Maps between manifolds

Let us define f : M → N a map from an m-dimensional manifold M with achart Ui, φii to a n-dimensional manifold N with a chart Vi, ψii. A pointp ∈M is mapped to a point f(p) ∈ N . Let us take Ui an open set containingp with the map φi and Vi an open set containing f(p) with the map ψi. Byusing the maps φi and ψi, f can be viewed as a map from Rm to Rn

F ≡ ψi f φ−1i : Rm → Rn

If we set φi(p) = x and ψi(f(p)) = y, we can write this map as

yj = f j(x)


*(x,y)

R^2*

p

N

S

SPHERE

FIGURE 4.2: 2-sphere

We say that f is C∞-differentiable if f = F jj=1,··· ,n is a C∞-differentiablefunction from Rm to Rn. Note that this definition is independent from theopen sets Ui and Vi containing respectively p and f(p). Indeed, let U ′i beanother open set including p with a map φ′i. Therefore, we have the otherrepresentative of f :

ψi f φ′−1i : Rm → Rn

As φi φ′−1i : Rm → Rm is a C∞-differentiable function by definition of a

smooth manifold M , therefore ψi f φ′−1i is if ψi f φ−1

i is.In the following, the space of C∞-differentiable functions from M to R isnoted C∞(M).

4.2.3 Tangent space

On the space C∞(M), we define a derivation:


DEFINITION 4.1 derivation A derivation D at a point x0 ∈ M is amap C∞(M)→ R such that

• D is linear: D(f + λg) = D(f) + λD(g) with λ ∈ R.

• D satisfies the Leibnitz rule: D(f.g) = D(f)g(x0) + f(x0)D(g).

The set of derivations forms a vector space and also a C∞(M)-module as theproduct of a function f ∈ C∞(M) with a derivation D is still a derivation.The vector space of all derivations at a point x0 is denoted by Tx0M andcalled the tangent space to the manifold M at a point x0.Let x1, · · ·xn be the local coordinates on the chart U containing the pointx0. The partial derivatives ( ∂

∂xi )i=1,··· ,n evaluated at x0 are independent andbelong to Tx0M . One can show that they form a basis for Tx0M :

THEOREM 4.1On a n-dimensional manifold M , Tx0M is a n-dimensional vector space gen-erated by

(∂∂xi

)i=1,··· ,n

Therefore an element X of Tx0M , called a vector field, can be uniquely de-composed as

X = Xi ∂

∂xi

Xi are the components of X in the basis ∂∂xi i=1,··· ,n.

Let us consider another chart U ′ with local coordinates x′1, · · ·x′n contain-ing the point x0. By using these new coordinates, the vector field can bedecomposed as X = Xi′ ∂

∂xi′. As we have

X = Xi′ ∂xi

∂xi′∂

∂xi= Xi ∂

∂xi

we find that the components of a vector field written in two different chartstransform covariantly as

Xi = Xi′ ∂xi

∂xi′(4.12)

4.2.4 Metric

A metric gij(x) written with the local coordinates x ≡ xii=1,··· ,n (corre-sponding to a particular chart U) is a symmetric non-degenerate and C∞-differentiable tensor. It allows us to measure the distance between infinitesi-mally nearby points p with coordinates xi and p+dp with coordinates xi+dxi

by

ds2 = gijdxidxj (4.13)


As p + dp is assumed to be infinitesimally close to the point p, it belongs tothe same chart and has a common system of coordinates with p.If a point p (and p + dp) belongs to two different charts U and U ′ then thedistance can be computed using two different systems of coordinates xi andxi′

= f(x) with f ∈ C∞(Rn,Rn). However, the result of the measure betweenp and p+ dp should be the same if the two points are considered to be on Uor U ′, meaning that

gij(x)dxidxj = gi′j′(x′)dxi′dxj

′(4.14)

As dxi′

= ∂xi′

∂xi dxi, we deduce that under a change of coordinates, the metric

is not invariant but on the contrary changes in a contravariant way by

gij(x) = gi′j′(x′)∂xi

′

∂xi∂xj

′

∂xj(4.15)

A manifold endowed with a metric is called a Riemannian manifold.From this metric, we can measure the length of a curve C. The basic ideais to divide this curve into infinitesimal pieces whose square of the lengthis given by (4.13). The rule (4.15) enables to glue the pieces belonging todifferent charts (i.e., different systems of coordinates). Let C : [0, 1] → Mbe a C1-differentiable or piecewise C1-differentiable curve (parameterized byxi(t)) joining the two points x(0) = x and x(1) = y. Its length l(C) is definedby

l(C) =∫ 1

0

√gij(x(t))

dxi(t)dt

dxj(t)dt

dt (4.16)

4.2.5 Cotangent space

We can associate to a vector space E its dual E∗ defined as the vector spaceof linear forms on E. Assuming that E is a finite n-dimensional vector spacegenerated by the basis eii=1,··· ,n, an element x ∈ E can be decomposeduniquely as

x = xiei

For a linear form l on E, we have by linearity that

l(x) = xil(ei)

Therefore, the map l is completely fixed if we know the value l(ei). E∗ is an-dimensional vector space with a (dual) basis ejj=1,··· ,n defined by

ej(ei) = δji


with δji the Kronecker symbol. l can therefore be decomposed over this basisas

l = liei

with l(ei) = li.Let M be a smooth manifold and x0 a point of M . The dual vector space toTx0M is called the cotangent vector space and noted T ∗x0

M . An element ofT ∗x0

M is a linear form ω which when applied to a vector field X in Tx0M givesa real number. Elements of T ∗x0

M are called one-form. A particular one-form(called exact one-form) is defined by

df(X) = X(f) , X ∈ Tx0M

with f ∈ C∞(M).If x1, · · · , xn are local coordinates of x0 and ( ∂

∂xi )i=1,··· ,n a basis of Tx0M ,then we write (dxi)i=1,··· ,n as the dual basis of T ∗x0

M as

dxj(

∂

∂xi

)=∂xj

∂xi= δji

By using this canonical basis, a one-form ω can be uniquely decomposed as

ω = ωidxi

For an exact one-form df , we have

df =∂f

∂xidxi

Note that if we use another system of coordinates x1′ , · · · , xn′ of x0, we have

ω = ωi′dxi′ = ωi′

∂xi′

∂xidxi

Therefore, the components of a one-form transform contravariantly under achange of coordinates as

ωi = ωi′∂xi

′

∂xi(4.17)

4.2.6 Tensors

In this section, we see how to map vectors to one-forms using the metric whichdefines an isomorphism between the tangent and cotangent vector spaces.Then, we will define general (r, p)-tensors where (1, 0) is a vector and (0, 1) isa one-form.


The data of a non-degenerate bilinear form g(·, ·) on a vector space E isequivalent to the data of an isomorphism between E and its dual E∗, g : E →E∗, according to the relation

g(X,Y ) = g(X)(Y )

By using a basis ei of E and its dual basis ei of E∗, the above-mentionedrelation can be written as

ei = gijej

with gij = g(ei, ej). Similarly, through multiplying the equation above by theinverse of the matrix [gij ], noted gij , we obtain

ei = gijej

Similar expressions can be obtained on the components of an element of E orE∗:

ω = ωiei = ωig

ijej ≡ ωiei

and therefore

ωi = gijωj

ωi = gijωj

Applying this isomorphism between the tangent space Tx0M and the cotan-gent space T ∗x0

M using the metric, we can map the components of a vectorfield Xi to the components of a one-form Xi by

Xi = gijXj

Xi = gijXj

The metric is therefore a machine to lower and raise the indices of vectors andone-forms.A tensor of type (r, p) is a multi-linear object which maps r elements of T ∗x0

Mand p elements of Tx0M to a real number. The set of tensors of type (r, p)forms a vector space noted T

(r,p)x0 M . An element T of T (r,p)

x0 M is written inthe basis ∂

∂xi i and dxii as

T = T i1···irj1···jp∂

∂xi1⊗ · · · ⊗ ∂

∂xirdxj1 ⊗ · · · ⊗ dxjp

By using another system of coordinates xi′, the components of T transform

as

Ti′1···i

′r

j′1···j′p= T i1···irj1···jp

∂xi′1

∂xi1· · · ∂x

i′r

∂xir

∂xj1

∂xj′1· · · ∂x

jr

∂xj′r

(4.18)


A special (0, p)-tensor is a p-differential form which is a totally antisymmetric(0, p)-tensor. Let us define the wedge product ∧ of p one-forms by the totallyantisymmetric tensor product

dxi1 ∧ · · · dxip =∑P∈Sp

sgn(P )dxiP (1) ⊗ · · · ⊗ dxiP (p) (4.19)

where P is an element of Sp, the symmetric group of order p and sgn(P ) = +1(resp. sgn(P ) = −1) for even (resp. odd) permutations.

Example 4.2

dxi ∧ dxj = dxi ⊗ dxj − dxj ⊗ dxi

The elements (4.19) form a basis of the vector space of r-forms, Ωr(M), andan element ω ∈ Ωr(M) is decomposed as

ω =1n!ωi1···indx

i1 ∧ · · · ∧ dxin

We can then define the exterior derivative d : Ωr(M) → Ωr+1(M) acting onr-forms as

dω =1n!∂kωi1···indx

k ∧ dxi1 ∧ · · · ∧ dxin

One easily checks that d is a linear operator and d2 = 0.

Example 4.3

The exterior derivative d on a 1-form A = Aidxi gives

dA = ∂jAidxj ∧ dxi

= (∂jAi − ∂iAj) dxj ⊗ dxi

Tensors, metrics, forms, vector fields have been locally defined on a neighbor-hood of x0 as elements of T (r,p)

x0 M and we have explained how these objectsare glued together (4.12), (4.17), (4.18) on the intersection of two patches.We can unify these definitions with the notion of vector bundles. The tensorfields will then be seen as sections of particular vector bundles.


4.2.7 Vector bundles

The main idea of a vector bundle (more generally of differential geometry)is to define a vector space over each chart of a manifold M . To a point x0

belonging to two different patches, we can associate two different vector spacesand we need to specify how to glue them.A real vector bundle E of rank m is defined as follows: one starts with anopen covering of M , Uαα∈A, and for each (α, β) ∈ A, a smooth transitionfunction

gαβ : Uα ∩ Uβ → GL(m,R)

where GL(m,R) is the space of invertible real matrix of dimension m. Weimpose that the function gαβ satisfies

gαα = 1m

and the co-cycle condition on the triple intersection Uα ∩ Uβ ∩ Uγ

gαβgβγgγα = 1m on Uα ∩ Uβ ∩ Uγ (4.20)

with 1m being the identity (linear) map on Rm.Let us denote E the set of all triples (α, p, v) ∈ A×M ×Rm such that p ∈ Uα.Let us define an equivalence relation ∼ on E by

(α, p, v) ∼ (β, q, w) ⇔ p = q ∈ Uα ∩ Uβv = gαβw (4.21)

Let us denote the equivalence classes of (α, p, v) by [α, p, v] and the set ofequivalence classes by E and define a projection map

π : E →M , π([α, p, v]) = p

Let us define Uα = π−1(Uα) and a bijection by

ψα = Uα → Uα × Rm , ψα([α, p, v]) = (p, v)

There is a unique manifold structure on E which makes ψα into a diffeomor-phism.A real vector bundle of rank m is a pair (E, π) constructed as above anda collection of transition functions gαβ which satisfy the co-cycle condition(4.20). The fiber of E over p ∈ M is the m-dimensional vector space Ep =π−1(p). When m = 1, the real vector bundle is called a line bundle (see Fig.4.3).A section σ of E is defined by its local representatives σ on each Uα:

σ|Uα ≡ σα : Uα → Rm


s_a

s_b

U_a

U_b

R

R

g_ab

s_a

s_b

Line Bundle

FIGURE 4.3: Line bundle

and they are related to each other by the formula

σα = gαβσβ (4.22)

on Uα ∩ Uβ . The smooth sections on E are denoted Γ(E).In the following, we give classical examples of real vector bundles.

Example 4.4 Trivial vector bundleThe simplest real bundle I of rank m which can be constructed on a manifoldM is the trivial vector bundle given by the transition function

gαβ = 1m

The co-cycle condition (4.20) is trivially satisfied. One can show that I isisomorphic to the product M × Rm.

Example 4.5 (Co)-Tangent vector bundleThe tangent space TM to a manifold M is a vector bundle of rank n. The


transition functions are given by

gα′α : Uα′ ∩ Uα → GL(n,R)

gα′α(x) =∂xα

′

∂xα

A vector field is then a section of TM .The cotangent vector bundle (of rank n) T ∗M is the dual vector bundle ofTM with a transition function given by

g∗α′α : Uα′ ∩ Uα → GL(n,R)

g∗α′α ≡ g−1†α′α =

∂xα

∂xα′

A one-form is a section of T ∗M .The tensor vector bundle of type (r, s) , ⊗rp=1TM ⊗sq=1 T

∗M , is then definedas the n-rank real vector bundle with transitions ⊗rp=1gα′α ⊗sq=1 g

∗α′α. A

tensor field is a section of the tensor vector bundle and transforms as (4.18)conformingly to (4.22).

4.2.8 Connection on a vector bundle

DEFINITION 4.2 Connection 1 A connection dA on a vector bundleE is a map

dA : Γ(E)→ Γ(T ∗M ⊗ E)

which satisfies the following axiom:

dA(fσ + τ) = df ⊗ σ + fdAσ + dAτ (4.23)

with f ∈ C∞(M), σ ∈ Γ(E) and τ ∈ Γ(E).Firstly, we will characterize a connection on a trivial bundle E = M × Rm.Then using that a vector bundle is locally trivial, we will generalize to ageneral vector bundle.Let us consider a trivial bundle E = M × Rm defined by the transition func-tions gαβ = Uα ∩ Uβ → GL(m,R) , gαβ = 1m. A section is then representedby a global vector-valued map σ : M → Rm which can be decomposed on theconstant sections ei:

e1 =

10...0

, e2 =

01...0

· · · , em =

00...1


σ(x) = σi(x)ei

By definition (4.23), we have

dAσ = dσi ⊗ ei + σidAei

As dAei should be an element of Γ(T ∗M ⊗ E), there exists a rank m matrixAji ∈ T ∗M of 1-form, Aji = Ajkidx

k, such that

dAei = Aji ⊗ ej

Therefore

dAσ = (dσi + σjAij)⊗ ei

and the components of dAσ ≡ (dAσ)iei are

(dAσ)i = dσi + σjAij

= (∂kσi +Aikjσj)dxk

We note the components of dA, dAσ ≡ ∇kσidxk ⊗ ei, and

∇kσi = ∂kσi +Aikjσ

j (4.24)

Similarly, on a non-trivial bundle, a connection can be characterized as follows:A section σ ∈ Γ(E) possesses a local representative σα which is a vector-valuedmap on Uα. So, proceeding as above, we can write

(dAσ)α = dσα +Aασα

with Aα a rank m matrix of 1-form on Uα. Now, we need to specify howto glue together these local connections (dAσ)α and (dAσ)β defined on twodifferent patches Uα and Uβ . For a section defined on Uα ∩ Uβ , we have

(dAσ)α = dσα +Aασα

(dAσ)β = dσβ +Aβσβ

By using the transition function gαβ , we impose that

σβ = gβασα

(dAσ)β = gβα(dAσ)α

It gives

Aβ = g−1αβdgαβ + g−1

αβAαgαβ (4.25)


DEFINITION 4.3 Connection 2 A connection on a real vector bundleE of rank m defined by a covering Uα : α ∈ A and transition functions gαβ :α, β ∈ A is a collection of first-order differential operators d + Aα : α ∈ Awhere d is the exterior derivative on Rm-valued functions and Aα is a rank mmatrix of one-forms on Uα which transform according to (4.25). When E isa line bundle, i.e., m = 1, Aα is called an Abelian connection.

Applying to the tangent vector bundle E, a connection dA is called a covari-ant derivative and Aijk is noted usually Γijk. The covariant derivative of thecomponents of a vector field X = Xi∂i is according to (4.24)

∇iXj = ∂iXj + ΓjikX

k (4.26)

The action of a covariant derivative on a vector field can be extended to any(r, s)-tensor fields by the rules:

1. If T is a tensor field of type (r, s), then ∇T is a tensor field of type(r, s+ 1).

2. ∇ is linear and commutes with contractions.

3. For any tensor fields T1 and T2, we have the Leibnitz rules

∇(T1 ⊗ T2) = ∇(T1)⊗ T2 + T1 ⊗∇(T2)

4. ∇f = df for any function f ∈ C∞(M).

If we contract a one-form ω with a vector field X, < ω,X > is a function onM : ωjXj . By using the rules 3 and 4, we have

∇i(ωjXj) = ∂i(ωjXj)= ∇i(ωj)Xj + ωj∇i(Xj)

Taking X = ∂j , we obtain

∇iωj = ∂iωj − Γkijωk

Note the sign − in front of the connection instead of the sign + in (4.26). Itis easy to generalize these results to a (q, p)-tensor field T

∇νTj1···jpi1···iq = ∂νT

j1···jpi1···iq + Γj1νkT

k···jpi1···iq + · · ·ΓjpνkT

j1···ki1···iq

−Γkνi1Tj1···jpk···iq − · · · − ΓkνiqT

j1···jpi1···k (4.27)

Example 4.6 Levi-Cevita connectionEndowing M with a Riemannian metric g, the Levi-Cevita connection Γ is

the unique symmetric connection on TM which preserves the metric

∇kgij = 0


This equation is equivalent to (using (4.27))

∂kgij − Γpkigpj − Γpkjgpi = 0 (4.28)

Cyclic permutations of (k, i, j) yield

∂jgki − Γpjkgpi − Γpikgpj = 0 (4.29)

∂igjk − Γpijgpk − Γpjigpk = 0 (4.30)

The combination of −(4.28)+(4.29)+(4.30) yields

−∂kgij + ∂jgki + ∂igjk

+(Γpki − Γpik)gpj + (Γpkj − Γpjk)gpi − (Γpij + Γpji)gpk = 0

If we impose that the connection is symmetric Γpij = Γpji (the antisymmetricpart is called the torsion), we obtain

−∂kgij + ∂jgki + ∂igjk − 2Γpijgpk = 0

Solving for Γpij , the connection is uniquely given by

Γpij =12gpk (−∂kgij + ∂jgki + ∂igjk) (4.31)

Γpij are called the Christoffel symbol.

4.2.9 Parallel gauge transport

Pullback bundle

If (E, π) is a vector bundle over M defined by a covering Uα : α ∈ A andtransition functions gαβ : α, β ∈ A and F : N → M a smooth map betweentwo manifolds N and M , the pullback bundle (F ∗E, π∗) is the vector bundleover N defined by the open covering F−1(Uα) : α ∈ A and the transitionfunctions gαβ = gαβ F : α, β ∈ A. It is easy to see that gαβ satisfies theco-cycle condition (4.20).If we define a connection dA on the vector bundle E, we can induce (we saypullback) a connection dπ∗A on the pullback bundle π∗E in the following way:Locally on Uα, dA is represented by

dA|Uα = d+Aα

with Aα = Aαkdxk. On the open covering F−1(Uα), the pullback connection

dπ∗A is defined as

dπ∗A|F−1(Uα) = d+ π∗Aα

with π∗Aα = Aαk∂Fk

∂xα dxα.


Applying this construction to a curve C = [0, 1]→M and a connection dA ona vector bundle (E, π) on M, we induce a pullback connection on C, dC∗A.Now, we consider the sections σ on C∗E which are preserved by the connectiondC∗A meaning that

dC∗Aσ = 0 (4.32)

Locally on C−1(Uα), this condition reads

dσα + (C∗Aα)σα = 0

Written in local coordinates, we have the first-order differential equation

dσα(t)dt

+ (C∗Adt)σα(t) = 0 (4.33)

It follows from the theory of ordinary differential equations (ODEs) that givenan element σ0 ∈ (C∗E)0, there is a unique solution of (4.33) which satisfiesthe initial condition σ(0) = σ0. We can define the isomorphism

P : (C∗E)0 → (C∗E)1

by setting P(σ0) = σ(1) where σ is the unique solution to (4.33) which satisfiesσ(0) = σ0. P is called the parallel gauge transport. When E is a line bundle,C∗Adt is no more a matrix but a real number. In this case, P is given by

P = e−R 10 C∗Adt(u)du

4.2.10 Geodesics

We apply the previous construction to the parallel transport of a vector, tan-gent to a curve C, along this curve. The ODE (4.33) will be called a geodesicequation and the curve C, a geodesic curve.Given a curve on a Riemannian manifold M , we may define the parallel(gauge) transport of a vector field tangent to a curve C using the Levi-Cevitaconnection:Let V be a vector field tangent to a curve C = [0, 1] → M . For simplicity,we assume that the curve is covered by a single chart (U , φ) with coordinatesxii=1,··· ,n. Therefore, the vector V tangent to the curve C can be writtenas

V |C =dxi(C(t))

dt

∂

∂xi

The (tangent) vector V is said to be parallel transport along its curve C(t) ifV satisfies the condition (4.32) (with ∇ the Levi-Cevita connection)

∇C∗V V = 0


This condition is written in terms of components as

d2xi

dt2+ Γijk

dxj

dt

dxk

dt= 0 (4.34)

Indeed, we obtain this ODE from (4.33) by observing that σ(t) ≡ dxi(t)dt i and

C∗Γdt = Γijk dxj

dt ik. This ODE is called the geodesic equation. We recall that

the Christoffel coefficients Γijk depend on the metric and its first derivativesby (see example 4.6)

Γpij =12gpk (−∂kgij + ∂jgki + ∂igjk) (4.35)

The curve C satisfying this equation is called a geodesic curve.Geodesics are, in some sense, the straightest possible curves in a Riemannianmanifold. To understand this point, we try to minimize the length (4.16) ofa curve C joining two points x(0) = x and x(1) = y:

minC

∫ 1

0

√gij(x(t))

dxi(t)dt

dxj(t)dt

dt (4.36)

By using the Euler-Lagrange equation giving the critical point of this func-tional, we obtain the geodesics equation (4.34).As the geodesics are the main mathematical objects which appear in theheat kernel expansion technique, we give below examples of metrics on two-dimensional manifolds and solve explicitly the geodesic equations. These met-rics will re-emerge naturally when we will discuss implied volatility asymp-totics of stochastic volatility models.

Example 4.7 Hyperbolic surfaceLet us consider the metric on the hyperbolic surface which is the complexupper half-plane H2 = x ∈ R, y > 0

ds2 =dx2 + dy2

y2(4.37)

We will give more information on this geometry when we will look at theSABR model in chapter 6. For the moment, we focus on the derivation of thegeodesics curves and the geodesic distance which are the two main ingredi-ents in the heat kernel expansion. There are only three non-zero Christoffelsymbols2(4.35) (x1 = x, x2 = y)

Γ112 = Γ2

22 = −Γ211 = −1

y

2Note that this computation can be easily done using Mathematica with the package tensorhttp://home.earthlink.net/ djmp/TensorialPage.html.


So the geodesic equations (4.34) reduce to

x− 2yxy = 0 (4.38)

y +1yx2 − 1

yy2 = 0 (4.39)

The dot means derivatives according to the parameter s which we have chosento parameterize the geodesic curve. First note that the geodesic for x (4.38)can be simplified to read

d

ds

(ln(x)− ln(y2)

)= 0

or

x = Cy2 (4.40)

for some unspecified constant C.If we choose C = 0, we have x constant and y is a vertical line from (4.39):

y(s) = y0ecs

x(s) = x0 (4.41)

with c an integration constant. We assume below that C 6= 0.The geodesic equation (4.39) for y is equivalent to

d

ds

(x2 + y2

y2

)= 0

that we integrate to

x2 + y2 = y2

where we have chosen an undetermined constant to 1. In fact, this equationis the definition of the parameter s (4.37). When we substitute in the knownvalue (4.40) of dx

ds , we find that3

dy

ds= y√

1− C2y2

Next, we use the chain rule to find

dxdsdyds

=dx

dy=

Cy√1− C2y2

3We have chosen the sign + in front of the square root.


and

(x− c) =∫ y

·

Cy′dy′√1− C2y2

= −√

1C2− y2

where c is another undetermined constant. Upon squaring both sides, we find

y2 + (x− c)2 =1C2

(4.42)

indicating that the geodesic for C 6= 0 is a semi-circle with radius 1C centered

at (c, 0).In conclusion, geodesics are semi-circles centered at (c, 0) on the x-axis andof radius 1

C (4.42) and vertical lines (4.41).As the geodesic curve has been parameterized by the affine coordinate s rep-resenting the length of the geodesic curve, the square of the geodesic distanced between two points (x1, y1) and (x2, y2) joining by a geodesic curve is

d2 ≡ (s2 − s1)2

=∫ y2

y1

dy′

y′√

1− C2y2

= lny2

y1− ln

(1 +

√1− C2y2

2

1 +√

1− C2y21

)

C is fixed by the constraint

x2 − x1 = −√

1C2− y2

2 +

√1C2− y2

1

which can be solved explicitly. Then after some simple algebraic transforma-tions, we obtain the final expression for the geodesic distance

d = cosh−1

(1 +

(x2 − x1)2 + (y2 − y1)2

2y2y1

)(4.43)

Example 4.8 Metric on a Riemann surface with one Killing vectorLet us consider a general two-dimensional Riemannian manifold (M, g). In alocal coordinate (u, v), the metric takes the form

ds2 = guudu2 + 2guvdudv + gvvdv

2 (4.44)


Note g the determinant of the metric, g = guugvv − g2uv and rewrite (4.44) as

ds2 = (√guudu+

guv + ı√g

√guu

dv)(√guudu+

guv − ı√g

√guu

dv)

According to the theory of differential equations, there exists an integratingfactor λ(u, v) = λ1(u, v) + ıλ2(u, v) such that

λ(√guudu+

guv + ı√g

√guu

dv) = dx+ ıdy

λ∗(√guudu+

guv − ı√g

√guu

dv) = dx− ıdy

Then in the new coordinates (x, y), we have by setting |λ|−2 = e2φ(x,y)

ds2 = e2φ(x,y)(dx2 + dy2) (4.45)

The coordinates (x, y) are called the isothermal coordinates.We consider in the following the most general metric on a Riemann surfacewith one Killing vector meaning that φ(x, y) only depends on x: φ(x, y) ≡12 ln(F (x)). So, we have

ds2 = F (y)(dx2 + dy2) (4.46)

The Christoffel symbols are

Γ112 = Γ2

22 = −Γ211 =

F ′(y)2F (y)

and the geodesic equation reads

x+F ′(y)F (y)

xy = 0 (4.47)

y − F ′(y)2F (y)

x2 +F ′(y)2F (y)

y2 = 0 (4.48)

The dot · (resp. prime) means derivative according to the parameter s (resp.y). The first equation (4.47) is

x = CF (y)−1 (4.49)

for some constant C. The geodesic equation (4.48) for y is equivalent to

d

ds

(F (y)(x2 + y2)

)= 0

that we integrate to

x2 + y2 =1

F (y)(4.50)


where we have chosen the undetermined constant to 1. In fact, this equationis the definition of the parameter s (4.46). From (4.49), the equation (4.50)simplifies to

dy

ds=

√F (y)− C2

F (y)

and from (4.49), we have

dx

dy=

C√F (y)− C2

As usual, the geodesic curve has been parameterized by s and therefore thegeodesic distance between the points (x1, y1) (reached at s1) and (x2, y2)(reached at s2) is

d ≡ |s2 − s1|

= |∫ y2

y1

F (y′)dy′√F (y′)− C2

|

with the constant C = C(x1, y1, x2, y2) determined by the equation

x2 − x1 =∫ y2

y1

C√F (y′)− C2

dy′

Cut-locus

Every geodesic can be extended in both directions indefinitely and every pairsof points can be connected by a distance-minimizing geodesic.4 For each unitvector5 e ∈ ToM , there is a unique geodesic Ce : [0,∞) → M such thatCe(0) = e. The exponential map exp : ToM →M is

exp(te) ≡ Ce(t) (4.51)

For small t, the geodesic Ce([0, t]) is the unique distance-minimizing geodesicbetween its endpoints. Let t(e) be the largest t such that the geodesicCe([0, t(e)]) is the unique distance-minimizing from Ce(0) to Ce(t(e)). Letus define

C0 = t(e)e : e ∈ ToM , |e| = 1

4Assuming that the Riemannian manifold is complete. This is the Hopf-Rinow theorem.Note that this result is not applicable for Lorentzian manifolds and this is why we mustlive with black holes!5||e||2 = gij(o)e

iej = 1.


Then the cut-locus is the set

Co = exp C0

The set within the cut-locus is the star-shaped domain

Eo = t(e)e : e ∈ ToM , 0 ≤ t < t(e) , |e| = 1

On M the set within the cut-locus is Eo = exp Eo.The cut-locus can be characterized with the following proposition

PROPOSITION 4.1

A point x ∈ Co if and only

• there exists a non-minimizing geodesic from o to x.

• or/and there exists two minimizing geodesics from o to x.

The injectivity radius at a point o is defined as the largest strictly positivenumber r such that every geodesic γ starting from o and of length l(γ) ≤ r isminimizing

inj(o) = mine∈ToM , |e|=1

t(e)

We have the following basic results [8]

THEOREM 4.2

1. The map exp : E0 → E0 is a diffeomorphism.

2. The cut-locus Co is a closed subset of measure zero.

3. If x ∈ Cy, then y ∈ Cx.

4. E0 and C0 are disjoint sets and M = E0 ∪ C0.

Example 4.9 Cut-locus of a 2-sphereOn S2, the cut-locus Co of a point o is its antipodal point: Co = −o. Forexample, the cut-locus of the North pole N is the South pole S. Indeed, thereexists an infinity of minimizing geodesics between N and S.


4.2.11 Curvature of a connection

The curvature of a connection dA on a rank m-vector bundle E is defined asthe map

d2A ≡ dA dA : Γ(E)→ Ω2(M)⊗ Γ(E)

with Ω2(M) the space of 2-forms on M . It is easy to verify that d2A is a well

defined tensor as

d2A(fσ + τ) = dA(df ⊗ σ + fdAσ + dAτ)

= d2fσ − dfdAσ + dfdAσ + fd2Aσ + d2

Aτ

= fd2Aσ + d2

Aτ

where we have used that d2 = 0. Locally on Uα, d2A is represented by a rank

m-matrix 2-form

Ωα = Ωαijdxi ∧ dxj

and

d2Aσα = Ωασα

As locally dA = d+Aα, we have

d2A = (d+Aα)(d+Aα)

= d2 + (dAα) +Aαd−Aαd+Aα ∧Aα

and finally

Ωα = dAα +Aα ∧Aα (4.52)

In local coordinates, the matrix-valued 2-form Ωα is

[Ωα] = Ωijkldxk ∧ dxl

and the connection is Aα = Aikjdxk. From (4.52), we obtain

Ωijkl = −∂lAikj + ∂kAilj +AikrA

rlj −AilrArkj (4.53)

As d2A is a tensor, we have

d2Aσβ = d2

A(gβασα)= gβαd

2Aσα

= gβαΩασα = Ωβσβ

Therefore on Uα ∩ Uβ , the curvatures Ωα and Ωβ glue together as

Ωβ = gβαΩαg−1βα


Example 4.10 Riemann tensor, Ricci tensor and scalar curvatureLet us take the Levi-Cevita connection on TM with Γikj the Christoffel sym-bols. The components (4.53) of the curvature Ωijkl are preferably noted Rijkland called the Riemann tensor

Rijkl = −∂lΓikj + ∂kΓilj + ΓikrΓrlj − ΓilrΓ

rkj

From this tensor, two other important tensors can be defined by contraction.The first one is the Ricci tensor obtained by contracting the upper indicesand the lower second indices in the Riemann tensor

Rjl ≡ Rijil

Then the scalar curvature is obtained by taking the trace of the Ricci tensor

R ≡ gijRij

4.2.12 Integration on a Riemannian manifold

On a n-dimensional Riemannian M , we can build a partition of unity:

DEFINITION 4.4 Partition of unity Given an atlas (Uα, φα) on an-dimensional manifold M , a set of C∞ functions εα is called a partition ofunity if

1. 0 ≤ εα ≤ 1.

2. the support of εα, i.e., the closure of the set p ∈ M : εα(p) 6= 0, iscontained in the corresponding Uα.

3.∑α εα = 1 for all p ∈M .

By using this partition of unity, we define the integral of a measurable functionf on M as ∫

M

fdµ(g) ≡∑α

∫φα(Uα)

(εα√gf)(φ−1

α (x))dnx

g is the determinant of the metric [gij ] and dnx ≡∏ni=1 dx

i is the Lebesguemeasure on Rn. In order to have a consistent definition, the integral aboveshould be independent of the partition of unity and of the atlas. The in-dependence with respect to the atlas comes from the fact that the measure√g(x)dnx is invariant under an arbitrary change of coordinates:


PROOF Indeed the metric changes as gij = gi′j′∂xi′

∂xi∂xj′

∂xj and therefore

g = det(gij) changes as√g = det

(∂xi′

∂xi

)√g′. Moreover, the element

∏ni=1 dx

i

changes as∏ni=1 dx

i = [det(∂xi′

∂xi

)]−1∏ni′=1 dx

i′ and we deduce the result.

DEFINITION 4.5 Density bundle We call |Λ|x the density bundleon M whose sections can be locally written as f(x)

√g with f(x) a smooth

function on M .

Now, everything is in place to define the heat kernel on a Riemannian manifoldand formulate the Minakshisundaram-Pleijel-De Witt-Gilkey theorem givingan expansion of the heat kernel in the short-time limit and representing anasymptotic solution for the conditional probability density to the backwardKolmogorov equation.

4.3 Heat kernel on a Riemannian manifold

In this section, the PDE (4.9) will be interpreted as a heat kernel on a generalsmooth n-dimensional Riemannian manifold M endowed with a metric gij(here we have that i, j = 1 · · ·n) and an Abelian connection A.

REMARK 4.3 Note that the coordinates αii (resp. xii) willbe noted xii (resp. yii) below in order to be consistent with ourprevious (geometric) notation.

The inverse of the metric gij is defined by

gij(x) =12ρijσi(x)σj(x)

Note that in this relation although two indices are repeated, there is no im-plicit summation over i and j as the result is a symmetric tensor dependentprecisely on these two indices. The metric (ρij inverse of ρij , i.e., ρijρjk = δik)is

gij(x) = 2ρij

σi(x)σj(x)(4.54)

The differential operator

D = bi(x)∂i + gij(x)∂ij

which appears in (4.9) is a second-order elliptic operator of Laplace type.


REMARK 4.4 Elliptic operator We recall to the reader the definitionof an elliptic operator. We need to introduce the symbol of D:The symbol of D is given by

σ(x, k) = bi(x)ki + gij(x)kikj

It corresponds to the Fourier transform of D when x is fixed. Its leadingsymbol is defined as

σ2(x, k) = gij(x)kikj

DEFINITION 4.6 Second-order elliptic operator A second-orderoperator is elliptic if for any open set Ω ⊂M , its corresponding leading sym-bol σ2(x, k) is always non-zero for non-zero k (i.e., non-degenerate quadraticform).

From the definition, we see that D is elliptic if and only if gij is a metric.

We can then show that there is a unique connection ∇ on L, a line bundleover M , and a unique smooth section Q(x) of End(L) ' L⊗ L∗ such that

D = gij∇i∇j +Q

= g−12 (∂i +Ai) g

12 gij (∂j +Aj) +Q (4.55)

Here g = det[gij ] and Ai are the components of an Abelian connection. Then,the backward Kolmogorov equation (4.9) can be written in the covariant way

∂p(τ, x|y)∂τ

= Dp(τ, x|y) (4.56)

If we take Ai = 0 , Q = 0 then D becomes the Laplace-Beltrami operator (orLaplacian)

∆ = g−12 ∂i

(g

12 gij∂j

)(4.57)

For this configuration, (4.56) will be called the Laplacian heat kernel equation.As expected, ∆ is invariant under a change of coordinates:

PROOF From xii=1,··· ,n to xi′i′=1,··· ,n, the metric gij transforms as

gi′j′(x′) = ∂i′xi∂j′x

jgij(x)

Then by plugging the expression for gi′j′(x′) into (4.57), we obtain

∆(g′) = ∆(g)


We may express the connection Ai and Q as a function of the drift bi and themetric gij by identifying in (4.55) the terms ∂i and ∂ij with those in (4.9).We find

Ai =12

(bi − g− 1

2 ∂j

(g

12 gij

))(4.58)

Q = gij (AiAj − bjAi − ∂jAi) (4.59)

Note that the Latin indices i,j · · · can be lowered or raised using the metricgij or its inverse gij as explained in 4.2.6. For example Ai = gijAj andbi = gijbj . The components Ai define locally a one-form A = Aidxi. Wededuce that under a change of coordinates xi

′(xj), Ai undergoes the vector

transformation Ai′∂ixi′

= Ai. Note that the components bi don’t transformas a vector (see 4.10). This results from the fact that the SDE (4.1) has beenderived using the Ito calculus and not the Stratonovich one (see section 4.4).The equation (4.58) can be re-written using the Christoffel symbol (4.35)

Ai =12(bi − gpqΓipq

)To summarize, a heat kernel equation on a Riemannian manifold M is con-structed from the following three pieces of geometric data:

1. A metric g on M , which determines the second-order piece.

2. A connection A on a line bundle L , which determines the first-orderpiece.

3. A section Q of the bundle End(L) ' L ⊗ L∗, which determines thezeroth-order piece.

The fundamental solution of the heat kernel equation (4.56), called a heatkernel, is defined as follows

DEFINITION 4.7 A heat kernel for a heat kernel equation (4.56) is acontinuous section p(τ, x|y) of the bundle6 (LL∗⊗|Λ|y) over R+×M ×M ,satisfying the following properties:

1. p(τ, x|y) is C1 with respect to τ , that is, ∂τp(τ, x|y) is continuous in(τ, x, y)

6Let E1 and E2 be two vector bundles on M and pr1 (resp. pr2) be the projections fromM ×M onto the first (resp. second) factor. We denote the external product E1 E2 the(pullback) vector bundle pr∗1E1 ⊗ pr∗2E2 over M ×M .


2. p(τ, x|y) is C2 with respect to x, that is the partial derivatives ∂2p(τ,x|y)∂xi∂xj

are continuous in (τ, x, y) for any coordinate system x.

3. p(τ, x|y) satisfies the boundary condition at τ = 0:

limτ→0

p(τ, x|y) = δ(x− y)

This boundary condition means that for any smooth section s of L, then

limτ→0

∫M

p(τ, x|y)s(y)dny = s(x)

In order to be considered as a probability density for a backward Kolmogorovequation, the heat kernel p(τ, x|y) should be a positive section and normalizedby ∫

M

p(τ, x|y)dny = 1 (4.60)

Note that it may happen that∫M

p(τ, x|y)dny < 1

Probabilistically it means that the Ito diffusion process xt following (4.1)associated to the backward Kolmogorov equation may not run for all timeand may go off the manifold in a finite amount of time. In the next chaptersabout local and stochastic volatility models, we discuss the conditions on themetric and the Abelian connection under which there is no explosion of theprocess xt in a finite amount of time. If the heat kernel satisfies (4.60), thepair (M,D) is called stochastically complete.Let us see now the simplest example of a heat kernel equation, that is theheat kernel on Rn for which there is an analytical solution.

Example 4.11 Heat kernel equation on RnOn Rn with the flat metric gij = δij , with a zero-Abelian connection and azero section Q = 0, the heat kernel equation (4.56) reduces to

∂p(τ, x|y)∂τ

= ∂2i p(τ, x|y)

It is easy to see that the heat kernel is

p(τ, x|y) =1

(4πτ)n2e−|x−y|2

4τ (4.61)

where |x − y|2 =∑ni=1(xi − yi)2 is the Euclidean distance which coincides

with the geodesic distance. We can verify the identity (4.60) and (Rn,∆) isstochastically complete.


4.4 Abelian connection and Stratonovich’s calculus

In our presentation, one of the motivations for introducing an Abelian con-nection was that A is a one-form and therefore transforms covariantly undera change of variables. Another possibility to achieve this goal is to use theStratonovich calculus.Let M , N be two continuous semi-martingales. The Stratonovich product is defined as follows

DEFINITION 4.8 Stratonovich product∫ T

0

Ms dNs =∫ T

0

MsdNs +12< M,N >t

with < M,N >t the quadratic variation (see definition 2.13). This writesformally

Mt dNt = MtdNt +12d < M,N >t (4.62)

By using the definition (4.62), an Ito diffusion SDE

dxi = bi(x)dt+ σi(x)dWi

dWidWj = ρij(t)dt

can be transformed into the Stratonovich diffusion SDE

dxi = Ais(x)dt+ σi(x) dWi (4.63)

where the drift Ais(x) in (4.63) is

Ais(x) = bi(x)− 12ρik(t)σk(x)∂kσi(x)

From the transformation of the drift (4.10) and the volatility (4.11) under achange of variables xi

′= f i

′(x), we obtain that the drift Asi changes covari-

antly as a vector field, i.e.,

Ai′

s = ∂ifi′Ais

In the Stratonovich calculus, the complicated drift in Ito’s lemma cancels andwe have

dC(t, x) = ∂tC(t, x)dt+ ∂xiC(t, x)dxi

where dxi is given by (4.63).


PROOF From Ito’s lemma, we have

dC(t, x) = ∂tC(t, x)dt+12ρijσ

iσj∂ijC(t, x) + σi∂iC(t, x)dWi

As

σi∂iC(t, x)dWi = σi∂iC(t, x) dWi −12ρkiσ

k∂k(σi∂iC(t, x)

)we get our final result.

Example 4.12 GBM processAs an example, the GBM process dft

ft= σdWt becomes the Stratonovich

diffusion

dftft

= −12σ2dt+ σ dWt

Note that there is a difference between Ais(x) and the Abelian connectionAi(x) which is given by

Ai =12

(bi +

12ρijσ

iσj∂jσ

p

σp− 1

2σi∂iσ

i +12(σi)2 ∂iσp

σp

)The main motivation for using Ai instead of Ais is that we can apply gaugetransformations on the heat kernel equation.

4.5 Gauge transformation

In order to reduce the complexity of the heat kernel equation (4.56), twotransformations can be used. The first one is the group of diffeomorphismDiff(M) which acts on the metric and the Abelian connection by

(f∗g)ij = gpk∂ifp(x)∂jfk(x)

(f∗A)i = Ap∂ifp(x) , f ∈ Diff(M)

The second one is a gauge transformation. Before introducing a geometricaldefinition, let us give a simple definition without any formalism.If we define a new section p′ ∈ (L L∗ ⊗ |Λ|y) by

p′(τ, x|y) = eχ(τ,x)−χ(0,y)p(τ, x|y) (4.64)


with χ(τ, x) a smooth function on R+ × M , then it is easy to check thatp′(τ, x|y) satisfies the same heat kernel equation as p(τ, x|y) (4.56) but witha new Abelian connection A′ = A′idxi and a new section Q′

A′i ≡ Ai − ∂iχ (4.65)Q′ ≡ Q+ ∂τχ

The transformation (4.64) is called a gauge transformation. The reader shouldbe warned that the transformation (4.65) only applies to the connection Aiwith lower indices. Moreover, the constant phase eχ(0,y) has been added in(4.64) so that p(τ, x|y) and p′(τ, x|y) satisfy the same boundary condition atτ = 0.This transformation is particularly useful in the sense that if the one-form Ais exact (meaning that there exists a smooth function χ such that A = dχ),then the new connection A′ (4.65) vanishes. The complexity of the heat kernelequation is consequently reduced as the Abelian connection has disappearedand the operator D has become a Laplace-Beltrami operator (4.57).In order to understand mathematically the origin of the gauge transformation,let us consider the group of automorphisms Aut(L) of the line bundle L. Wehave seen previously that the line bundle L can be defined as the set L ofall triples (α, p, v) ∈ A ×M × R+ modulo the equivalence relation ∼ (4.21).An automorphism of L can be seen as a map on L which commutes with theprojection π of L. Therefore an element g ∈ Aut(L) maps the triple (α, p, v)to (α, p, g.v) with g.v ∈ R+, i.e., it leaves the fiber unchanged. This elementg can be seen locally as a map from Uα to R+. Moreover it induces an actionon the Abelian connection by (4.25) which reduces to (4.65) for

g = e−χ(τ,x)

Interpretation of a gauge transformation as a change of mea-sure

In this section, we show with a simple example that a gauge transformationcan be interpreted as a change of measure according to the Girsanov theorem.For the sake of simplicity, we assume that we have a one-dimensional Itoprocess Xt given by

dXt = dWt

with Wt a one-dimensional Brownian motion. We have the initial conditionXt = x. The option fair value C(t, x) = EP[f(XT )|Ft] satisfies the PDE

∂tC(t, x) +12∂2xC(t, x) = 0


with the terminal condition C(T, x) = f(XT ). We apply a time-independentgauge transformation on the function (section) C(t, x):

C′(t, x) = eΛ(x)C(t, x)

C′(t, x) satisfies the PDE

∂tC′(t, x) +12∂2xC′(t, x)− ∂xΛ∂xC′(t, x) +

12((∂xΛ)2 − ∂2

xΛ)C′(t, x) = 0

with the boundary condition C′(T, x) = eΛ(xT )f(XT ). By using the Feynman-Kac theorem, we obtain the identity

EP[f(XT )|Ft] = e−Λ(x)EP[eΛ(X′T )f(X ′T )e12

R Tt ((∂xΛ(X′s))

2−∂2xΛ(X′s))ds|Ft]

with dX ′t = dWt − ∂xΛ(X ′t)dt. By using that

Λ(X ′T ) = Λ(x) +∫ T

t

∂xΛ(X ′s) (dWs − ∂xΛ(X ′s)ds) +12

∫ T

t

∂2xΛ(X ′s)ds

= Λ(x) +∫ T

t

∂xΛ(X ′s)dWs +12

∫ T

t

∂2xΛ(X ′s)ds−

∫ T

t

(∂xΛ(X ′s))2ds

the identity above becomes

EP[f(XT )|Ft] = EP[f(X ′T )eR Tt∂xΛ(X′s)dWs− 1

2

R Tt

(∂xΛ(X′s))2ds|Ft]

(4.66)

This relation can be obtained according to a change of measure from P to thenew measure P′ defined by the Radon-Nikodym derivative

dP′

dP|FT = e

R T0 ∂xΛ(X′t)dWt− 1

2

R T0 (∂xΛ(X′t))

2dt

and (4.66) can be written as

EP[f(XT )|Ft] = EP′ [f(X ′T )|Ft]

4.6 Heat kernel expansion

The asymptotic resolution of a heat kernel (4.56) in the short time is an im-portant problem in theoretical physics and in mathematics. In physics, itcorresponds to the solution of an Euclidean Schrodinger equation on a fixedspace-time background [13] and in mathematics, the heat kernel, correspond-ing to the determination of the spectrum of the Laplacian, can give topological


information (e.g., the Atiyah-Singer index theorem) [19]. The following theo-rem proved by DeWitt-Gilkey-Pleijel-Minakshisundaram (in short the DeWitttheorem) gives the complete asymptotic solution for a heat kernel on a Rie-mannian manifold. Beforehand, let us introduce the cut-off function φ:

DEFINITION 4.9 Cut-off If φ : R+ → [0, 1] is a smooth function suchthat

φ(s) = 1 if s <ε2

4

φ(s) = 0 if s >ε2

4with ε ∈ R∗+, we will call φ a cut-off function.

THEOREM 4.3 Heat kernel expansion [59]Let M be a Riemannian n-dimensional manifold and p(τ, x|y) be the heat

kernel of the heat kernel equation (4.56). Then there exists smooth sectionsan(x, y) ∈ Γ(M ×M,L L∗) such that for every N > n

2 , pN (τ, x|y) definedby the formula

pN (τ, x|y) = φ(d(x, y)2)

√g(y)

(4πτ)n2

√∆(x, y)P(y, x)e−

σ(x,y)2τ

N∑n=0

an(x, y)τn

(4.67)

is asymptotic to p(τ, x|y):

‖ ∂kτ(p(τ, x|y)− pN (τ, x|y)

)‖l= O

(tN−

n2−

l2−k)

with ‖ φ ‖l= supk≤l supx∈Rn | dk

dxkφ(x) |.

• φ is a cut-off function where ε is chosen smaller than the injectivityradius of the manifold M .

• σ(x, y) is the Synge world function equal to one half of the square ofgeodesic distance d(x, y) between x and y for the metric g.

• ∆(x, y) is the so-called Van Vleck-Morette determinant

∆(x, y) = g(x)−12 det

(−∂

2σ(x, y)∂x∂y

)g(y)−

12 (4.68)

with g(x) = det[gij(x, x)].

• P(y, x) ∈ Γ(M ×M,L L∗) is the parallel transport of the Abelianconnection along the geodesic curve C from the point y to x

P(y, x) = e−RC(y,x)Aidx

i

(4.69)


• The functions ai(x, y), called the heat kernel coefficients, are smoothsections Γ(M ×M,L L∗). The first coefficient is simple

a0(x, y) = 1 , ∀ (x, y) ∈M ×M

The other coefficients are more complex. However, when evaluated onthe diagonal x = y, they depend on geometric invariants such as thescalar curvature R. As for the non-diagonal coefficients, they can becomputed as a Taylor series when x is in a neighborhood of y.

The first diagonal coefficients are fairly easy to compute by hand. Re-cently an(x, x) has been computed up to the order n = 8. The formulasbecome exponentially more complicated as n increases. For example,the a6 formula has 46 terms. The first diagonal coefficients are givenbelow [51]

a1(x, x) = P (x) ≡ 16R+Q(x) (4.70)

a2(x, x) =1

180(RijklR

ijkl −RijRij)

+12P 2

+112FijF ij +

16

∆Q+130

∆R (4.71)

with Rijkl the Riemann tensor, Rij the Ricci tensor, R the scalar cur-vature and Fij = ∂iAj − ∂jAi the curvature associated to the Abelianconnection A.

REMARK 4.5 When p is truncated at the N th order, we call the solutionan asymptotic solution of order N .

Let us, before giving a sketch of the proof of the theorem (4.3), explain howto use the heat kernel expansion (4.67) with a simple example, namely theGBM process.

Example 4.13 GBM processThe SDE is dft = σftdW with the initial condition f0 = f . By using thedefinition(4.54), we obtain the following one-dimensional metric

gff =2

σ2f2

When written with the coordinate s =√

2 ln(f)σ , the metric is flat gss = 1

and all the heat kernel coefficients depending on the Riemann tensor vanish.The geodesic distance between two points s0 and s is given by the classicalEuclidean distance

d(s0, s) = |s− s0|


and the Synge function is

σ(s0, s) =12

(s− s0)2

In the old coordinate [f ], σ(f0, f) is equal to

σ(f0, f) =1σ2

ln(f

f0

)2

Furthermore, the connection A (4.58) and the section Q (4.59) are given by

A = − 12fdf

Q = −σ2

8

Therefore the parallel transport P(f0, f) is given by

P(f0, f) = e12 ln“ff0

”

By plugging these expressions into (4.67), we obtain the following first-orderasymptotic solution for the log-normal process

p1(τ, f |f0) =1

f(2πσ20τ)

12e−

ln( ff0

)2

2σ20τ− 1

2 ln“ff0

”(1− σ2τ

8

)(4.72)

REMARK 4.6 Note that in the last equation, we have replaced thenotation f (resp. f0) by f (resp. f0) according to our convention (remark4.3).

At the second-order, the second heat kernel coefficient is given by

a2 =12Q2 =

σ4τ2

128

and the asymptotic solution is

p2(τ, f |f0) =1

f(2πσ2τ)12e−

ln( ff0 )2

2σ2τ− 1

2 ln“ff0

”(1− σ2τ

8+σ4τ2

128

)(4.73)

We end this example with the use of gauge transformations to solve the heatkernel:We note that A and Q are exact, meaning that

A = dΛQ = −∂τΛ


with Λ = − 12 ln(f) + σ2τ

8 . Modulo a gauge transformation

p′ = e− 1

2 ln“ff0

”+σ2τ

8 pσf0√

2

p′ satisfies the (Laplacian) heat kernel on R

∂2sp′ = ∂τp

′

The constant factor σf0√2

has been added in order to have the initial conditionlimτ→0 p

′(τ, s) = δ(s− s0). The solution is the normal distribution (4.61). So

the exact solution for p = p′e12 ln( ff0

)−σ2τ8 is given by (see remark 4.6)

p(τ, f |f0) =1

f(2πσ2τ)12e−

(ln( ff0

)+σ2τ2 )2

2σ2τ (4.74)

To be thorough, we conclude this section with a brief sketch of the derivationof the heat kernel expansion.

PROOF We start with the Schwinger-DeWitt ansatz

p(t, x|y) =

√g(y)

(4πt)n2

∆(x, y)12P(y, x)e−

σ(x,y)2t Ω(t, x|y)

(4.75)

By plugging (4.75) into the heat kernel equation (4.56), we derive a PDEsatisfied by the function Ω(t, x|y)

∂tΩ =(−1tσi∇i + P−1∆−

12 (D +Q)∆

12P)

Ω (4.76)

with ∇i = ∂i+Ai and σi = ∇iσ, σi = gijσj . The regular boundary conditionis Ω(t, x|x) = 1. We solve this equation by writing the function Ω as a formalseries in t:

Ω(t, x|y) =∞∑n=0

an(x, y)tn (4.77)

By plugging this series (4.77) into (4.76) and identifying the coefficients in tn,we obtain an infinite system of coupled ODEs:

a0 = 1

(1 +1kσi∇i)ak = P−1∆−

12 (D +Q)∆

12P ak−1

kk 6= 0

The calculation of the heat kernel coefficients in the general case of arbi-trary background offers a complex technical problem. The Schwinger-DeWitt


method is quite simple but is not effective at higher orders. By means of it,only the two lowest order terms were calculated. For other advanced methodssee [51].

From the DeWitt theorem, we see that the most difficult part in obtaining theasymptotic expression for the conditional probability is to derive the geodesicsdistance.

Heat kernel on a time-dependent Riemannian manifold

In most financial models, the drift and volatility terms can explicitly dependon time. In this case, we obtain a time-dependent metric, Abelian connectionand section Q. It is therefore useful to generalize the heat kernel expansionfrom the previous section to the case of a time-dependent metric defined by

gij(t, x) = 2ρij(t)

σi(t, x)σj(t, x)(4.78)

This is the purpose of this section.We consider the time-dependent heat kernel equation

∂

∂tp(t, x|y) = Dp(t, x|y) (4.79)

where the differential operator D is a time-dependent family of operators ofLaplace type given by

D = bi(t, x)∂i + gij(t, x)∂ij (4.80)

Let ; (resp. , t) denote the multiple covariant differentiation according to theLevi-Cevita connection (resp. to the time t). We expand D in a Taylor seriesexpansion in t to write D invariantly in the form

Du = Du+∑r>0

tr(Gijr u;ij + F iru;i +Qr

)with the operator D depending on the connection Ai and the smooth sectionQ given by (4.55) (with gij ≡ gij(t = 0) and bi ≡ bi(t = 0)). The tensor Gij1is given by

Gij1 (x) = gij,t (0, x) (4.81)

=ρij,t(0)

2σi(0, x)σj(0, x) + ρij(0)σi,t(0, x)σj(0, x)

The asymptotic resolution of the heat kernel (4.79) in the short time limit in atime-dependent background is an important problem in quantum cosmology.When the spacetime varies slowly, the time-dependent metric describing the


cosmological evolution can be expanded in a Taylor series with respect to t. Inthis situation, the index r is related to the adiabatic order [4]. The subsequenttheorem obtained in [92, 93] gives the complete asymptotic solution for thetime-dependent heat kernel on a Riemannian manifold.

THEOREM 4.4

Let M be a Riemannian n-dimensional manifold and p(t, x|y) be the heatkernel of the time-dependent heat kernel equation (4.79). Then there existssmooth sections an(x, y) ∈ Γ(M ×M,L L∗) such that for every N > n

2 ,pN (t, x|y) defined by the formula

pN (t, x|y) = φ(d(x, y)2)

√g(y)

(4πt)n2

√∆(x, y)P(y, x)e−

σ(x,y)2t

N∑n=0

an(x, y)tn

(4.82)

is asymptotic to p(x, y, t):

‖ ∂kτ(p(t, x|y)− pN (t, x|y)

)‖l= O

(tN−

n2−

l2−k)

with ‖ φ ‖l= supk≤l supx∈Rn | dk

dxkφ(x)|.

σ, ∆ and P are computed with the metric gij(t = 0) and the connectionA(t = 0).

The diagonal heat kernel coefficients ai(x, x) depend on geometric invariantssuch as the scalar curvature R. The coefficients an have been computed upto the fourth-order (a0(x, y) = 1). The first coefficient is given by

a1(x, x) =16R+Q− 1

4G1,ii

where G1,ii ≡ gij(t = 0)Gij1 .

4.7 Hypo-elliptic operator and Hormander’s theorem

4.7.1 Hypo-elliptic operator

As seen in the previous sections, the heat kernel expansion can be applied ifD is an elliptic operator. However, in some situations such as the pricing ofpath-dependent options, D is no longer elliptic.


Example 4.14 Asian optionsFor the pricing of an Asian option in the case of a Black-Scholes model, themarket model is given by

dSt = σStdWt (4.83)dAt = Stdt

where St is the asset price and At =∫ t

0Ssds. Here we have taken a zero

interest rate for the sake of simplicity. The second-order differential operatorD is

D =12σ2S2∂2

S + S∂A

Its leading symbol σ2(S,A, k1, k2) = 12σ

2S2k21 is a degenerate quadratic form

and D is not elliptic. This situation precisely appears when our market modelinvolves path-dependent variables which are not driven by a Brownian motion.

The heat kernel expansion can be extended to an hypo-elliptic operator [56].

DEFINITION 4.10 Hypo-ellipticity A differential operator D is hypo-elliptic if and only if the condition Du is C∞ in an open set Ω ⊂ M impliesthat u is C∞ in Ω.

In particular it can be proved that an elliptic operator is an hypo-ellipticoperator.The definition above is difficult to use except when we have an explicit solutionof the equation Du = f . There exists a sufficient (not necessary) condition toprove that D is an hypo-elliptic operator as given by the Hormander theorem.

4.7.2 Hormander’s theorem

We consider a market model (2.20) written using the Stratonovich calculus:

dxit = Ais(xt)dt+m∑j=1

σij(xt) dWjt , i = 1 · · ·n

We assume that the coefficients of the SDE above are infinitely differentiablewith bounded derivatives of all orders and do not depend on time. We definethe m+ 1 n-dimensional vector fields

V0 = Ais∂iVj = σij∂i , j = 1, · · ·m


The operator D can be put in the Hormander form

D =12

V0 +m∑j=1

VjVj

To formulate the Hormander theorem, we consider the Lie bracket (see exer-

cise 4.3) between two vector fields X = Xi∂i and Y = Y i∂i:

[X,Y ] = (Xi∂iYj − Y i∂iXj)∂j

THEOREM 4.5 Hormander [33]If the vector space spanned by the vector fields

V1, · · ·Vm , [Vi, Vj ] , 0 ≤ i, j ≤ m (4.84)[Vi, [Vj , Vk]] , 0 ≤ i, j, k ≤ m · · ·

at a point x0 is Rn, then D is an hypo-elliptic operator. Furthermore, for anyt > 0, the process xt has an infinitely differentiable density.

Note that if D is an elliptic operator then the Hormander condition (4.84)holds.

Example 4.15 Asian optionsFrom (4.83), we have the two vector fields

V0 = −σ2

2S∂S + S∂A

V1 = σS∂S

As [V0, V1] = −σS∂A, the Hormander condition (4.84) holds for S 6= 0. There-fore D is an hypo-elliptic operator and (St, At) has an infinitely differentiabledensity.

Example 4.16 Heat kernel on Heisenberg groupWe consider the following SDE which corresponds to a Brownian motion onR2 and its Levy area (see Appendix B)

dx = dWX

dy = dWY

dt = 2(y dWX − x dWY

)The vector fields associated to this SDE are

X = ∂x + 2y∂tY = ∂y − 2x∂t


D is hypo-elliptic in the Hormander sense as (X,Y, T ≡ ∂t) generates theHeisenberg Lie algebra:7

[X,Y ] ≡ −4T[X,T ] = 0 , [Y, T ] = 0

As a consequence, the Markov process admits a smooth density with respectto the Lebesgue measure on R3. The (hypo-elliptic) heat kernel reads

∂τp(τ, Z, T |0) = Dp(τ, Z, T |0)

with D = 12

(X2 + Y 2

)and Z ≡ x+ ıy. This heat kernel is solvable [90] and

gives the joint probability law for WX ,WY ,∫ (WY dWX −WY dWY

):

p(τ, Z, T |0) =1

(2πτ)2

∫ ∞−∞

dk2k

sinh 2keikTτ e−

|Z|22τk

2tanh 2k dk

4.8 Problems

Exercises 4.1 Conformal metrics on a Riemann surfaceLet us consider the metric on a Riemann surface

ds2 = F (y)(dx2 + dy2)

1. Compute the Christoffel symbols Γkij .

2. Prove that the geodesic equations are given by the ODEs (4.47) and(4.48).

3. Compute the Riemann curvature Rpijk.

4. Compute the Ricci curvature Rij .

5. Compute the scalar curvature R.

7In Quantum Mechanics, this reads [X,P ] = ı~.


Exercises 4.2 Christoffel symbolsProve that under a change of coordinates from xi to xi′, the Christoffelsymbols do not transform in a covariant way but into

Γk′

i′j′ =∂xk

′

∂xp∂2xp

∂xi′∂xj′︸︷︷︸+Γpqr∂xq

∂xi′∂xr

∂xj′∂xk

′

∂xp

Non-covariant term

Exercises 4.3 Lie algebraA vector space G is called a Lie algebra if there exists a product (called Lie

bracket) [·, ·] : G × G → G with the following properties

• Antisymmetry: [x, y] = −[y, x] ,∀x, y ∈ G

• Jacobi identity: [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 ,∀x, y, z ∈ G

1. Prove that the space of real antisymmetric n-dimensional matrix

U(n) = A ∈ GL(n,R) , A† = −A

is a (finite) Lie algebra with the bracket

[A,B] = AB −BA

2. Prove that the space of vector fields of TM is an (infinite) Lie algebrawith the Lie bracket

[X,Y ] = (Xi∂iYj − Y i∂iXj)∂j

where we have set X = Xi∂i and Y = Y i∂i.

Exercises 4.4 Asian option and Heisenberg algebraAn Asian call option on a single stock with strike K is defined by the followingpayoff at the maturity date T

max

(1T

∫ T

0

fudu−K, 0

)

with fu the stock forward at time u (we take a zero interest rate for the sake ofsimplicity). We explain in this exercise a link with the Heisenberg Lie algebra.We note At =

∫ t0fudu. The SDEs generated by the processes (ft, At) in the

Black-Scholes model are

dftft

= σdWt

dAt = ftdt


1. Convert the Ito diffusion in a Stratonovich diffusiondftft

= −12σ2dt+ σ dWt

dAt = ftdt

2. Write the diffusion operator in the Hormander form.

3. Convert these SDEs into ODEs by replacing the Brownian Wt by thepath ω± = ±t.

dftft

= (−12σ2 ± σ)dt

dAt = ftdt

4. These flows (see B.6 in appendix B) are generated by the following vectorfields

V± = f∂A + (−12σ2 ± σ)f∂f

Prove that the vector fields V+, V−, V+− ≡ [V+, V−] satisfy the followingLie algebra (see exercise 4.3), called Heisenberg algebra

[V+−, V+] = 0 , [V+−, V−] = 0

Exercises 4.5 Mehler’s formulaIn this exercise, we give an example of a one-dimensional heat kernel equation(HKE) which admits an analytical solution. We consider the following HKE

∂tp(t, x|x0) = (∂2x −

r2x2

16− f)p(t, x|x0) (4.85)

with r and f two constants.

1. By scaling the variables x and t and doing a time-dependent gaugetransform, show that the HKE (4.85) can be reduced to

∂tp(t, x|x0) = (∂2x − x2)p(t, x|x0)

This PDE corresponds to a HKE with a zero Abelian connection A anda quadratic potential Q (called harmonic potential). By observing thatthe operator D is quadratic in differentiation and multiplication, we seeka solution which is a Gaussian function of x and x0. In addition, thesolution must clearly be symmetric in x and x0 since the operator D isself-adjoint. We therefore try the following ansatz

p(t, x|x0) = exp(a(t)

x2

2+ b(t)xx0 + a(t)x2

0 + c(t))


2. Prove that the coefficients a(t), b(t) and c(t) satisfy the following ODEs

a(t)2

= a(t)2 − 1 = b(t)2

c(t) = a(t)

3. Prove that the solutions are given by

a(t) = − coth(2t+ C)b(t) = csc(2t+ C)

c(t) = −12

ln sinh(2t+ C) +D

with C and D two integration constants.

4. By using the initial condition p(0, x|x0) = δ(x − x0), show that thevalues of the integration constants are

C = 0D = ln(2π)−

12

5. Finally, prove that the solution of (4.85) is

p(t, x|x0) =

√t r2

4πt sinh(t r2 )

exp(− r

8t

(coth(t

r

2)(x2 + x2

0)− 2 csc(tr

2)xx0

)− tf

)

Chapter 5

Local Volatility Models andGeometry of Real Curves

Abstract The existence of an implied volatility indicates that the Black-Scholes assumption that assets have a constant volatility should be relaxed.The simplest extension of the log-normal Black-Scholes model is to assumethat assets still follow a one-dimensional Ito diffusion process but with avolatility function σloc(t, f) depending on the underlying forward f and thetime t. As shown by Dupire [81], prices of European call-put options deter-mine the diffusion term σloc(t, f) uniquely.In this chapter, we show how to apply the theorem (4.4) to find an asymp-totic solution to the backward Kolmogorov (Black-Scholes) equation and de-rive an asymptotic implied volatility in the context of local volatility models(LVM). Before moving on to the general case in the second section, we considerin the first section a specific separable local volatility function: σloc(t, f) =A(t)C(f).Throughout this chapter, deterministic interest rates are assumedfor the sake of simplicity.

5.1 Separable local volatility model

Let us consider an asset with a price St whose forward is ft = StPtT

. As theproduct of the forward with the bond PtT is a traded asset, the forward ftshould be a local martingale under the forward measure PT . It follows adriftless process and we assume the following dynamics

dft = A(t)C(ft)dWt (5.1)

C : R+ → R+ and A : R+ → R+ are continuous deterministic functions. Wealso assume that C(0) = 0 which ensures that the forward cannot go belowzero.After a change of local time

t′(t) =∫ t

0

A2(s)ds

123


TABLE 5.1: Example of separable LVmodels satisfying C(0) = 0.

LV Model C(f)Black-Scholes f

Quadratic f(af + b)CEV fβ , β > 0

LCEV f min(fβ−1, εβ−1) , ε > 0Exponential f(1 + ae−bf ) , a, b > 0

we obtain the simpler time-homogeneous SDE1

dft′ = C(ft′)dWt′ (5.2)

Below, we write t instead of t′.In table (5.1), we list examples of commonly used separable local volatilitymodels. For most of these models (except Black-Scholes), the function C(f)does not satisfy the global Lipschitz condition (2.17). Therefore we cannotconclude that the SDE (5.2) admits a unique strong solution.This point will be discussed below: it is shown that these models admit aweak solution up to an explosion time.

5.1.1 Weak solution

Let T ∈ (0,∞) be a fixed time horizon and D an open connected subset ofRn. We consider the n-dimensional SDE

dxt = b(t, xt)dt+m∑j=1

σj(t, xt)dWjt (5.3)

for continuous functions b : [0, T ] × D → Rn, σj : [0, T ] × D → Rn , j =1, · · · ,m with an m-dimensional Brownian motion W j

t j=1,··· ,m.

THEOREM 5.1 KunitaIf the coefficients b and σj, j = 1, · · · ,m on [0, T ]×D are locally Lipschitz-

continuous in x, uniformly in t, i.e.,

‖ G(t, x)−G(t, y) ‖≤ Kn ‖ x− y ‖ , ∀ ‖ x ‖, ‖ y ‖≤ n , ∀ t ∈ [0, T ]

with G ∈ b, σ1, · · · , σm, then the SDE (5.3) has a unique weak solution upto an explosion time.

The local Lipschitz condition is satisfied for all models listed in table 5.1 (ex-cept for the Constant Elasticity of Variance (CEV) model 0 < β < 1). From

1Use the formal rule < dWt′ , dWt′ >= dt′ = A(t)2 < dWt, dWt >= A(t)2dt.

Local Volatility Models and Geometry of Real Curves 125

this theorem, we conclude that these models admit a unique weak solution upto an explosion time τ that will be computed in the section 5.1.2.

REMARK 5.1 Note that the linear growth condition

∃K > 0 such that ||G(t, x)||2 ≤ K(1 + ||x||2

), ∀ x > 0

ensures that the process does not explode (i.e., τ =∞ a.s.).

CEV model

As the local Lipschitz condition is not satisfied for the CEV model for 0 <β < 1 at f = 0, the theorem (5.1) can not be used. As the CEV process willoften appear in the remainder, here is a list of a few useful results about theCEV model below [80], [44].

LEMMA 5.1

dft = fβt dWt , β > 0 (5.4)

1. All solutions to (5.4) are non-explosive.

2. For β ≥ 12 , the SDE has a unique solution.

3. For 0 < β < 1, f = 0 is an attainable boundary.

4. For β ≥ 1, f = 0 is an unattainable boundary.

5. For 0 < β < 12 , the SDE (5.4) does not have a unique solution unless

a separate boundary condition is specified for the boundary behavior atf = 0.

6. For 12 ≤ β ≤ 1, the SDE (5.4) has an unique solution with an absorbing

condition at f = 0.

7. If ft=0 > 0, the solution is positive ∀t > 0.

According to (5), if 0 < β < 12 , the behavior at f = 0 is not unique and

requires us to choose between the two possible boundary conditions: absorbingor reflecting. If we require ft to be a martingale even when starting at f = 0,we should have an absorbing condition for f = 0 [80]. Indeed a reflectingboundary condition leads to an arbitrage opportunity: if there is a positiveprobability of reaching zero, one just has to wait for that event to happenand buy at zero cost the forward which will have a strictly positive value aninstant later due to the reflecting boundary.


For 0 < β < 1, the origin is an attainable boundary, not such a nice featurefrom a modeling point of view. To overcome this drawback, we introduce theLCEV model [44]

C(f) = f min(fβ−1, εβ−1

), ε > 0 (5.5)

ε is a small number for β < 1 and a large number when β > 1.

5.1.2 Non-explosion and martingality

Being a driftless process (5.2), ft is a local martingale as imposed by the no-arbitrage condition. Whilst strictly local martingale discounted stock pricesdo not involve strict arbitrage opportunities, they can create numerical prob-lems, for example, causing the inability to price derivatives using the pricingformula (2.31). The Monte-Carlo pricing of derivatives is therefore inaccurate.Moreover, the put-call parity is no longer valid as the forward is not pricedconveniently [29], [114]:

PROPOSITION 5.1If ft is a strict local martingale then the put-call parity is no longer valid:

EPT [max(fT −K, 0)|Ft]− EPT [max(K − fT , 0)|Ft] 6= f0 −K

PROOF The payoff of a call option at the maturity date T with strike Kcan be decomposed as

max(fT −K, 0) = max(K − fT , 0) + fT −K

The first term max(K − fT , 0) is the payoff of a European put option. Aswell as being a bounded local martingale, it is also a true martingale [27].Consequently, max(ft−K, 0) is a martingale if and only if ft is a martingale.In this case, we obtain the put-call parity

C = P + f0 −K

with C (resp. P) the fair value of a call (resp. put) option.

In order to preserve the put-call parity and the ability to use the pricingformula (2.31), it is required that the forward is not only a local martingalebut also a true martingale. Let us see under which conditions a positive localmartingale can be a martingale.We have

EPT [fT |Ft] = ftEPT [e−12

R Tta2sds+

R TtasdWs |Ft] (5.6)


with at = C(ft)ft

the volatility of the forward. Under PT , at satisfies the SDE

dat =C(f)2

2

d2(C(f)f

)df2

dt+ C(f)d(C(f)f

)df

dWt (5.7)

The exponential term in (5.6) is the Radon-Nikodym derivative

Mt ≡dPf

dPT|Ft= e−

12

R t0 a

2sds+

R t0 asdWs

corresponding to the change of measure from the forward measure PT to thespot measure Pf associated to the forward itself. Therefore, assuming thatMt is a well-defined martingale, when doing the change of measure from PTto Pf , we obtain

EPT [fT |Ft] = ftEPT [MT

Mt|Ft]

= ftEPf [1|Ft] = ft

and ft is a martingale. In the section on Girsanov’s theorem 2.7, we haveencountered sufficient conditions (2.39) and (2.40) to ensure that Mt is notonly a local martingale but also a true martingale. The following theoremgives the necessary and sufficient condition [140, 107, 47].

THEOREM 5.2Let us suppose that at is the unique non-exploding (at infinity) strong (orweak) solution of the SDE under PT

dat = b(at)dt+ σ(at)dWt

where b(·) and σ(·) are continuous function and σ(·)2 > 0. Then, the expo-nential local martingale Mt defined by

Mt = e−12

R t0 a

2sds+

R t0 asdZs

is a martingale if and only if there is a non-exploding weak solution of theSDE

dat = (b(at) + ρatσ(at)) dt+ σ(at)dBt (5.8)

where dZtdBt = ρdt(= dWtdZt).

PROOF⇒: If Mt is a martingale, we define the probability measure Pf by the Radon-Nikodym derivative: dPf

dPT = Mt. From the change of measure from PT to Pf ,


at satisfies the SDE (5.8). at does not explode under Pf as Pf is equivalentto PT . Besides, at does not explode under PT by definition.⇐: Let us denote τn = inft ∈ R+ : at ≥ n and τ∞ = limn→∞ τn theexplosion time for at. As at does not explode under PT , we have 1τ∞>T = 1PT -a.s. and

EPT [fT ] = EPT [fT 1τ∞>T ] = EPT [fT limn→∞

1τn>T ] (5.9)

By observing that

0 ≤ fT 1τn>T ≤ fT 1τn+1>T , n = 1, 2, . . .

Thanks to the monotone convergence theorem [26], we conclude that we canpermute the mean-value and the lim sign in (5.9)

EPT [fT ] = limn→∞

EPT [fT 1τn>T ]

= f0 limn→∞

EPT [MT 1τn>T ]

Let us define Mnt = Mt∧τn .2 We have

EPT [MT 1τn>T ] = EPT [MnT 1τn>T ]

As the coefficients in Mnt are bounded, Mn

t is a martingale (the Novikovcondition is satisfied). Therefore, if we define the measure Pfn with the Radon-Nikodym derivative dPfn

dPT = Mnt then

EPT [MnT 1τn>T ] = EPfn [1τn>T ]

and

EPT [fT ] = f0 limn→∞

EPfn [1τn>T ]

As the process at satisfies the same SDE under Pfn and Pf up to an explosiontime τ∞, we have

EPT [fT ] = f0 limn→∞

EPf [1τn>T ]

= f0EPf [1τ∞>T ]

The monotone convergence theorem was used in the last equality. As at doesnot explode under Pf , we deduce that EPT [fT ] = f0 and fT is a martingaleunder PT .

2t1 ∧ t2 = min(t1, t2).


According to the theorem above, ft is a martingale if and only if the volatilityprocess at does not explode in the measure PT (5.7) and Pf (5.10): In thespot measure Pf , we have

dat = C(f)2

12

d2(C(f)f

)df2

+1f

d(C(f)f

)df

dt+ C(f)d(C(f)f

)df

dW ft

(5.10)

The SDEs (5.7) and (5.10) correspond to one-dimensional Ito diffusion pro-cesses for which there exists a necessary and sufficient criteria to test if thereare non-exploding solutions (Theorem VI. 3.2 in [24]):

THEOREM 5.3 Feller non-explosion testLet I = (c1, c2) with −∞ ≤ c1 < c2 ≤ ∞. Let σ(·) and b(·) be continuous

functions on I such that σ(x) > 0 ∀x ∈ I. Define for fixed c ∈ I

s(x) =∫ x

c

e−2R yc

b(z)σ(z)2

dzdy

l(x) =∫ x

c

s′(y)dy∫ y

c

1σ(z)2s(z)′

dz

Let Px be the diffusion measure of an Ito process Xt starting from x wherethe infinitesimal generator is

L =12σ(x)2 d

2

dx2+ b(x)

d

dx

Below, we refer to

e = inft ≥ 0 : Xt /∈ (c1, c2)

as the exit time from I.

• Px[e =∞] = 1 if and only if l(c1) = l(c2) =∞.

• Px[e < ∞] < 1 if and only if one of the following three conditions issatisfied

1. l(c1) <∞ and l(c2) <∞.2. l(c1) <∞ and s(c2) =∞.3. l(c2) <∞ and s(c1) = −∞.

Applying the criteria to the processes (5.7) and (5.10), we obtain

lPT (a) =∫ φ−1(a)

φ−1(c)

df

∫ f

φ−1(c)

dx

C(x)2(5.11)

lPf (a) =∫ φ−1(a)

φ−1(c)

df

f2

∫ f

φ−1(c)

dxx2

C(x)2(5.12)


TABLE 5.2: Feller criteria for the CEVmodel.

0 < β < 12

12 ≤ β < 1 β > 1

lPT (f = 0) 6=∞ 6=∞ ∞lPT (f =∞) ∞ ∞ ∞lPf (f = 0) ∞ ∞ 6=∞lPf (f =∞) ∞ ∞ 6=∞

where φ−1(·) is the inverse of the monotone function φ(f) = C(f)f .

Example 5.1 CEV modelApplying the formula above (5.11,5.12) to the CEV model for which C(x) =xβ , φ−1(a) = a

1β−1 , β 6= 1, we obtain

• For all β ∈ R/1, 12

lPT (a) =1

(1− 2β)[f2−2β

2− 2β− φ−1(c)1−2βf ]φ

−1(a)φ−1(c)

lPf (a) =1

(3− 2β)[f2−2β

2− 2β+φ−1(c)3−2β

f]φ−1(a)φ−1(c)

• For β = 12

lPT (a) = [−f(1 + lnφ−1(c)

)+ f ln f ]φ

−1(a)φ−1(c)

lPf (a) =12

[

(φ−1(c)

)2f

+ f ]φ−1(a)φ−1(c)

From table 5.2, we deduce the properties (1), (3) and (4) in the lemma 5.1and that the CEV model defines a martingale if and only if β ≤ 1.

Example 5.2 Quadratic volatility modelIn a quadratic volatility model, the forward is driven under PT by

dft = ft(aft + b)dWt , a 6= 0 b 6= 0 (5.13)

These models have been studied in detail by [145]. Applying the Feller criteria(in particular formula (5.11) and (5.12)), we obtain that

lPT (f = 0) < ∞ , lPT (f =∞) =∞lPf (f = 0) =∞ , lPf (f =∞) <∞

We conclude that the SDE (5.13) under PT does not explode to infinity con-trary to the intuition but converges to f = 0 PT -a.s. Moreover, ft is not atrue martingale but only a local martingale.


5.1.3 Real curve

In our geometrical framework, the model (5.2) corresponds to a (one-dimen-sional) real curve endowed with the time-independent metric

gff =2

C(f)2

For the new coordinate u(f) =√

2∫ ff0

dxC(x) , the metric is flat: guu = 1. The

distance is given by the classical Euclidean distance

d(u, u′)2 = |u− u′|2 = 2

(∫ f

f ′

dx

C(x)

)2

(5.14)

The connection A (4.58) and the section Q (4.59) are given by

A = −12d lnC(f)

Q(f) =C(f)2

4

((C′′(f)

C(f)

)− 1

2

(C ′(f)C(f)

)2)

(5.15)

As A is an exact form, the parallel transport between two points f ′ and f isgiven by

P(f ′, f) = e−R ff′ A =

√C(f)C(f ′)

(5.16)

Moreover, the first (4.70) and second heat kernel coefficients (4.71) (in thelimit f ′ = f) are given by

a1(f, f) = Q(f) (5.17)

a2(f, f) =12

(Q(f)2 +

∆Q(f)3

)(5.18)

with the Laplacian

∆ = ∂2u =

C(f)2

∂f (C(f)∂f )

In reality, to compute the conditional probability density p(t, f |f0) using theheat kernel expansion, we need the non-diagonal coefficients a1(f, f0) anda2(f, f0). From the diagonal heat kernel coefficients (5.17) and (5.18), weuse the following approximation for the non-diagonal terms that we justify insection 5.2.2.1

a1(f, f0) = a1(fav, fav) (5.19)a2(f, f0) = a2(fav, fav) (5.20)


with fav = f0+f2 . By plugging the expressions (5.14), (5.15), (5.16), (5.19),

(5.20) in the heat kernel expansion (4.67), we obtain the probability densityfor the old coordinate f at the second-order in time (see remark 4.6)

p(t′, f |f0) =1

C(f)√

2πt′

√C(f0)C(f)

e−σ(f,f0)

2t′ (5.21)(1 +Q(fav)t′ +

12

(Q2(fav) +

∆Q(fav)3

)t′2)

with

σ(f, f0) =

(∫ f0

f

dx

C(x)

)2

and

t′ =∫ t

0

A(s)2ds

Whilst this result has already been obtained in [99] using a more direct ap-proach, our method produces a quicker answer.Specializing our asymptotic solution (5.21) to the CEV model, we obtain

Example 5.3 CEV modelFor the CEV model (C(f) = fβ , A(t) = σ), the asymptotic solution at thesecond-order to the backward Kolmogorov equation is from (5.21)

p(t, f |f0) =f−β√2πt′

e−(R f0f df′f′−β)2

2t′

(f0

f

) β2

(5.22)(1 +

18f2β−2

av β(β − 2)t′ +β(β − 2)(3β − 2)(3β − 4)

128f4β−4

av (t′)2)

with t′ = σ2t. We can check that this asymptotic solution for β = 1 reproducesthe asymptotic solution previously given in the case of the GBM process(4.73).In the case of the CEV model, an exact solution (with absorbing conditionat f = 0) exists for the Kolmogorov equation [76] that we derive in chapter 9using a spectral decomposition

p(t, f |f0) =f

12−2β

(1− β)t′√f0e− f

2(1−β)+f2(1−β)0

2(1−β)2t′ I 12(1−β)

((ff0)1−β

(1− β)2t′

)(5.23)

where Iν(z) is the modified Bessel function of the first kind given by

Iν(z) =(

12z

)ν ∞∑k=0

(14z

2)k

k!Γ(ν + k + 1)


Γ(·) is the gamma function.This exact solution will be used to test the validity of our asymptotic solu-tion at the second-order. In the following graphs (Fig. 5.1, Fig. 5.2), theasymptotic conditional probability (5.22) has been plotted against the exactsolution (5.23) for β = 0.33 and β = 0.6. There is a good match between theexact and approximate solutions.

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

0% 100% 200% 300% 400% 500%

Prob_HKE0Prob_HKE1Prob_Exact

FIGURE 5.1: Comparison of the asymptotic solution at the first-order(resp. second-order) against the exact solution (5.23). f0 = 1, σ = 0.3,β = 0.33, τ = 10 years.

In the next paragraph, we focus on the general case where we assume that thevolatility function σloc(t, f) is a general function of the forward and the time.This model is called a local volatility model (LVM). What is particular withthis LVM is that the market is complete and the model can be automaticallycalibrated to the initial implied volatility as coded in the Dupire formula.


0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

0% 100% 200% 300% 400% 500%

Prob_HKE0Prob_HKE1Prob_Exact

FIGURE 5.2: Comparison of the asymptotic solution at the first-order(resp. second-order) against the exact solution (5.23). f0 = 1, σ = 0.3,β = 0.6, τ = 10 years.

5.2 Local volatility model

5.2.1 Dupire’s formula

We derive the Dupire local volatility formula [81]. We assume that the assetprice St follows the following SDE in the risk-neutral measure P

dSt = rtStdt+ Stσ(t, St)dWt

with σ(·, ·) a continuous function of time t and St satisfying the uniformellipticity condition: there exists a λ ∈ R+ such that

λ−1 ≤ σ(t, S) ≤ λ

As a result ft admits a non-explosive strong solution and ft is a true martin-gale.


We try to choose the local volatility (LV) σ(t, St) such that when we priceEuropean call options C(K,T |S0) for any strike price K and any maturity T ,we match the market prices Cmkt(K,T |S0).By applying Ito-Tanaka’s formula [27] (which is a generalization of the Itoformula for non-smooth functions) on the payoff max(St −K, 0), we obtain

dmax(St −K, 0) = 1(St −K) (rtStdt+ Stσ(t, St)dWt)

+12K2σ(t,K)2δ(St −K)dt (5.24)

with δ(·) the Dirac function. Then, by taking the mean value operator EP[·|F0]on both sides of equation (5.24), we have

dEP[max(St −K, 0)|F0] = rtEP[1(St −K)St|F0]dt

+12K2σ2(t,K)EP[δ(St −K)|F0]dt

As the fair value of a call option with a maturity t and a strike K is

C(t,K|S0) = P0tEP[max(St −K, 0)|F0]

we obtain

dC(K, t|S0) = rtdt(−C(t,K|S0) + P0tEP[1(St −K)St|F0])

+12P0tK

2σ2(t,K)EP[δ(St −K)|F0]dt

Using the following identities

∂2C(t,K|S0)∂K2

= P0tEP[δ(St −K)|F0]

EP[1(St −K)St|F0] = EP[max(St −K, 0)|Ft] +KEP[1(St −K)|F0]∂C(t,K|S0)

∂K= −P0tEP[1(St −K)|F0]

we finally obtain

dC(t,K|S0) =(−rtK

∂C(t,K|S0)∂K

+12K2σ2(t,K)

∂2C(K, t|S0)∂K2

)dt

Inverting this formula, we obtain the Dupire LV

DEFINITION 5.1 Local volatility The Dupire LV, denoted σloc(t, S),is defined as

σloc(t,K)2 ≡ 2∂C(t,K|S0)

∂t + rtK∂C(t,K|S0)

∂K

K2 ∂2C(t,K|S0)∂2K

(5.25)


We have shown that European call options C(t,K|S0) for every strike K ∈[0,∞) and maturity t ∈ [0,∞) are automatically calibrated (therefore so isthe initial implied volatility) if the instantaneous volatility σ(t, S) is equalto the Dupire LV σloc(t, S) defined by (5.25). By definition of the impliedvolatility, we have

Cmkt(T,K|S, t) = CBS(K,T, σBSt(K,T )|S, t)

By substituting the expression (3.1) in (5.25), we obtain a relation betweenthe implied volatility and the Dupire LV. A straightforward computation gives

σloc(T, y)2 = (5.26)∂ω(T,y)∂T

1− yω(T,y)

∂ω(T,y)∂y + 1

4

(− 1

4 −1

ω(T,y) + y2

ω(T,y)2

)(∂ω(T,y)∂y

)2

+ 12∂2ω∂y2

with ω(T, y) ≡ σBS(K,T |S, t)2(T − t) and y = ln(Kf0

).

In Fig. (5.3), we have plotted the market implied volatility (one year maturity)for the index SP500 versus the Dupire local volatility.

Example 5.4 Merton modelWhen we have an implied volatility with no skew, i.e., ∂ω∂y = 0, the expression(5.26) reduces to

σloc(T )2 =∂ω(T )∂T

Inverting this equation, we deduce that the initial implied volatility (with noskew) is perfectly calibrated with a Merton model characterized by a time-dependent volatility σloc(·) such that

σ2BS(T |t) =

1T − t

∫ T

t

σloc(s)2ds

In the following, we obtain an asymptotic relation in the short-time limitbetween the LV function and the implied volatility using the heat kernel ex-pansion on R with a time-dependent metric. This map will be particularlyuseful when discussing stochastic volatility models in chapter 6.

5.2.2 Local volatility and asymptotic implied volatility

Let us assume that the forward ft follows a LVM in the forward measure PT

df = C(t, f)dWt ; f(t = 0) = f0 (5.27)


IV versus LV

-

10.00

20.00

30.00

40.00

50.00

60.00

30% 50% 70% 90% 110% 130% 150% 170% 190%

Strike

Market IVLV

FIGURE 5.3: Market implied volatility (SP500, 3-March-2008) versusDupire local volatility (multiplied by ×100). T = 1 year. Note that thelocal skew is twice the implied volatility skew (see remark 5.2).

Repeating what was done in the previous section, we apply the Ito-Tanakalemma on the payoff max(ft −K, 0) and obtain

dmax(ft −K, 0) = 1(ft −K)dft +12C(t,K)2δ(ft −K)dt (5.28)

Taking the mean value operator EPT [·|F0] on both sides of this equation (5.28)and integrating over the time from t = 0 to T , we obtain that the fair valueof a European call option (with a maturity date T and a strike K)

C(T,K|f0)P0T

= EPT [max(fT −K, 0)|F0]

is given by

C(T,K|f0) = P0T

(max(f0 −K, 0) +

12

∫ T

0

C(t,K)2p(t,K|f0)dt

)(5.29)


p(t,K|f0) is the conditional probability associated to the process (5.27) in theforward measure PT which satisfies the forward Kolmogorov equation

∂tp(t, f |f0) =12∂2f

(C(t, f)2p(t, f |f0)

)=

12C2∂2

fp(t, f |f0) + 2C∂fC∂fp(t, f |f0)

+(C∂2

fC + (∂fC)2)p(t, f |f0)

with the terminal condition limt→0 p(t, f |f0) = δ(f − f0).In our geometrical framework, the LV model corresponds to a (one-dimensio-nal) real curve endowed with the time-dependent metric

gff (t) =2

C(t, f)2

For the new coordinate u(f) =√

2∫ ff0

dxC(x) (with C(f) ≡ C(0, f)), the metric

at t = 0 is flat

guu = 1

The distance between two points f ′ and f is then given by the classical Eu-clidean distance

d(f, f0) = |u(f)− u(f0)| =√

2|∫ f0

fdx

C(x)| (5.30)

The connection A (4.58) and the function Q (4.59) are given by

A =32d lnC(f) (5.31)

Q =C(f)2

4

((C′′(f)

C(f)

)− 1

2

(C ′(f)C(f)

)2)

(5.32)

As the Abelian connection is an exact form, the parallel transport betweenthe point f ′ and f is obtained by direct integration

P(f0, f) =

√C(f0)C(f)

1C(f)

Furthermore, the first heat kernel diagonal coefficient a1(f, f) is given by

a1(f, f) = Q(f)− 14G(f)

with the coefficient G(f) (4.81) equal to

G(f) = 2∂t lnC(0, f) (5.33)


By using the same approximation as for the separable LVM, the non-diagonalfirst heat kernel coefficient is approximated by

a1(K, f) = Q(fav)− 14G(fav) (5.34)

with fav = K+f2 . From the heat kernel expansion on a time-dependent mani-

fold (4.82), the first-order conditional probability at time t is then

p(τ,K|0, f) =1

C(K)√

2πτ

√C(f)C(K)

e−(R fK dx

C(x) )2

2τ

(1 +

(Q(fav)− G (fav)

4

)τ

)By plugging this expression in (5.29) and using that

C(t, f)2 = C(f)2 (1 + tG(f)) + o(t2)

we obtain

C(T,K|f0)P0T

= max(f0 −K, 0) + (5.35)√C(K)C(f0)

2

∫ T

0

dτ1√2πτ

e−(R f0K dx

C(x) )2

2τ

(1 +

(Q(fav) +

3G(fav)4

)τ

)The integration over τ can be performed and we obtain

PROPOSITION 5.2The value of a European call option C(T,K|f0), at the first-order in the ma-turity T , is for the local volatility model (5.27),

P−10T C(T,K|f0) = max(f0 −K, 0) +

√C(K)C(f0)T

2√

2π(5.36)(

H1(ω) +(Q(fav) +

3G(fav)4

)TH2(ω)

)with

H1(ω) = 2(e−ω

2+√πω2(N(

√2|ω|)− 1)

)H2(ω) =

23

(e−ω

2(1− 2ω2)− 2|ω|3

√π(N(

√2|ω|)− 1)

)ω =

∫ K

f0

df ′√2TC(f ′)

with Q(f) and G(f) given respectively by (5.32) and (5.33).

In the case of a constant volatility C(t, f) = σf , the formula above reduces to


Example 5.5 Black-Scholes model

P−10T C(T,K|f0) = max(f0 −K, 0) +

√Kf0σ2T

2√

2π

(H1(ω)− σ2T

8H2(ω)

)(5.37)

with ω =ln( Kf0

)√

2Tσ.

By identifying the formula (5.36) with the same formula obtained with animplied volatility σ = σBS(K,T ) (5.37), we deduce that the implied volatilityσBS(K,T ) at the first-order satisfies the non-linear equation

σBS(K,T ) =

√C(K)C(f0)√

Kf0

H1(ω)H1(ω)

(5.38)(1 +

(Q(fav) +

3G(fav)4

)TH2(ω)H1(ω)

+σ2

BS(K,T )T8

H2(ω)H1(ω)

)

with ω =ln( Kf0

)√

2TσBS(K,T ). At the zero-order (i.e., independent of the maturity

T ), we obtain ω = ω , i.e.,

limT→0

σBS(T,K) =ln(Kf0

)∫Kf0

df ′

C(f ′)

(5.39)

The formula (5.39) has already been found in [59], [46] and we will call it theBBF relation in the following. Then using the recurrence equation (5.38), weobtain at the first-order

σBS(K,T ) =ln(Kf0

)∫Kf0

df ′

C(f ′)

1 +T

3

18

ln(Kf0)∫K

f0

df ′

C(f ′)

2

+Q(fav) +3G(fav)

4

(5.40)

This expression can be approximated by

σBS(K,T ) 'ln(Kf0

)∫Kf0

df ′

C(f ′)

(1 +C2(fav)T

24

(2C ′′(fav)C(fav)

−(C ′(fav)C(fav)

)2

+1f2

av

)

+ T∂tC(0, fav)

2C(fav))

In the case of a time-homogeneous LV function C(t, f) = C(f), we reproducethe asymptotic implied volatility obtained by [99].


REMARK 5.2 Local skew versus implied volatility skew From theBBF formula (5.39), we obtain that the local skew Sloc ≡ f0∂fσ(f0) (σ(f) =C(f)f ) is twice the implied volatility skew: Sloc = 2S as observed in Fig. 5.3.

Example 5.6 CEV modelFor the CEV model, C(f) = fβ , A(t) = σ, the first-order asymptotic implied

volatility is from (5.40)

σBS(K,T ) =

√T ′

T

(1− β) ln Kf0

K1−β − f1−β0

(1 +

(β − 1)2T ′

24f2β−2

av

)(5.41)

with T ′ = σ2T . In the case of the CEV model, an analytical formula for thefair value of a call option exists [76]: For 0 < β < 1, we have

C(T,K|f0)P0T

= f0

(1− χ(a, b+ 2, c)2

)−Kχ(c, b, a)

(5.42)

and for β > 1

C(T,K|f0)P0T

= f0

(1− χ(c,−b, a)2

)−Kχ(a, 2− b, a)

The variables a, b, c are given by

a =K2(1−β)

(1− β)2σ2T, b =

11− β

, c =f

2(1−β)0

(1− β)2σ2T

and χ is the non-central Chi-squared distribution.This exact solution is used to test the validity of our asymptotic impliedvolatility. In Fig. 5.4 and Fig. 5.5, the asymptotic implied volatility (5.40) isplotted against the exact solution (5.42). We obtain a good match betweenthe exact and approximate solutions.

The formula (5.40) will be useful for deriving an implied volatility for astochastic volatility model. More precisely, we will compute the LV asso-ciated to the stochastic volatility model (the square of the LV function is themean value of the square of the stochastic volatility conditional to the spot)and finally we will use the relation (5.40) in order to obtain the correspondingimplied volatility.In the next section, we present an alternative method to derive an impliedvolatility from a LV. This second derivation will justify the approximation(5.34) that we have used to compute the non-diagonal heat kernel coefficientsfrom the diagonal coefficients. A similar computation can be found in [46].


20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

0% 50% 100% 150% 200% 250%

IV_1 IV_Exact

IV_0

FIGURE 5.4: Comparison of the asymptotic implied volatility (5.41) atthe zero-order (resp. first-order) against the exact solution (5.42). f0 = 1,σ = 0.3, τ = 10 years, β = 0.33.

5.2.2.1 Alternative method

We seek a small T expansion of the implied volatility of the form

σBS(y, T ) =∞∑i=0

T iσi(y) (5.43)

with the moneyness y ≡ ln(Kf0

). Note that (5.43) only admits integer powers

of T . Similarly, we do a Taylor expansion of the Dupire LV function aroundT = 0

σloc(T, y) =∞∑i=0

T iσi,loc(y) (5.44)


20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

55.00%

0% 50% 100% 150% 200% 250%

IV_1 IV_Exact

IV_0

FIGURE 5.5: Comparison of the asymptotic implied volatility (5.41) atthe zero-order (resp. first-order) against the exact solution (5.42). f0 = 1,σ = 0.3%, τ = 10 years, β = 0.6.

We recall the Dupire formula (5.26) giving a map between the LV functionand the implied volatility

σloc(T, y)2 =σ2

BS + 2σBST∂TσBS(1− y

σBS∂yσBS

)2

− σ2BST

2

4 (∂yσBS)2 + TσBS∂2yσBS

Substituting (5.43) and (5.44) into the equation above and matching term oforder T 0 gives

σ0,loc(y)2 =σ0(y)2

(1− y∂y lnσ0(y))2

Taking the square-root of the equation above and rearranging leads to thefirst-order ODE

1σ0,loc(y)

= ∂y

(y

σ0(y)

)(5.45)


Solving (5.45), subject to the boundary condition that the limit must be finitefor y goes to zero, leads to

σ0(y) =y∫ y

0dy′

σ0,loc(y′)

In the K coordinate, we have

σ0(K) =ln(Kf0

)∫Kf0

df ′

f ′σ0,loc(f ′)

which is nothing but the BBF relation (5.39).Next, matching the term T 1 and rearranging, we obtain the first-order ODEfor the function φ(y) ≡ σ1(y)

σ0(y)

y∂yφ(y) + 2σ0(y)σ0,loc(y)

φ(y) =σ0,loc(y)

2∂2yσ0(y) +

σ0(y)σ1,loc(y)σ0,loc(y)2

Solving this first-order ODE, subject to the boundary condition that the limitmust be finite for y goes to zero, leads to

φ(y) = −12

(σ0(y)y

)2

(ln(

σ0(y)2

σ0,loc(y)σ0,loc(0)

)−∫ y

0

dy′σ1,loc(y′)σ0,loc(y′)

∂y′

(y′

σ0(y′)

)2)

Finally, in the forward coordinate, we obtain that the implied volatility at thefirst-order is given by

σBS(K,T ) = σ0(K)

1− T

2(∫K

f0

df ′

f ′σ0,loc(f ′)

)2

ln(

σ0(K)2

σ0,loc(K)σ0,loc(f0)

)−∫ K

f0

σ1,loc(f ′)σ0,loc(f ′)

∂f ′

(ln f ′

f0

σ0(f ′)

)2

df ′

Approximating the last integral by (fav = f0+K

2 )

∫ K

f0

σ1,loc(f ′)σ0,loc(f ′)

∂f ′

(ln f ′

f0

σ0(f ′)

)2

df ′ ' σ1,loc(fav)σ0,loc(fav)

∫ K

f0

∂f ′

(ln f ′

f0

σ0(f ′)

)2

df ′

=σ1,loc(fav)σ0,loc(fav)

(ln K

f0

σ0(K)

)2


we obtain

σBS(K,T ) = σ0(K)

1− T

2(∫K

f0

df ′

f ′σ0,loc(f ′)

)2 ln(

σ0(K)2


)

+T

2σ1,loc(fav)σ0,loc(fav)

)As

1

2(∫K

f0

df ′

f ′σ0,loc(f ′)

)2 ln(

σ0(K)2


)'

C2(fav)24

(2C ′′(fav)C(fav)

−(C ′(fav)C(fav)

)2

+1f2

av

)with C(f) ≡ fσ0,loc(f), we reproduce our previous result (5.40) and thereforejustify the approximations (5.19), (5.20) and (5.34).In the next paragraph, following [18], we derive another expression for theimplied volatility in terms of the LV function. This expression is simplerto use than (5.40) in order to study the large-time behavior of the impliedvolatility.

5.3 Implied volatility from local volatility

The PDEs satisfied by a European call option in the Black-Scholes and LVmodels are

∂CBS(t, f)∂t

+12σ2

BSf2 ∂

2CBS(t, f)∂f2

= 0

∂C(t, f)∂t

+12σloc(t, f)2f2 ∂

2C(t, f)∂f2

= 0

Setting Φ(t, f) ≡ C(t, f)− CBS(t, f) and

ΓBS(t, f) ≡ ∂2CBS(t, f)∂f2

the so-called Gamma of a European call option in the Black-Scholes model,we obtain that Φ satisfies

∂Φ(t, f)∂t

+12σloc(t, f)2f2 ∂

2Φ(t, f)∂f2

(5.46)

+12f2ΓBS(t, f)

(σloc(t, f)2 − σ2

BS

)= 0


subject to the terminal condition φ(T, f) = 0 as the two options have thesame payoff max(fT −K, 0) at the maturity T . By using the Feynman-Kactheorem (with source), the solution at t = 0 of (5.46) can be represented inthe integral form by

Φ(0, f0) =12

∫ T

0

dt

∫ ∞0

dff2(σloc(t, f)2 − σ2

BS

)ΓBS(t, f)p(t, f |f0)

By definition of the implied volatility, we have

Φ(0, f0) = 0

and we obtain the following exact relation between the implied volatility andthe LV function

σ2BS(K,T ) =

∫ T0

∫∞0f2σloc(t, f)2ΓBS(t, f)p(t, f |f0)dtdf∫ T

0

∫∞0f2ΓBS(t, f)p(t, f |f0)dtdf

(5.47)

By using this expression (5.47), we obtain the following approximation be-tween the local and the implied volatilities

PROPOSITION 5.3Assuming that in the forward measure PT , the forward satisfies the driftlessprocess dft = ft (σ + ε(t, ft)) dWt, then the implied volatility can be approxi-mated at the first-order in ε by

σBS(K,T )2 ≈ σ2 + 2σ∫ 1

0

dt

∫ ∞−∞

du√2πe−

u22 ε

(tT, f0e

t ln Kf0

+√t(1−t)σu

)

PROOF At the zero-order in ε, the conditional probability for the forwardis a log-normal distribution

p(t, f |f0) =1

f√

2πtσe−

„ln ff0

+σ2t2

«2

2σ2t

The exact Black-Scholes Gamma is

ΓBS(t, f) =1

f√

2π(T − t)σBS(K,T )e−

ln fK

+σBS(K,T )2(T−t)

2

!2

2σBS(K,T )2(T−t)

By using that at the zero-order σBS(K,T ) = σ, the product of the Gammaand the conditional probability is

f2ΓBS(t, f)p(t, f |f0) =1√

2π(T − t)σe−

ln fK

+σ2(T−t)2

!2

2σ2(T−t)1√

2πtσe−

„ln ff0

+σ2t2

«2

2σ2t

(5.48)


By plugging (5.48) into (5.47), we obtain at the first-order in ε

σBS(K,T )2 = σ2 + 2σ

∫ T0dt∫∞

0df 1√

t(T−t)ε(t, f)e−

ln fK

+σ2(T−t)2

!2

2σ2(T−t) e−

„ln ff0

+σ2t2

«2

2σ2t

∫ T0dt∫∞

0df 1√

t(T−t)e−

„ln fK

+σ2(T−t)2

«2

2σ2(T−t) e−(ln f

f0+σ2t

2 )2

2σ2t

Doing a change of variable y = ln ff0

, we get

σBS(K,T )2 = σ2 + 2σ∫ T0dt∫∞−∞ dy 1√

t(T−t)ε(t, f0e

y)e−( 1T−t+ 1

t )(y− t

Tln Kf0 )2

2σ2

∫ T0dt∫∞−∞ dy 1√

t(T−t)e−( 1

T−t+ 1t )

(y− tT

ln Kf0 )2

2σ2

With a new change of variable from y to u = 1σ

√(1

T−t + 1t

)(y − t

T ln Kf0

),

we get our result.

Example 5.7 Skew averagingWe consider the following local volatility

σloc(t, f) = σ + λ(t) lnf

f0(5.49)

By using the proposition 5.3, we get the implied volatility at the first-orderin λ

σBS(K,T )2 = σ2 +2σT 2

lnK

f0

∫ T

0

tλ(t)dt (5.50)

We try to recover the same implied volatility using a time-independent localvolatility

σloc(t, f) = σ + λ lnf

f0(5.51)

λ can be interpreted as twice the skew S: From the BBF formula (5.39), wehave at the zero-order in the maturity T

σBS(f, T ) = σ + λ lnfav

f0


and

S ≡ K∂KσBS(K,T )|K=f0 =λ

2

From (5.50), the implied volatilities associated to the LV models (5.49) and(5.51) are equivalent at the first-order in λ if and only if

λ =2T 2

∫ T

0

tλ(t)dt

This is called a skew averaging which was previously obtained in [136] usinga different but close enough approach.

Exercises 5.1 Generalized skew averagingLet us consider the LV model

df = f

(σ +

n∑i=1

λi(t)(

lnf

f0

)i)dWt

By using the proposition 5.3, prove that this model produces an equivalentimplied at the first-order in λi that the time-homogeneous LV model definedby

df = f

(σ +

n∑i=1

λi

(lnf

f0

)i)dWt

Specify the parameters λi.

Chapter 6

Stochastic Volatility Models andGeometry of Complex Curves

The shortest path between two truths in the real domain passesthrough the complex domain.— J. Hadamard

Abstract In this chapter we pursue the application of the heat kernelexpansion for stochastic volatility models (SVM). In our geometric framework,a SVM corresponds to a complex curve, also called Riemann surfaces. Byusing the classification of conformal metrics on a Riemann surface, we showthat SVMs fall into two classes. In particular, the SABR model correspondsto the Poincare hyperbolic surface.We derive the first-order asymptotics for implied volatility for any time-homogeneous SVM. This general formula, particularly useful for calibrationpurposes, reproduces and improves the well-known asymptotic implied volatil-ity in the case of the SABR model. This expression only depends on the geo-metric objects (metric, connection) characterizing a specific SVM. We applythis formula to the SABR model with a mean-reverting drift and the Hestonmodel.Finally, in order to show the strength of our geometrical framework, we give anexact solution to the Kolmogorov equation for the normal (resp. log-normal)SABR model. The solutions are connected to the Laplacian heat kernel onthe two-dimensional (resp. three-dimensional) hyperbolic surface.Deterministic interest rates will be assumed throughout this chap-ter.

6.1 Stochastic volatility models and Riemann surfaces

6.1.1 Stochastic volatility models

As previously discussed, a LVM can automatically be calibrated to the initialimplied volatility via the Dupire formula. However, the dynamics producedby the LVM is not very realistic. For example, the LVM process is not time-homogeneous and therefore the dynamics is not invariant by time translation.

149


An alternative to produce a better dynamics for the implied volatility is tointroduce time-homogeneous SVMs.A SVM is defined as a set of two correlated SDEs: one for the forward ftand one for the instantaneous stochastic volatility at. As the latter variable isnot directly observable on the market (this is not a traded asset), the marketmodel is incomplete in opposition to the case of LVMs. As a consequence, therisk-neutral measure P is not unique here according to the theorem 2.8.We define a general (one-factor) SVM in the forward measure PT by theprocess

dft = atC(ft)dWt (6.1)dat = b(at)dt+ σ(at)dZt (6.2)

dWtdZt = ρdt

with Zt and Wt two correlated Brownian motions and with the initial valuesft=0 = f0 and at=0 = α. σ is called by the practitioners the volatility ofvolatility (or in short vol of vol).As usualWt can be decomposed over a basis of uncorrelated Brownian motionsZt, Z⊥t :

Wt = ρZt +√

1− ρ2Z⊥t

Note that as at is not a traded asset, at is not a local martingale and we havea priori a drift term b(a). We assume that b(·) and σ(·) are only functions ofthe volatility process at. We could assume that b(·) and σ(·) depend on theforward ft as well but we do not pursue this route as examples we look at donot exhibit this dependence.Below is a list of commonly used SVMs (see Table 6.1).

Non-explosion and martingality

As in section 5.1.2, we preserve the put-call parity and ensure the abilityto price derivatives using the pricing formula (2.31) if ft defines not only alocal martingale but also a martingale. According to the theorem 5.2, ft isa martingale if and only if the instantaneous (log-normal) volatility processξt = C(ft)

ftat does not explode under the measure Pf associated to the forward

as numeraire.Under PT , we have

dξ = aC(f)∂f

(C(f)f

)dW +

C(f)f

σ(a)dZ (6.3)

+(C(f)f

b(a) + ∂f

(C(f)f

)C(f)aρσ(a) +

a3

2C(f)2∂2

f

(C(f)f

))dt

Stochastic Volatility Models and Geometry of Complex Curves 151

TABLE 6.1: Example of SVMs.Name SDE

Stein-Stein dftft

= atdWt

dat = λ(at − a)dt+ ζdZt , dWtdZt = 0Geometric dft

ft= atdWt

dat = λatdt+ ζatαdZt , dWtdZt = ρdt

3/2-model dftft

= atdWt

da2t = λ(a2

t − va4t )dt+ ζa3

tαdZt , dWtdZt = ρdt

SABR dftft

= atfβ−1t dWt

dat = νatdZ2 , dWtdZt = ρdt

Scott-Chesney dftft

= eyftdWt

dy = λ (ey − y) dt+ ζdZt , dWtdZt = 0Heston dft

ft= atdWt

da2t = λ(v − a2

t )dt+ ζatdZt , dWtdZt = ρdt

and under Pf

dξ = aC(f)∂fC(f)f

dW f +C(f)f

σ(a)dZf + (6.4)(C(f)f

b(a) + ∂f

(C(f)f

)C(f)aρσ(a) +

a3

2C(f)2∂2

f

(C(f)f

)+ξ(aC(f)∂f

(C(f)f

)+ ρ

C(f)f

σ(a)))

dt

For the special case when C(f) = f (ξ = a), the above-mentioned SDEsreduce to

dat = b(at)dt+ σ(at)dZt under PT (6.5)

dat = (b(at) + ρatσ(at)) dt+ σ(at)dZft under Pf (6.6)

In this case the Feller non-explosion criterion 5.3 can be used to check if theprocesses (6.5,6.6) do not explode. We apply this test to the log-normal SABRand Heston models.

Example 6.1 Log-normal SABR modelThe log-normal SABR model [99] is defined by the following SDEs under PT

dft = atftdWt

dat = νatdZt

Using (6.6), the process ξt = at follows under Pf

dat = ρνa2tdt+ νatdZ

ft (6.7)


Therefore, we have

lPT (a) = − 1ν2

(lna

c− a

c+ 1)

(6.8)

lPf (a) =1ν2

∫ ρa

ρc

e−vdv

∫ v

ρc

u−2eudu (6.9)

with ρ = 2ρν ≤ 0. We deduce that at does not explode under PT and a = 0 is

an unattainable boundary. Moreover at does not explode under Pf if and onlyif ρ ≤ 0. In conclusion, the log-normal SABR model defines a martingale ifand only if ρ ≤ 0. This condition is satisfied when we calibrate this model tothe market implied volatility (ρ < 0 and ν > 0). It corresponds to a negativeskew.

Example 6.2 Heston modelThe Heston model [106] is defined by the following SDEs under PT

dft = atftdWt

da2t = λ

(v − a2

t

)+ ζatdZt

By using (6.6), the process vt ≡ a2t follows under Pf

dvt = ((−λ+ ρζ)vt + λv) + ζ√vtdZ

ft (6.10)

Therefore, we have

lPT (v) =1ζ2

∫ v

c

e−2λyζ2 y

2λvζ2 dy

∫ y

c

e2λzζ2 z

− 2λvζ2−1dz

lPf (v) =1ζ2

∫ v

c

e2(−λ+ρζ)y

ζ2 y2λvζ2 dy

∫ y

c

e− 2(−λ+ρζ)z

ζ2 z− 2λv

ζ2−1dz

As lPT (∞) = ∞, we deduce that at does not explode under PT . a = 0 is anunattainable boundary if and only if 2λv > ζ2 as in this case lPT (0) = ∞.Moreover at does not explode under Pf and ft is a true positive martingale.

In the general case (i.e., C(f) 6= f), it is difficult to check that the two-dimensional processes (6.3,6.4) do not explode as the Feller criterion is notapplicable anymore. In this case, one can use the Hasminskii non-explosiontest [38] which gives a sufficient criterion for a multi-dimensional diffusionprocess:


σij(t, xt)dWjt , i = 1, · · · , n (6.11)

with W jt being uncorrelated Brownian motions.


THEOREM 6.1 Hasminskii non-explosion testSuppose that for each T , there exists an r > 0 and continuous functionsAT : [r,∞) → (0,∞) and BT : (r,∞) → (0,∞) such that for ρ > (2r)

12 ,

0 ≤ t ≤ T and√∑n

i=1(xi)2 = ρ,

AT

(ρ2

2

)≥

n∑i,j=1

m∑k=1

xiσikσjkx

j

n∑i,j=1

m∑k=1

xiσikσjkx

jBT

(ρ2

2

)≥

n∑i=1

m∑k=1

σikσik + 2

n∑i=1

xibi(t, x)

and ∫ ∞r

CT (ρ)−1dρ

∫ ρ

r

CT (σ)AT (σ)−1dσ =∞

where

CT (ρ) = eR ρrBT (σ)dσ

Then the process xt (6.11) does not explode.

In our geometrical framework, a SVM corresponds to a two-dimensional Rie-mannian manifold, called also a Riemann surface. Surprisingly, there is aclassification of Riemann surfaces, corresponding to a classification of SVMs.

6.1.2 Riemann surfaces

On a Riemann surface we can show that a metric can always be locally writtenin a neighborhood of a point (using the so-called isothermal coordinates)

gij(x) = eφ(x)δij , i, j = 1, 2 (6.12)

and therefore a metric is locally conformally flat. The coordinates x = xiare called the isothermal coordinates (see example 4.8). Furthermore, twometrics on a Riemann surface, gij and hij (in local coordinates), are calledconformally equivalent if there exists a (globally defined) smooth functionφ(x) such that

gij(x) = eφ(x)hij(x)

The following theorem follows from the observations above:

THEOREM 6.2 UniformizationEvery metric on a simply connected Riemann surface1 is conformally equiv-

alent to a metric of constant scalar curvature R:

1The non-simply connected Riemann surfaces can also be classified by taking the doublecover.


1. R = +1: the Riemann sphere S2.

2. R = 0: the complex plane C.

3. R = −1: the upper half plane H2 = z ∈ C | =(z) > 0, called also thePoincare hyperbolic surface.

By the uniformization theorem, all surfaces fall into one of these three types.We conclude that there are a priori three types of SVMs (modulo the confor-mal equivalence). If we discard the two-sphere S2 corresponding to a compactmanifold, we are left with two classes: the flat manifold C and the upper half-plane H2. The SVMs corresponding to H2 and C envelop these universalityclasses and thus are generic frameworks allowing all possible behaviors of im-plied volatility.The metric associated to a SVM defined by (6.1,6.2) is (using (4.54))

ds2 = gijdxidxj

=2

a2(1− ρ2)

(df2

C(f)2− 2ρ

adfda

C(f)σ(a)+a2da2

σ(a)2

)Let us introduce the variable q(f) =

∫ ff0

df ′

C(f ′) and ξ(a) =∫ a·

uσ(u)du. We have

ds2 =2

a2(1− ρ2)(dq2 − 2ρdqdξ + dξ2

)Completing the square with the new coordinates

x = q(f)− ρξ(a) (6.13)

y = (1− ρ2)12 ξ(a) (6.14)

the metric becomes in the coordinates [x, y] (i.e., isothermal coordinates)

ds2 = eφ(y)(dx2 + dy2) (6.15)

with the conformal factor

F (y) ≡ eφ(y) =2

a(y)2(1− ρ2)(6.16)

Note that this metric exhibits a Killing vector ∂x as the conformal factordoes not depend on the coordinate x. In the example 4.8, we have computedexplicitly the geodesic distance for such metrics. We reproduce the resulthere:The geodesic distance d between two points (x1, y1) and (x2, y2) is

d =∫ y2

y1

F (y′)dy′√F (y′)− C2

(6.17)


TABLE 6.2: Example of metrics for SVMs. ∼means modulo a multiplicative constant factor.

Name Conformal factor Scalar curvatureGeometric F (y) ∼ y−2 R = −1

3/2-model F (y) ∼ e−2y√1−ρ2 R = 0

SABR F (y) ∼ y−2 R = −1Heston F (y) ∼ y−1 R = −2a−2 < 0

with the constant C = C(x1, y1, x2, y2) determined by the equation

x2 − x1 =∫ y2

y1

C√F (y′)− C2

dy′ (6.18)

Using the results from exercise 4.1, the Ricci and scalar curvatures are givenby

Rij = −F (y)F ′′(y)− (F ′(y))2

2F (y)2δij

R = −F (y)F ′′(y)− (F ′(y))2

F (y)3

Plugging the expression (6.16) for F in the equations above, we obtain

Rij =σ(a)2

(1− ρ2)a3

(σ′(a)σ(a)

− 2a

)δij

R =σ(a)2

a

(σ′(a)σ(a)

− 2a

)(6.19)

In particular, for a volatility of volatility given by σ(a) = ap , we have

Rij =1

(1− ρ2)a2p−4(p− 2)δij

R = a2p−2(p− 2) (6.20)

and the conformal factor is

F (y) =2(1− ρ2)

p−12−p y

2p−2

(2− p)2

2−p, ∀p 6= 2

=2

(1− ρ2)e−2y√1−ρ2 , p = 2

Table 6.2 shows the conformal factors and the curvatures associated to theSVMs listed in table 6.1.


The metric associated to the SABR model is the metric on H2 and the 3/2-model corresponds to the metric on C ≈ R2. The scalar curvature is alwaysnon-negative and therefore we do not have a SVM exhibiting the metric onS2. A SVM is therefore connected to a non-compact Riemann surface.

REMARK 6.1 Heston model The most commonly used SVM, theHeston model, has a negative scalar curvature diverging at a = 0. This is atrue singularity2 and the manifold is not complete.The heat kernel expansion explained in chapter 4 is no more applicable as longas the volatility process can reach the singularity a = 0. As seen in example6.2, we see that a = 0 is not reached if and only if

2λv > ζ2

Cut-locus and Cartan-Hadamard manifold

In our discussion on the heat kernel, we have seen that the short-time behaviorof the fundamental solution p(t, x|y) to the heat kernel is valid when x andy are not on each other cut-locus. Surprising, for commonly used SVMs, theunderlying manifold is a Cartan-Hadamard manifold for which the cut-locusis empty.

DEFINITION 6.1 Cartan-Hadamard manifold Let M be a Riemannsurface. If the scalar curvature3 is non-positive then M is called a Cartan-Hadamard manifold. For such a manifold, the cut-locus is empty [22].

It means that the map exp as defined in (4.51) realized a diffeomorphismof Rn to M . Two points on M can then be joined by a unique minimizinggeodesic. As shown by the expression for the scalar curvature (6.19), a SVM isa Cartan-Hadamard manifold (assuming that there is no singularity at a = 0)if the volatility of volatility satisfies the following inequality

aσ′(a)σ(a)

≤ 2

In the next section, we explain the link between LV and SV models.

2The Heston model behaves as an (Euclidean) black hole.3For a general n-dimensional manifold, the scalar curvature is replaced by the sectionalcurvature.


6.1.3 Associated local volatility model

For a generality purpose, we assume that the forward follows the driftlessprocess (in the forward measure PT )

dft = ftσtdWt (6.21)

where σt is a Ft adapted process that can depend on time t, on the forwardft and other Markov processes noted generally at. This model can be viewedas a general SVM. In our case, we have assumed that σt = at

C(ft)ft

.Applying the Ito-Tanaka formula on the payoff max(ft −K, 0), we obtain

dmax(ft −K, 0) = 1(ft −K)ftσtdWt +12f2t σ

2t δ(ft −K)dt

Then, taking the mean value operator EPT [·|F0] on both sides of this equation,we have

dEPT [max(ft −K, 0)|F0] =K2

2EPT [σ2

t δ(ft −K)|F0]dt (6.22)

We recall that for the Dupire LVM, we have found in the previous chapter

dEPT [max(ft −K, 0)|F0] =K2σ2

loc(t,K)2

EPT [δ(ft −K)|F0]dt

(6.23)

Comparing (6.22) and (6.23), we obtain that the square of the Dupire localvolatility function is equal to the mean of the square of the stochastic volatilitywhen the forward is fixed to the strike

σ2loc(t,K) =

EPT [σ2t δ(ft −K)|F0]

EPT [δ(ft −K)|F0]

≡ EP[σ2t |ft = K] (6.24)

This derivation obtained by Dupire in [81] can be put on a rigorous groundand we obtain

THEOREM 6.3 Gyongy [98]Let Xt = Xi

ti=1,···n be an n-dimensional Ito process

dXit = bitdt+

m∑i=1

σit,jdWjt

where bit and σt ≡ [σij ] are bounded Ft-adapted processes. We assume that σtsatisfies the uniform ellipticity condition

∃C ∈ R∗+ such as σtσ†t ≥ C1


with † the transpose and 1 the m×m identity matrix.Then there exists bounded measurable functions Σ : R+×Rn → Mm,n(R) andB : R+ × Rn → Rn defined ∀(t, x) ∈ R+ × Rn by

Σ(t, x)Σ(t, x)† = E[σtσ†t |xt = x]

B(t, x) = E[bt|xt = x]

such that the following SDE

dxit = Bi(t, xt)dt+m∑j=1

Σ(t, xt)ijdWj , x0 = X0

has a weak solution with the same one-dimensional marginals as Xt.

REMARK 6.2 For most commonly used SVMs, the boundness and el-lipticity conditions are not satisfied. This technical complication will be over-looked in the following.

The formula (6.24) involving the Dirac function means

EP[σ2t |ft = K] ≡ lim

ε→0+

EP[σ2t 1(ft −K ∈ (−ε, ε))]

EP[1(ft −K ∈ (−ε, ε))]

with 1(·) the Heaviside function. Assuming that σt = σ(t, ft, at) with at aIto process, then by definition of the conditional mean value, the expressionabove becomes

σloc(t,K)2 =

∫∞0σ(t,K, a)2p(t,K, a|f0, α)da∫∞

0p(t,K, a|f0, α)da

where p(t,K, a|f0, α) is the conditional probability associated to the process(6.21). Therefore we have shown that a SVM can be calibrated exactly tothe initial implied volatility if the stochastic volatility satisfies (6.24). Equiv-alently, a LV model and a SV model, which satisfies the relation (6.24), havethe same marginals.For example, let us assume that σt = at

C(t,ft)ft

with at a Ito process andC(t, ft) a function depending on the forward and the time. The impliedvolatility is matched if C(t, ft) satisfies

K2σloc(t,K)2 = C(t,K)2EPT [a2t |ft = K]

This model, although automatically calibrated to the IV with

C(t,K)2 =K2σloc(t,K)2

EPT [a2t |ft = K]


without specifying the dynamics for the process at, is however not time-homogeneous. Consequently we impose that a general SVM is defined bythe processes (6.1) and (6.2).In the next paragraph, using our relation between a LVM and a SVM, weobtain an asymptotic implied volatility for a general SVM defined by (6.1)and (6.2). Before explaining the derivation, we state the result.

6.1.4 First-order asymptotics of implied volatility

The general asymptotic implied volatility at the first-order for any time-homogeneous SVM, depending implicitly on the metric gij (4.54) and theconnection Ai (4.58) on our Riemann surface, is given by

σBS(K,T ) =ln K

f0∫Kf0

df ′√2gff (amin)

(1 (6.25)

+gff (amin)T

12

(−3

4

(∂fg

ff (amin)gff (amin)

)2

+∂2fgff (amin)

gff (amin)+

1fav2

)

+gff

′(amin)T

2gff (amin)φ′′(amin)

(ln(∆gP2)′(amin)− φ′′′(amin)

φ′′(amin)+gff

′′(amin)

gff ′(amin)

))

with fav ≡ f0+K2 and with amin the volatility a which minimizes the geodesic

distance d(a, fav|α, f0) on the Riemann surface (φ = d2). ∆ is the Van Vleck-Morette determinant (4.68), g is the determinant of the metric and P is theparallel gauge transport (4.69). The prime symbol ′ indicates a derivativeaccording to a. The formula (6.25) is particularly useful as we can use itto rapidly calibrate any SVM. In section 6.3 (resp. 6.5), we apply it to theλ-SABR model (resp. Heston model). In order to use the formula, the onlycomputation needed is the calculation of the geodesic distance. This wasachieved for our class of SVMs in equation (6.17).

PROOF A time-homogeneous SVM is coded by a metric g on a Riemannsurface, an Abelian connection A and a section Q. This is the necessarydata to apply the heat kernel expansion and deduce from it the asymptoticformula for the probability density at the first-order. We obtain (x = (f0, α),y = (f, a))

p(τ, x|y) =

√g(y)

(4πτ)

√∆(y, x)P(y, x)e−

d2(x,y)4τ (1 + a1(y, x)τ) (6.26)

The computation of an asymptotic expression for the implied volatility in-volves two steps. The first step consists in computing the LV function σ(T, f)associated to the SVM. In the second step, we deduce the implied volatilityfrom the LV function.


As seen previously (6.24), the mean value of the square of the SV when theforward is fixed is the LV given by

σ(T, f)2 = C(f)2EPT [a2T |fT = f ]

= f2σloc(T, f)2

By definition of the mean value, we have

σ(T, f)2 =2∫∞

0gff (f, a)p(T, f0, α|f, a)da∫∞0p(T, f0, α|f, a)da

(6.27)

p(T, x|y) is the conditional probability given in the short-time limit at thefirst-order by (6.26). We set φ(x, y) = d(x, y)2. By plugging our asymptoticexpression for the conditional probability (6.26) in (6.27), we obtain

σ(T, f)2 =

∫∞0f(T, a)eεφ(a)da∫∞

0h(T, a)eεφ(a)da

(6.28)

with φ(a) = d2(x, y), h(T, a) =√g√

∆(x, y)P(y, x)(1 + a1(y, x)T ), f(T, a) =2h(T, a)gff and ε = − 1

4T . Using the saddle-point method (see appendix A),we find an asymptotic expression for the LV:At the zero-order, σ2 is given by

σ(0, f)2 = 2gff (amin)

with amin the stochastic volatility which minimizes the geodesic distance onour Riemann surface:

amin ≡ a | minaφ(a) (6.29)

At the first-order, we find the following expression for the numerator in (6.28)(see Appendix A for a sketch of the proof)∫ ∞

0

f(T, a)eεφ(a)da =

√2π

−εφ′′(amin)f(T, amin)eεφ(amin) (1

+1ε

(− f ′′(0, amin)

2f(0, amin)φ′′(amin)+

φ(4)(amin)8φ′′(amin)2

+f ′(0, amin)φ′′′(amin)2f(0, amin)φ′′(amin)2

− 5(φ′′′(amin))2

24(φ′′(amin))3

))The prime symbol ′ indicates a derivative according to a. Computing thedenominator in (6.28) in a similar way, we obtain the first-order correction tothe LV

σ(T, f)2 = 2gff (amin) (1

+1ε

(− 1

2φ′′(amin)

(f ′′(amin)f(amin)

− h′′(amin)h(amin)

)+

φ′′′(amin)2φ′′(amin)2

(f ′(amin)f(amin)

− h′(amin)h(amin)

)))


Plugging the expression for f and g, we finally obtain

σ(T, f) =√

2gff (amin) (1+

T

φ′′(amin)

(gff

′(amin)

gff (amin)

(ln(∆gP2)′(amin)− φ′′′(amin)

φ′′(amin)

)+gff

′′(amin)

gff (amin)

))This expression depends solely on the metric and the connection A on our Rie-mann surface and not on the first heat kernel coefficient a1(y, x) the contribu-tion of which has disappeared when we have taken the ratio of the numeratorand denominator.The final step is to use the (asymptotic) relation between a LV function σ(t, f)and the implied volatility (5.40) that we have obtained in the previous chapterusing the heat kernel expansion on a time-dependent one-dimensional real line.This gives (6.25).

REMARK 6.3 Boundary conditions Note that in our proof, we havedisregarded the boundary conditions. In the heat kernel expansion, theseboundary conditions only affect the heat kernel coefficients [92]. As a result,our formula (6.25) does not depend on some specific boundary conditions.

Computation of the saddle-point amin

By definition, the saddle-point amin minimizes the geodesic distance when theforward and the strike are fixed: ∂d(amin)

∂a = 0. From the exact expression(6.17) for the geodesic distance, we obtain

F (ymin)√F (ymin)− C(ymin)2

+∫ ymin

y0

F (y)(F (y)− C(ymin))

32C(ymin)∂yC(ymin) = 0

(6.30)

where ymin =√

1− ρ2ζ(amin). Differentiating the equation (6.18) with re-spect to a, we get

C(ymin)√F (ymin)− C(ymin)2

+ ∂yC(ymin)(∫ ymin

y0

F (y)dy(F (y)− C(ymin))

32

)= − ρ√

1− ρ2

Using (6.30), the equation above simplifies and reduces to

C(ymin)2 = F (ymin)(1− ρ2

)and therefore from (6.18), amin satisfies the following equation

q(K)− ρ (ζ(amin)− ζ(α)) = ± a2min√

1− ρ2

∫ √1−ρ2

√1−ρ2αamin

u2

σ

(amin√1−ρ2

u

)√1− u2

du

(6.31)


6.2 Put-Call duality

Let us have a look at the general duality that the implied volatility associatedto a SVM should satisfy. It turns out that this relation allows to constrainthe asymptotic series (in time) for the implied volatility.In the forward measure PT , a European call option, with strike K and matu-rity T , is given by

C(T,K|f0) = P0TEPT [max(fT −K, 0)]

which can be rewritten as

C(T,K|f0) = P0TKEPT [fT max(1K− 1fT, 0)] (6.32)

Using the measure Pf associated to the martingale ft (6.1), the expressionabove becomes

C(T,K|f0) = P0T f0KEPf [max(1K− 1fT, 0)] (6.33)

Setting Xt = 1ft

and applying Ito’s lemma, we have that Xt satisfies in themeasure PT

dXt

Xt= a2

t

(C(ft)ft

)2

dt− atC(ft)ft

dWt

Thanks to a change of measure, Xt satisfies the SDE in the measure Pf

dXt = −atX2t C(

1Xt

)dWt

dat =(b(at) +XtC(

1Xt

)ρatσ(at))dt+ σ(at)dZt

It does not come as a surprise that Xt is a martingale: As the product of Xt

with ft is 1, Xt is driftless in the measure associated to the numeraire ft (i.e.,Pf ).The forementioned SDEs for Xt and at define a SVM, dual to the SVM definedby (6.1, 6.2). It will be called the SV model II, and (6.1,6.2) the SV model I.From the relation (6.33), we obtain

CI(T,K|f0) = f0KPII(T,1K| 1f0

) (6.34)

with C(T,K|f0) (resp. P(T,K|f0)) the fair value of a call (resp. put) option.From the Black-Scholes formula (3.1), we deduce that the implied volatilities


for the SV models I and II are the same when the strike K and the forwardf0 are inverted

σIBS(T,K|f0) = σII

BS(T,1K| 1f0

) (6.35)

REMARK 6.4 Zero-correlation case When the correlation is zeroand C(f) = f (i.e., conditional on a, we have a log-normal process for theforward), the processes ft and Xt satisfy the same SDE (except that dWt

is changed into −dWt). Moreover, in this case, the implied volatility is afunction of the moneyness y = ln(Kf0

) as shown in 3.2. Therefore, the relation(6.35) reduces to

σBS(T, y) = σBS(T,−y) (6.36)

meaning that the implied volatility is a symmetric function in y.

REMARK 6.5 Put-Call duality for LVMs For a LVM (a = 1), theput-call duality becomes

σIBS(T,K|f0) = σII

BS(T,1K| 1f0

)

where I (resp. II) is associated to the LV model dft = C(ft)dWt (resp. dft =f2t C( 1

ft)dWt). As a check of our asymptotic relation between a local volatility

and an implied volatility, we can show that the relation (5.40) is preservedunder the put-call duality. In particular this is the case for the BBF relation(5.39) and the potential Q. More generally, the implied volatility asymptoticsseries depend on quantities invariant under the (projective) transformation

C(f)→ f2C(1f

)

f0,K →1f0,

1K

The invariant projective functions have been classified in [2] and are

I2 = 2C ′′C − C ′2 , I3 = 2C ′′′C2 , · · · , In = CI ′n−1

I2 is precisely the potential Q.

REMARK 6.6 Put-Call symmetry for LVMs We assume that thelocal volatility satisfies the relation

f2C(1f

) = C(f) (6.37)


In this case, models I and II are identical and we obtain the so-called put-callsymmetry

CI(T,K|f0) = f0KPI(T,1K| 1f0

) (6.38)

As an application, we can derive a closed form formula for a down-and-outcall option, DO(T,K,B|f0), in this model. We recall that a down-and-outcall pays a call option with strike K at maturity T if the barrier level B isnot reached by the asset during the interval [0, T ]. Here, we take f0 > B andK > B. We have

DO(T,K,B|f0) = CI(T,K|f0)− f0

BCI(T,K|B

2

f0)

The reader can check as an exercise (see the similar exercise 2.8) that

DO(t, f) ≡ CI(T,K|t, f)− f

BCI(T,K|t, B

2

f)

is indeed a solution of the Black-Scholes PDE:

∂tDO +12C(f)2∂fDO = 0 , ∀f > B

subject to C(·) satisfying (6.37). Moreover the boundary conditions

DO(T, f) = max(K − f, 0)1f>B

and DO(t, B) = 0 are satisfied: Indeed, if the asset does not reach the barrierB then the second call CI(T,K|B

2

f0) is worthless. Furthermore, if the asset

reaches the barrier, the two calls cancel.Finally, using (6.38), we obtain a static replication (i.e., model-independentmodulo the relation (6.37)) for the value of a down-and-out call

DO(T,K,B|f0) = CI(T,K|f0)− K

BPI(T,

B2

K|f0)

6.3 λ-SABR model and hyperbolic geometry

We introduce the λ-SABR SVM. This model corresponds to the hyperbolicsurface H2. By applying the formula (6.25), we obtain an asymptotic impliedvolatility.


6.3.1 λ-SABR model

The volatility at is not a tradable asset. Consequently, at can have a drift inthe forward measure PT . A popular choice is to make the volatility processmean-reverting. We investigate the λ-SABR model defined by the followingSDE (this model has also been considered in [60] with C(f) = f)

dft = atC(ft)dWt

dat = λ(at − λ)dt+ νatdZt

C(f) = fβ , a0 = α , ft=0 = f0

where Wt and Zt are two Brownian processes with correlation ρ ∈ (−1, 1).The model depends on 6 (constant) parameters: α, β, λ, λ, ν and ρ. Forλ = 0, the λ-SABR model degenerates into the SABR model introduced in[100].

Conditional to the volatility at, ft is a CEV process. From lemma 5.1, wehave that for β < 1, ft can reach 0 with a positive probability. In orderto preserve the martingality condition (β ≤ 1/2) we assume an absorbingboundary condition at 0 as usual.

In the following section, we present our asymptotic smile for the λ-SABRmodel and postpone the derivation to the next section.

6.3.2 Asymptotic implied volatility for the λ-SABR

The asymptotic implied volatility (with strike f , maturity date T and forwardf0) at the first-order associated to the stochastic λ-SABR model is

σBS(f, T ) =ln f

f0

vol(f)

(1 + σ1

(f + f0

2

)T

)(6.39)

with

σ1(f) =(C(f)amin(f))2

24

(1f2

+2∂ff (C(f)amin(f))

C(f)amin(f)−(∂f (C(f)amin(f))C(f)amin(f)

)2)

+αν2 ln(P)′(f)(1− ρ2) sinh(d(f))

2d(f)


with

vol(f) =1ν

ln

(qν + αρ+

√α2 + q2ν2 + 2qανρα(1 + ρ)

)amin(f) =

√α2 + 2ανρq + ν2q2

d(f) = cosh−1

(−qνρ− αρ2 + amin(f)

α(1− ρ2)

)q =

f (1−β) − f (1−β)0

(1− β)(β 6= 1); , q = ln

f

f0(β = 1)

Besides, we have

ln(P

PSABR

)′(f) =

λ

ν2

(λ(α− amin)aminα

+ lnamin

α(6.40)

− ρ√1− ρ2

(G0(θ2, A0)θ′2 −G0(θ1, A0)θ′1 +A′0 (G1(θ2)−G1(θ1)))

)with

ln(PSABR

)′(f) = − βρ

2√

1− ρ2(1− β)(F0(θ2, A,B)θ′2

−F0(θ1, A,B)θ′1 −A′ (F1(θ2, A,B)− F1(θ1, A,B)))

and with

G1(x) =(

ln(

tanx

2

))G0(x, a) = 1 +

a

sinx

A0 = − λ√

1− ρ2

amin, A′0 =

λ√

1− ρ2(αρ+ νq)a2

min(νq + ρ(α− amin))

F0(x, a, b) =cos(x)

a+ cos(x) + b sin(x)

F1(x, a, b) =∫ x cos(θ)

(a+ cos(θ) + b sin(θ))2 dθ

tan θ2 = −α√

1− ρ2

αρ+ νq, θ′2 =

√1− ρ2

νq + ρ(α− amin)

tan θ1 = −√

1− ρ2

ρ, θ′2 = θ′1

A =f1−βν

(1− β)amin, A′ =

fν(αρ+ νq) + fβ(β − 1)a2min

(β − 1)fβa2min (νq + ρ (α− amin))

B =ρ√

1− ρ2


Note that although F1(x, a, b) can be integrated analytically, we do not repro-duce this lengthy expression here.

6.3.3 Derivation

In order to use our general formula for the implied volatility (6.25), we com-pute the geodesic distance and the connection associated to the λ-SABRmodel in the next subsection. We show that the λ-SABR metric is diffeo-morphic equivalent to the metric on H2, the so-called hyperbolic Poincareplane.

Hyperbolic Poincare plane

Let us introduce the variable q =∫ ff0

df ′

C(f ′) . As shown in paragraph (6.1.2),by introducing the new coordinates

x = νq − ρa (6.41)

y = (1− ρ2)12 a (6.42)

the metric is the standard hyperbolic metric on the Poincare half-plane H2 inthe coordinates [x, y]

ds2 =2ν2

dx2 + dy2

y2(6.43)

The unusual factor 2ν2 in front of the metric (6.43) can be eliminated by scaling

the time τ ′ = ν2

2 τ in the heat kernel (4.56) (and Q becomes 2ν2Q). This is

what we will use in the following.As the connection between the λ-SABR model and H2 is quite intriguing, weinvestigate several useful properties of the hyperbolic space (for example thegeodesics).

Isometry PSL(2,R)

By introducing the complex variable z = x+ ıy, the metric becomes

ds2 =dzdz

=(z)2

In this coordinate system, it can be shown that PSL(2,R)4 is an isometry,meaning that the distance is preserved. The action of PSL(2,R) on z istransitive and given by

z′ =az + b

cz + d

4PSL(2,R) = SL(2,R)/Z2 with SL(2,R) the group of two by two real matrices with deter-minant one. Z2 acts on A ∈ SL(2,R) by Z2A = −A.


By transitive we mean that given two complex points z and z′, there exists aunique element A ∈ PSL(2,R) such that z′ = A.z.Note that when the real coefficients (a, b, c, d) are replaced by (−a,−b,−c,−d),the action is unchanged and this is why we have taken the quotient of the Liegroup SL(2,R) by the discrete group Z2 to get a transitive action.

Models and coordinates for the hyperbolic plane.

There are four models (i.e., geometric representations) commonly used forhyperbolic geometry: the Klein model, the Poincare disc model, the upperhalf-plane model and the Minkowski model. We explain below the relationbetween the last three ones.Let us define a Moebius transformation T as an element of PSL(2,R) which isuniquely given by its values at 3 points: T (0) = 1, T (i) = 0 and T (∞) = −1.If =(z) > 0 then |T (z)| < 1 so T maps the upper half-plane on the Poincaredisk

D = z = x+ ıy ∈ C | |z| ≤ 1

Then if we define x0 = 1+|z|21−|z|2 , x1 = 2<(z)

1−|z|2 , x3 = 2=(z)1−|z|2 , we obtain that D is

mapped to the Minkowski pseudo-sphere

−x20 + x2

1 + x22 = −1

On this space, we have the Lorentzian metric

ds2 = −dx20 + dx2

1 + dx23

We can then deduce the induced metric. On the upper-half plane, this gives(6.43) (without the scale factor 2

ν2 ).

Geodesics

In the upper half-plane, the geodesics correspond to vertical lines and tosemi-circles centered on the horizon =(z) = 0, and in the Poincare disk D thegeodesics are circles orthogonal to D (Fig. 6.1).In the example 4.7, we have obtained that the geodesic distance (invariantunder PSL(2,R)) between the complex points z = x+ ıy, z′ = x′ + ıy′ on H2

is given by

d(z, z′) = cosh−1

(1 +|z − z′|2

2yy′

)(6.44)

This specific expression for the geodesic distance (6.44) allows us to find thatthe saddle-point amin which minimizes the geodesic distance when the forwardis fixed to the strike has the following expression

amin(f) =√α2 + 2ανρq + ν2q2 (6.45)


Poincare Disk

Upper half plane

Moebius Transformation

Geodesics

Geodesic

Geodesic

FIGURE 6.1: Poincare disk D and upper half-plane H2 with somegeodesics. In the upper half-plane, the geodesics correspond to vertical linesand to semi-circles centered on the horizon =(z) = 0 and in D the geodesicsare circles orthogonal to D.

Finally, by using this explicit expression for the geodesic distance, the VanVleck-Morette determinant is

∆(z, z′) =d(z, z′)

sinh(d(z, z′))

As ∆ only depends on the geodesic distance d and d is minimized for a = amin,we have

(ln ∆)′(amin) = 0


Furthermore, we have the following identities (using Mathematica)

d(amin) = cosh−1

(−qνρ− αρ2 + amin

α(1− ρ2)

)(6.46)

aminφ′′(amin) =

2d(amin)α(1− ρ2) sinh(d(amin))

(6.47)

φ′′′(amin)φ′′(amin)

= − 3amin

(6.48)

(ln g)′(amin) = − 4amin

(6.49)

This formula (6.46) has already been obtained independently in [60].

Connection for the λ-SABR

In the coordinates [a, f ], the connection A is

A =1

2(1− ρ2)

(2λλρ− 2λρa− νa2C ′(f)

νC(f)a2df +

−2λλ+ 2λa+ νρa2C ′(f)ν2a2

da

)(6.50)

In the new coordinates [x, y], the connection above is given by

A = ASABR +λ(λ√

1− ρ2 − y)

ν2y2√

1− ρ2

(ρdx−

√1− ρ2dy

)with ASABR the connection for the SABR model given by

ASABR = − d lnC(f)2(1− ρ2)

+ρβ

2√

1− ρ2(1− β)

dy(x+ ρ√

1−ρ2y + νf1−β

0(1−β)

) for β 6= 1

ASABR = − dx

2(1− ρ2)νfor β = 1

The pullback of the connection on a geodesic curve C satisfying (x−x0(a, f))2+y2 = R2(a, f) is given by (β 6= 1)5

i∗A = i∗ASABR +λ(−λ

√1− ρ2 + y)ν2y2

(ρy√

(R2(a, f)− y2)(1− ρ2)+ 1

)dy

and with

i∗ASABR =ρβ

2√

1− ρ2(1− β)

dy

(x0(a, f) +√R2(a, f)− y2 + ρ√

1−ρ2y)

− d lnC(f)2(1− ρ2)

5The case β = 1 will be treated in the next section.


with i : C → H2 the embedding of the geodesic C on the Poincare plane andx0 = x0 + νf1−β

0(1−β) . We have used that i∗dx = − ydy√

R2−y2. Note that the two

constants x0 and R are determined by using the fact that the two pointsz1 = −ρα + i

√1− ρ2α and z2 = ν

∫ ff0

df ′

C(f ′) − ρa + i√

1− ρ2a pass throughthe geodesic curves. The algebraic equations giving R and x0 can be exactlysolved:

x0(a, f) =x2

1 − x22 + y2

1 − y22

2(x1 − x2)

R(a, f)2 = y21 +

((x1 − x2)2 − y2

1 + y22

)24(x1 − x2)2

Using the polar coordinates x − x0 = R cos(θ), y = R sin(θ), we obtain thatthe parallel gauge transport is

ln(P)(a) =λ

ν2

(∫ y2

y1

λ√

1− ρ2 − yy2

dy − ρ√1− ρ2

∫ θ2

θ1

(1 +

A0

sin(θ)

)dθ

)+ ln(PSABR)(a)

with

lnPSABR(z, z′) = − βρ

2√

1− ρ2(1− β)

∫ θ2

θ1

cos(θ)dθ(cos(θ) + x0

R +B sin(θ))

+1

2(1− ρ2)lnC(f)C(f0)

with θi(a, f) = arctan(

yixi−x0

), i = 1, 2, B = ρ√

1−ρ2and A0 = − λ

√1−ρ2

R .

The two integration bounds θ1 and θ2 explicitly depend on a and the coefficientA ≡ x0

R . Doing the integration over θ, we obtain (6.40). Note that accordingto remark 4.3, we have interchanged f0 (resp. α) with f (resp. a).By plugging all these results (6.40,6.46,6.47,6.48, 6.49) in (6.25), we obtain ourfinal expression for the asymptotic smile (6.39) at the first-order associatedto the stochastic λ-SABR model.Below, we have plotted the probability density for the λ-SABR (see Fig. 6.2).Note that the density remains positive.

REMARK 6.7 SABR original formula We can now see how theHagan-al asymptotic smile [100] formula can be obtained in the case λ = 0and show that our formula gives a better approximation.First we approximate amin (6.57) by the following expression

amin ' α+ qρν


Lambda=-10%, lambdabar=0%, f0=5.37%, tau=5 y, nu=27.48%, rho=11.48%, beta=30%

0

10

20

30

40

50

60

2% 3% 4% 5% 6% 7% 8% 9% 10%

AsymptoticPDEHagan

FIGURE 6.2: Probability density p(K,T |f0) = ∂2C(T,K)∂2K . Asymptotic so-

lution vs numerical solution (PDE solver). The Hagan-al formula has beenplotted to see the impact of the mean-reverting term. Here f0 is a swap spotand α has been fixed such that the Black volatility αfβ−1

0 = 30%.

In the same way, we have

sinh(d(amin))d(amin)

' 1

Furthermore, for λ = 0, the connection (6.50) reduces to

ASABR =1

2(1− ρ2)

(−d ln(C(f)) +

ρ

ν∂fCda

)Therefore, the parallel gauge transport is obtained by integrating this one-form along a geodesic C

PSABR = e1

2(1−ρ2)

“− ln

C(f)C(f0) +

RCρν ∂fCda

”


The component Af of the connection is an exact form and therefore haseasily been integrated. The result doesn’t depend on the geodesic curve butonly on its endpoints. However, this is not the case for the component Aa.By approximating (fav = f0+f

2 ) ∂′fC(f ′) ' ∂fC(fav) , the component Aabecomes an exact form and can therefore be integrated by∫

C

ρ

ν∂fCda '

ρ

ν∂fC(fav)(a− α)

Finally, by plugging these approximations into our formula (6.25), we repro-duce the Hagan-al original formula [100]

σBS(K,T ) =ln K

f0

vol(K)

(1 + σ1(

K + f0

2)T)

(6.51)

with

σ1(f) =(αC(f))2

24

(1f2

+2∂ffC(f)C(f)

−(∂fC(f)C(f)

)2)

+αν∂fC(f)ρ

4+

2− 3ρ2

24ν2

Therefore, the Hagan-al formula corresponds to the approximation of theAbelian connection by an exact form. The latter can be integrated outsidethe parametrization of the geodesic curve.

In the following graphs (see Figs. 6.3, 6.4, 6.5) we have compared the accu-racy of our formula (6.39) (denoted H2 formula) against the Hagan-al formula(6.51) using a Monte-Carlo pricer. As the vol of vol ν is large (i.e., ν = 100%),we are outside the domain of validity of the perturbation expansion when thematurity τ is greater than two years. In this case, the dimensionless param-eter ν2τ

2 is greater than 1. We remark that our asymptotic formula performsbetter than the Hagan-al expansion. In this context, we have observed ex-perimentally that in our asymptotic expansion (6.39), C(f)amin should bereplaced by C(f) (α+ qρν) if q > 0. This is what we have used.

REMARK 6.8 H2-model In the previous section, we have seen that theλ-SABR model corresponds to the geometry of H2. This space is particularlynice in the sense that the geodesic distance and the geodesic curves are knownanalytically. A similar result holds if we assume that C(f) is a general function(C(f) = fβ for λ-SABR). In the following, we will try to fix this arbitraryfunction in order to fit the short-term smile. In this case, we can use ourgeneral asymptotic smile formula at the zero-order: The short-term smile will


Implied Vol 1Year

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0.50

0.60

0.70

0.80

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Strike

Hagan-alH^2MC

FIGURE 6.3: Implied volatility for the SABR model τ = 1Y. α = 0.2,ρ = −0.7, ν2

2 τ = 0.5 and β = 1.

be automatically calibrated by construction if the short-term local volatilityσloc(f) is

σloc(f) = C(f)amin(f) (6.52)

with

amin(f)2 = α2 + 2ρανq + ν2q2

q =∫ f

f0

df ′

C(f ′)

By short-term, we mean a maturity date less than 1 year in practice. Solving(6.52) according to q, we obtain

νq = −ραν +

√α2(−1 + ρ2) +

σloc(f)2

C(f)2

and if we derive under f , we have ( ψ(f) ≡ σloc(f)C(f) )

dψ√ψ2 − α2(1− ρ2)

=ν

σloc(f)df


Implied Vol 5Year

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Strike

Hagan-alH^2MC


2 τ = 2.5 and β = 1.

Solving this ODE, we obtain that C(f) is fixed to

C(f) =σloc(f)

α√

1− ρ2 cosh(ν∫ ff0

df ′

σloc(f ′)

)If ν = 0, we have C(f) = σloc(f) and we reproduce the Dupire formula.Using the BBF formula (5.39), we have

C(f) =fσBS(f)

(1− f ln( ff0

)σ′BS(f)σBS(f)

)α√

1− ρ2 cosh(ν

ln( ff0)

σBS(f)

)The short term smile is automatically calibrated when using this function inthe case of the λ-SABR model.

In the following, we will show how to find an exact solution to the SABR modelwith β = 0 and β = 1 for the conditional probability. These exact solutionswill allow us to test the validity of our asymptotic solution. A (complete)treatment of solvable stochastic volatility models will be done in chapter 9.


Implied Vol 10Year

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Strike

Hagan-alH^2MC


2 τ = 5 and β = 1.

6.4 Analytical solution for the normal and log-normalSABR model

6.4.1 Normal SABR model and Laplacian on H2

For the SABR model, the connection A and the function Q are given by

A =1

2 (1− ρ2)

(−d ln(C) +

ρ

ν∂fCda

)(6.53)

Q =a2

4

(C∂2

fC −(∂fC)2

2 (1− ρ2)

)(6.54)

For β = 0, the function Q and the potential A vanish. p satisfies a heat kernelequation where the differential operator D reduces to the Laplace-Beltramioperator on H2:

∂p

∂τ ′= ∆H2p (6.55)

≡ y2(∂2x + ∂2

y)p (6.56)


with τ ′ = ν2τ2 and the coordinates [x, y] defined by (6.41,6.42) (with q =

f − f0). Therefore solving the Kolmogorov equation for the normal SABRmodel (i.e., β = 0) is equivalent to solving this (Laplacian) heat kernel onH2. Surprisingly, there is an analytical solution for the heat kernel on H2

(6.55) found by McKean [116]. It is connected to the Selberg trace formula[21]. The exact conditional probability density p depends on the hyperbolicdistance d(z, z′) and is given by

p(τ ′, d) =√

2e−τ′4

(4πτ ′)32

∫ ∞d(z,z′)

be−b2

4τ′

(cosh b− cosh d(z, z′))12db

The conditional probability in the old coordinates [a, f ] is

p(τ ′, f, a)dfda =ν√

1− ρ2

dfda

a2

√2e−

τ′4

(4πτ ′)32

∫ ∞d(z,z′)

be−b2

4τ′

(cosh b− cosh d(z, z′))12db

where the term ν√1−ρ2a2

corresponds to the invariant measure√g on H2. We

have compared this exact solution (Fig. 6.6) with a numerical solution (PDE)of the normal SABR model and found an agreement. A similar result wasobtained independently in [101] for the conditional probability.The value of a European option is

C(T,K|f0) = max(f0 −K, 0) +12

∫ T

0

dτ

∫ ∞0

da a2p(τ, x1 = K, a|α)

This expression can be obtained by applying the Ito-Tanaka lemma on thepayoff max(ft −K, 0). In order to integrate over a, we use a small trick: weinterchange the order of integration over b and a. The half space b ≥ d witha arbitrary is then mapped to the half-strip amin ≤ a ≤ amax and b ≥ lmin

where6

(amax − amin)2

4(1− ρ2)= −(K − f0)2ν2 − α

(α+ 2νρ(K − f0) + αρ2

)+ α cosh b

(2ρ ((K − f0)ν + αρ) + α(1− ρ2) cosh b

)(6.57)

amin + amax

2= ρ (ν(K − f0) + αρ) + α(1− ρ2) cosh b (6.58)

cosh lmin =−ρ ((K − f0)ν + αρ) +

√α2 + 2ανρ(K − f0) + ν2(K − f0)2

α(1− ρ2)(6.59)

6All these algebraic computations have been done with Mathematica.


f0=1, alpha=1.23%, nu=27.48%, rho=11.48%, beta=0%, tau=10 y

0

2

4

6

8

10

12

80% 85% 90% 95% 100% 105% 110% 115% 120%

PdeExact

FIGURE 6.6: Exact conditional probability for the normal SABR modelversus numerical PDE.

Performing the integration according to a leads to an exact solution for aEuropean call option for the normal SABR model

C(T,K|f0) =√

2

ν√

1− ρ2

∫ ν2T2

0

dte−

t4

(4πt)32

∫ ∞lmin

b(amax − amin)e−b24t

√cosh b− cosh lmin

db

+ max(f0 −K, 0)

with amin, amax and lmin defined by (6.57,6.58,6.59).

6.4.2 Log-normal SABR model and Laplacian on H3

A similar computation can be carried out for the log-normal SABR model(i.e., β = 1). By using (6.53), the Abelian connection reduces to

A =1

2(1− ρ2)

(−d ln f +

ρ

νda)


The potential A is exact, meaning there exists a smooth function Λ such thatA = dΛ with

Λ(f, a) =1

2(1− ρ2)

(− ln(f) +

ρ

νa)

Furthermore, using (6.54), we have

Q = − a2

8(1− ρ2)= − y2

8(1− ρ2)2

Applying an Abelian gauge transformation (4.64)

p′ = eΛ(f,a)−Λ(f0,α)p

we find that p′ satisfies the following equation

y2

(∂2x + ∂2

y −1

4ν2(1− ρ2)2

)p′ = ∂τ ′p

′ (6.60)

How do we solve this equation? It turns out that the solution corresponds insome fancy way to the solution of the (Laplacian) heat kernel on the threedimensional hyperbolic space H3.

Geometry H3

This space can be represented as the upper-half space

H3 = x = (x1, x2, x3)|x3 > 0

In these coordinates, the metric takes the following form

ds2 =(dx2

1 + dx22 + dx2

3)x2

3

and the geodesic distance between two points x and x′ in H3 is given by7

cosh(d(x, x′)) = 1 +|x− x′|2

2x3x′3

As in H2, the geodesics are straight vertical lines or semi-circles orthogonalto the boundary of the upper-half space. An interesting property, useful tosolve the heat kernel, is that the group of isometries of H3 is PSL(2,C).8 Ifwe represent a point p ∈ H3 as a quaternion9 whose fourth components equal

7| · | is the Euclidean distance in R3.8PSL(2,C) is identical to PSL(2,R), except that the real field is replaced by the complexfield.9The quaternionic field is generated by the unit element 1 and the basis i, j, k which satisfythe multiplication table i.j = k and the other cyclic products.


zero, then the action of an element g ∈ PSL(2,C) on H3 can be described bythe formula

p′ = g.p =ap+ b

cp+ d(6.61)

with p = x11 + x2i + x3j.The Laplacian on H3 in the coordinates [x1, x2, x3] is given by

∆H3 = x23(∂2

x1+ ∂2

x2+ ∂2

x3)

and the (Laplacian) heat kernel equation is

∂τ ′p′ = ∆H3p′

Analytical probability density

The exact solution for the conditional probability density p′(τ ′, x|x′), depend-ing on the geodesic distance d(x, x′), is [95]

p′(τ ′, x|x′) =1

(4πτ ′)32

d(x, x′)sinh(d(x, x′))

e−τ′− d(x,x′)2

4τ′

Let us apply a Fourier transformation on p along the coordinate x1 (or equiv-alently x2)

p(τ ′, k, x2, x3|x′) =∫ ∞−∞

dx1√2πe−ikx1p′(τ ′, x|x′)

Then p′ satisfies the following PDE

∂τ ′ p′ = x2

3(−k2 + ∂2x2

+ ∂2x3

)p′ (6.62)

By comparing (6.62) with (6.60), we deduce that the exact solution for theconditional probability (see remark 4.3) for the SABR model with β = 1 is(with x ≡ x2 = ν ln f0

f − ρα, y ≡ x3 =√

1− ρ2α, k ≡ 12ν(1−ρ2) , x′1 = 0,

x′2 = −ρa, x′3 =√

1− ρ2a)

p′(x, y, x′, y′, τ ′) = e1

2(1−ρ2)(ln(

f0f )+ ρ

ν (a−α))∫ ∞−∞

dx1√2π

1(4πτ ′)

32

d(x, x′)sinh(d(x, x′))

e−τ′− d(x,x′)2

4τ′ e− ix1

2ν(1−ρ2)

A previous solution for the SABR model with β = 1 was obtained by [29],although only in terms of Gauss hypergeometric series.


6.5 Heston model: a toy black hole

In this section, we look at the Heston model, a typical example of SVMfrequently used by practitioners. The dynamics of the forward and the in-stantaneous volatility is

dftft

= atdWt

da2t = λ(v − a2

t )dt+ ζatdZt , dWtdZt = ρdt

or equivalently

dat =(−λ

2

(1− v

a2t

)− ζ

8a2t

)atdt+

ζ

2dZt

The model has five parameters: at=0 ≡ α, λ, v, ζ and ρ. The time τ = 1λ is a

cutoff which separates the short from the long maturities. In [61], the mainproperties of the Heston model are examined carefully. In particular, thismodel admits an analytical characteristic function. As a consequence the fairvalue of a call option can be written as the one-dimensional Fourier transformof the characteristic function [106]. This feature is one of the main reasonswhy the Heston model has drawn the attention of the practitioners despitemany disadvantages such as its delicate numerical simulation [43].

6.5.1 Analytical call option

The PDE satisfied by a call option C is

∂τC =12v∂yC +

ζ2

2v∂2vC + ζvρ∂vyC − λ(v − v)∂vC −

12v∂yC

using the variables τ = T−t, v = a2 and the moneyness y = ln ff0

. By analogywith the Black-Scholes formula, we guess a solution of the form

C(τ, y, v) = K (eyP1(τ, y, v)− P0(τ, y, v)) (6.63)

with P0 and P1 two-independent solutions. Plugging (6.63) into the pricingPDE, we obtain the PDEs

∂τPi =12v∂2yPi −

(12− j)v∂yPi +

12ζ2v∂2

vPi + ρζv∂xvPi

+ (a− bjv) ∂vPj

with a = λv and bj = λ − jρζ , j = 0, 1. In order to satisfy the terminalcondition, these PDEs are subject to the terminal condition Pj(0, y, v) = 1(y).


By applying a Fourier transform with respect to the variable y

Pj(τ, k, v) =∫ ∞−∞

dy√2πe−ıkyPj(τ, y, v)

we have10

∂τ Pj = v(αjPj − βj∂vPj + γ∂2

v Pj

)+ a∂vPj (6.64)

with

α = −k2

2− ık

2+ ıju

β = λ− ρζj − ρζık

γ =ζ2

2

As an ansatz for the solution, we guess

Pj = Aj(τ, k)eBj(τ,k)v

By plugging this guess into (6.64), we obtain that Aj and Bj satisfy twoRiccati ODEs that can be solved. Finally, we obtain

Pj(τ, y, v) =12

+1π

∫ ∞0

du<(eCj(τ,v)v+Dj(τ,u)v+iuy

iu

)with

D(τ, u) = r−1− e−dτ

1− ge−dτ

C(τ, u) = λ

(r−τ −

2ζ2

ln(

1− ge−dτ

1− g

))where we define

r± ≡β ±

√β2 − 4αγ2γ

≡ β ± dζ2

g ≡ r−r+

The integration over u can be achieved with a numerical scheme. Note thatthis integration can be tricky as the integrand is typically of oscillatory naturegiving rise to numerical instability. A careful treatment of this problem isachieved in [112].

10Here we follow closely the notations in [18].


6.5.2 Asymptotic implied volatility

By specifying our general results in section 6.1.2, we obtain that the Hestonmodel corresponds to the metric

ds2 =2√

1− ρ2ζ

dx2 + dy2

y

with the coordinates

x = lnf

f0− ρa

2

ζ

y =√

1− ρ2a2

ζ

Note that as explained in remark 6.1, the metric is smooth if the volatilitycan not reach a = 0. This is satisfied if and only if

2λv > ζ2 (6.65)

Unfortunately, this condition is not usually satisfied when the Heston model iscalibrated to market implied volatility surfaces. Although the use of the heatkernel expansion can appear quite problematic in this case, we conjecture thatour general first-order asymptotic implied volatility (6.25) is still valid whenthe condition (6.65) is not satisfied. For an other (probabilistic) approachvalidating this computation, the reader can consult [83].

Geodesic distance

The geodesic distance d (see (6.17)) between two points (x1 = −ρα2

ζ , y1 =√1− ρ2 α

2

ζ ) and (x2, y2) is

d =2δζC

[arcsin(√u)]C

2 y2ζδ

C2 y1ζδ

(6.66)

where δ = 2√1−ρ2

. The constant C = C(x1, y1, x2, y2) > 0 is determined by

the equation

sign((x2 − x1)(y2 − y1))(x2 − x1) =δ

ζC2[−√u(1− u) + arcsin(

√u)]

C2ζy2δ

C2ζy1δ

By doing the change of variables C2ζyiδ = sin2 θi , θi=1,2 ∈ [0, π), we get

d = 2

√δ

ζ

√yi

sin θi| θ2 − θ1 |

sign((x2 − x1)(y2 − y1))x2 − x1

y2sin2 θ2 = [−sin θ| cos θ|+ θ]θ2θ1


with sin θ2sin θ1

=√

y2y1

.

REMARK 6.9 Computation of amin

From (6.31), the saddle-point amin satisfies the equation

k − ρ

ζ

(a2

min − α2)

= ± a2min√

1− ρ2ζ[arcsin(u)− u

√1− u2]

√1−ρ2√

1−ρ2αamin

with k = ln Kf0

. We set sin θ ≡√

1− ρ2 αamin

, θ ∈ (0, π). This gives

sin2 θ(k)√1− ρ2

(kζ

α2+ ρ

)± (θ(k)− sin θ(k)| cos θ(k)|) = ±φ

with φ = arccos ρ. The zero-order implied volatility (5.39) is then

limT→0

σBS(T,K) =α√

1− ρ2k∫ k0dy sin θ(y)

with k = ln Kf0

.

Parallel transport

Throughout this paragraph, let us note i : C → Σ, the immersion of thegeodesic curve C on the non-compact Riemann surface Σ ' R× R∗+.From the definition (4.58), the Abelian connection is given by

A = − d ln f(1− ρ2)

(12− ρλ

ζ

(1− v

a2

))+

ada

(1− ρ2)ζ

(ρ− 2λ

ζ

(1− v

a2

))In the new coordinates [x, y], we obtain

A = − dx

(1− ρ2)

(12− ρλ

ζ

(1−

√1− ρ2

v

ζy

))− λdy

ζ√

1− ρ2

(1−

√1− ρ2

v

ζy

)The pullback of the Abelian connection on the geodesic curve C is

i∗A = − 1(1− ρ2)

√y

δC2ζ − y

dy

(12− ρλ

ζ

(1−

√1− ρ2

v

ζy

))

− λdy

ζ√

1− ρ2

(1−

√1− ρ2

v

ζy

)where we have used that i∗dx =

√y

δC2ζ−ydy. We deduce that the parallel

gauge transport is

lnP(x, y) =(

12− ρλ

ζ

)x2 − x1

1− ρ2+λ

ζ

y2 − y1√1− ρ2

− vλ

ζ2lny2

y1

+2ρλv

ζ2√

1− ρ2[arctan

√y

δC2ζ − y

]y2y1


Furthermore, we have

− ln g′(amin) = − 2amin

From the expression for the geodesic distance and the parallel gauge transport,we can deduce a first-order asymptotics for the implied volatility. The VanVleck-Morette determinant is computed numerically.

6.6 Problems

Exercises 6.1 Mixing solution and Hull-White decompositionFor general SVMs, there is no closed-form formula for European call options.

To circumvent this difficulty, we can use asymptotic methods as described inthis chapter. However, for long maturity date, such methods are no longerapplicable and one needs to rely on Monte-Carlo (MC) simulation. When theforward conditional to the instantaneous volatility is a log-normal Ito process,the MC simulation can be considerably simplified: This is called the Mixingsolution [29].Let us consider the following SVM defined by

dft = atft

(ρdZt +

√1− ρ2dWt

)da = b(at)dt+ σ(at)dZt

with Wt and Zt two independent Brownian motions and the initial conditionsft=0 = f0 and a0 = α.

1. Using Ito’s formula, prove that

d ln ft = −12a2tdt+ at

(ρdZt +

√1− ρ2dWt

)2. Conditional to the path of the second Brownian Zt (with filtration FZ),ft is a log-normal process with mean mt and variance Vt. Prove that

f0 ≡ E[ft|FZ ] = f0e− 1

2ρ2 R t

0 a2sds+ρ

R t0 asdZs

Vt = (1− ρ2)∫ t

0

a2sds


3. Deduce that the fair value C at time t of a European call option withstrike K and maturity date T ≥ t is

C = EP[CBS(K,T,

√(1− ρ2)

1T − t

∫ T

t

a2sds|f0, t)|Ft]

(6.67)

with CBS the Black-Scholes formula as given by (3.1). This formulais called the mixing solution. For ρ = 0, we obtain the Hull-Whitedecomposition [109]:

C = EP[CBS(K,T,

√1

T − t

∫ T

t

a2sds|f0, t)|Ft] (6.68)

From (6.67) and (6.68), we see that the MC pricing of a call option onlyrequires the simulation of the volatility at. The forward ft has beenintegrated out.

Exercises 6.2 Variance swapCompute the variance swap for the Heston and SABR model:

VST =1T

∫ T

0

EP[a2s]ds

Chapter 7

Multi-Asset European Option andFlat Geometry

Abstract A standard method to price a multi-asset European option in-corporating an implied volatility is to use a local volatility Monte-Carlo com-putation. Although straightforward, this method is quite time-consuming,particularly when the number of assets is large and we evaluate the Greeks.Applying our geometrical framework to this multi-dimensional problem, weexplain how to obtain accurate approximations of multi-asset European op-tions.We use the heat kernel expansion to obtain an asymptotic solution to theKolmogorov equation for a n-dimensional local volatility model. The resultingmanifold is the flat Euclidean space Rn. We present two applications. Thefirst application we look at is the derivation of an asymptotic implied volatilityfor a basket option. In particular, we try to reconstruct the basket impliedvolatility from the implied volatility of each asset. In the second application,we obtain accurate approximation for Collateralized Commodity Obligations(CCO), which are recent commodity derivatives that mimic the CollateralizedDebt Obligations (CDO).

7.1 Local volatility models and flat geometry

In the forward measure PT , each forward f it (i = 1, · · · , n) is a local martingaleand we assume that they follow a local volatility model

df it = Ci(t, f it )dWi ; dWidWj = ρijdt (7.1)

with a deterministic rate and with the initial condition f it=0 = f i0. The metric(4.78) at t = 0 underlying this model is

ds2 = 2ρijdf i

Ci(f i)df j

Cj(f i)(7.2)

187


where we have set Ci(f i) ≡ Ci(0, f i). By using the Cholesky decomposition,we write the inverse of the correlation matrix as

ρ−1 = L†L

or in components

ρij = LkiLkj

By convention ρij denotes the components of the inverse of the correlationmatrix. [L]ik is a n× n-matrix. Similarly the correlation ρ can be written as

ρ = L−1(L−1

)†or in components

ρij = LikLjk

Here Lij are the components of the inverse of the Cholesky matrix L.If we introduce the new coordinates

ui(f) = Lij

∫ fj

fj0

dxj

Cj(xj)(7.3)

we obtain that the metric (7.2) (at t = 0) is flat (the factor 2 is introducedfor a convenience purpose)

ds2 = 2duidui

The geodesic distance between the two points f0 ≡ f i0i and f ≡ f ii isthen given by the Euclidean distance

d(u)2 = 2u.u ≡ 2n∑i=1

(ui)2 (7.4)

After some algebraic manipulations, the connection A (4.58) which appearsin the time-dependent heat kernel expansion (4.79) is given by

A = −12ρij

∂jCj(f j)

Ci(f i)df i (7.5)

In the new coordinates u, the connection (7.5) is given by

A = −12LkjLpj

∂ ln(Cj(u))∂up

duk (7.6)

Note that unlike the one-dimensional case, the connection (7.6) is not an exactform and to obtain the parallel gauge transport we need to pullback this form

Multi-Asset European Option and Flat Geometry 189

on a geodesic curve. The geodesic curve joining together the spot forward f0

(i.e., u = 0) and the forward f (i.e., u) is a straight line (that we parameterizeby λ ∈ [0, 1]) on the flat manifold Rn

uk(λ) = ukλ

By expanding at the first-order ∂ ln(Cj(u))∂up

around u = 0, we deduce that theconnection pullback to this curve is approximated by

A ' −12Lkj

(∂jC

j(f j0 ) + λCj(f j0 )∂2jC

j(f j0 )Ljpup)ukdλ

From this expression, we derive the parallel gauge transport from the pointf0 to f

P(f0, f) ≈ e 12u†Cu+u†B (7.7)

with the n-dimensional matrix [C]kp and the vector [B]k defined by

Ckp =14Cj(f j0 )∂2

jCj(f j0 )(LkjLjp + LpjL

jk) (7.8)

Bk =12Lkj∂jC

j(f j0 ) (7.9)

By plugging the expression for the geodesic distance (7.4) and the parallelgauge transport (7.7) in the time-dependent heat kernel expansion (4.82), weobtain the conditional probability density p(t, u|0) at the first-order

p(t, u|0) =e−u†(I−tC)u

2t −u†B

(2πt)n2

(1 + a1(u, 0)t+ o

(t2))

(7.10)

We will use this asymptotic solution to price a basket option and a Collateral-ized Commodity Obligation (CCO). Note the minus sign in front of B as f0 iswritten as f in our geometric notation (see remark 4.3). As for the first heatkernel coefficient a1(u, 0) between the point u and u = 0, it is not writtenexplicitly as it does not take any part in the calculation.

7.2 Basket option

A basket option is a European option the payoff of which is linked to a portfolioor “basket” of assets. The basket can be any weighted sum of assets as longas all the weights are positive. Basket options are usually cash settled. Aclassical example is a call option on the France CAC 40 stock index.


Basket options are also popular for hedging foreign exchange risk. A corpora-tion with multiple currency exposures can hedge the combined exposure lessexpensively by purchasing a basket option than by purchasing options in eachcurrency individually.The payoff of a basket option, based on an index of n assets, with strike Kand maturity T , is given by

max

(n∑i=1

ωiSiT −K, 0

)

with SiT the stock price at maturity and ωi the weight of the asset i. Byconvention, the weights ωi are normalized by

∑ni=1 ωi = 1. Let us denote the

following dimensionless parameters

ωi =ωiS

i0∑n

i=1 ωiSi0

; K =K∑n

i=1 ωiSi0

f it =Sit

Si0PtT; At =

n∑i=1

ωifit (7.11)

with Si0 the spot price of asset i, Sit the price at time t and PtT the value ofthe bond quoted at t and expiring at T . In the forward measure PT , the fairvalue of a basket call option equals

C = P0TEPT [max

(n∑i=1

ωiSiT −K, 0

)|F0] (7.12)

=n∑i=1

ωiSi0P0TEPT [max

(AT − K, 0

)|F0] (7.13)

This representation (7.13) is introduced because it appears to give a betterapproximation of a basket option than formula (7.12).Basket options are usually priced by treating the basket value AT as a singleasset satisfying a log-normal process with a constant volatility σBS. Therefore,the fair value (7.13) becomes equivalent to the fair value of a European calloption on a single asset following a log-normal process. This leads to a Black-Scholes formula

C =n∑i=1

ωiSi0P0TCBS(K,T, σBS|A0, 0) (7.14)

From this formula, we can define the basket implied volatility as the volatilityσBS which when put in the formula above reproduces the market price. Weshow in the following how to derive an asymptotic implied volatility for abasket in the context of a multi-dimensional local volatility model (7.1). Wewill check this result against a Monte-pricer and the classical second-moment


matching approximation that we review in the next section. Note that evenif each asset has a constant volatility (i.e., no skew), the resulting basketimplied volatility is skewed by the fact that a weighted sum of log-normalrandom variables is not log-normal. More generally, we try to understandhow to construct the basket implied volatility from the implied volatility ofeach of its constituents. This problem was explored for the first time in [50].

7.2.1 Basket local volatility

We assume that each forward f it = SitPtT

follows a local volatility model (7.1).The process fi (see Eq. (7.11)) satisfies the SDE under PT

df it = Ci(t, f it )dWi ; dWidWj = ρijdt (7.15)

with Ci(t, f it ) = Ci(t,Si0fit )

Si0. Being the sum of all assets, the index At is a traded

asset and the process At is a local martingale under the forward measure PT .From the SDEs (7.15), the process At satisfies the SDE under PT

dAt =n∑i=1

ωiCi(t, f it )dWi (7.16)

As discussed in section (6.1.3) in chapter 6 the process (7.16) has the samemarginals as the local volatility model

dAt = C(t, At)dWt (7.17)

where the square of the Dupire local volatility function C(t, A) is equal to thesquare of the volatility in (7.16) when the index A is fixed to

∑Ni=1 ωif

it at

time t

C(t, A)2 =n∑

i,j=1

ρijωiωjEPT [Ci(t, f it )Cj(t, f jt )|A =

n∑i=1

ωifit ] (7.18)

For a completeness purpose, we give another proof.

PROOF Applying Ito-Tanaka’s lemma on the payoff max(At −K, 0), weobtain

dmax(At −K, 0) =12δ(At −K)

n∑i,j=1

ρijωiωjCi(t, f it )C

j(t, f jt )dt

+ 1(At −K)dAt

Taking the operator EPt [·|F0] on both sides of this equation, we have

dEPt [max(At −K, 0)|F0] =12

n∑i,j=1

ωiωjρijEPt [CiCjδ(At −K)]dt (7.19)


By definition of Dupire local volatility function, we have

dEPt [max(At −K, 0)|F0] =12C(t,K)2EPt [δ(At −K)]dt (7.20)

Identifying the two equations (7.19) and (7.20), we obtain the expected rela-tion (7.18).

By definition of the mean value in (7.18), the basket local volatility is

C(t, A)2 =n∑

i,j=1

ρijωiωj

∫B C

i(t, f i)Cj(t, f j)p(t, f |f0)∏k dfk∫

B p(t, f |f0)∏k dfk

(7.21)

with p(t, f |f0) the conditional probability associated to the SDEs (7.15) att and B the hyperplane defined by the linear equation

∑ni=1 ωif

i = A. Asusual the computation of an asymptotic implied volatility will be done in twosteps. First of all, from the definition (7.21), we calculate a local volatility atthe first-order in time using our first-order asymptotic conditional probabilitydensity (7.10). Then, as a second step, we use our map between an asymptoticimplied volatility and an asymptotic local volatility (5.40).

PROPOSITION 7.1The basket local volatility is approximated at the first-order in time by

C(t, A)2 =n∑

i,j=1

ρijωiωjCi(t, f∗i )Cj(t, f∗j )

(1 +

t

2Tr(M ijD0)

)with

[M ]ijtp = ∂i

(Ci∂iCi

)LitLip + 2∂iCi∂jCjLipLjt + ∂j

(Cj∂jCj

)LjtLjp |fi=f0

i

(7.22)

Dt = (1− tC)−1 − (1− tC)−1ωω† (1− tC)−1(

ω† (1− tC)−1ω) (7.23)

[ω]i=1,··· ,n =n∑j=1

ωj fj0σ

jBSL

ji

f∗i

f0i

= eσiBSL

ip(u∗)p

u∗ = (1− tC)−1

−tB + ω

(A− A0 + tω† (1− tC)−1

B)

(ω† (1− tC)−1

ω)

(7.24)

[C] and [B] are given by (7.8), (7.9). σiBS denotes the ATM implied volatilityfor asset i and Ci ≡ Ci(0, fi).


Note that D0 ≡ limt→0Dt = 1− ωω†

ω†ω.

PROOF From (7.21), the local volatility function is

C(t, A)2 =n∑

i,j=1

ρijωiωj

∫B C

i(t, u)Cj(t, u)p(t, u|0)du∫B p(t, u|0)du

where du ≡∏ni=1 dui. By plugging the first-order solution to the backward

Kolmogorov equation as given in (7.10) in the equation above, we have

C(t, A)2 =n∑

i,j=1

ρijωiωj

∫B C

i(t, u)Cj(t, u)e−12tu†(1−tC)u−u†Bdu∫

B e− 1

2tu†(1−tC)u−u†Bdu

Note that at this order, the first heat kernel coefficient does not contribute.The new coordinates u for which the metric at t = 0 is flat is

ui =n∑j=1

Lij

∫ fj

fj0

dx

Cj(0, x)

By using the BBF formula (5.39), we have

ln

(f j

f j0

)=

n∑i=1

σjBS(0, f j)Ljiui

By assuming that ln(fj

fj0

)≈ fi

fi0− 1, we obtain the relation

A− A0 =n∑j=1

ωj fj0

(f j

f j0− 1

)≈

n∑j=1

ωj fj0 ln

(f j

f j0

)

=n∑

i,j=1

ωj fj0σ

jBS(0, f j)Ljiui ≈

n∑i,j=1

ωj fj0σ

jBS(0, f j0 )Ljiui

Setting βj = ωj fj0σ

jBS(0, f j0 ) and ωi =

∑nj=1 βjL

ji, the hyperplane B is thenrepresented by the following linear form (in the u coordinates)

B : u†ω = A− A0

The Dirac function over the submanifold B can be replaced by its Fouriertransform

δ(u†ω − A+ A0

)=

12π

∫ ∞−∞

eıq(u†ω−A+A0)dq


We obtain

C(t, A)2 =n∑

i,j=1

ρijωiωj

∫Rn+1 C

i(t, u)Cj(t, u)e−12tu†(1−tC)u−u†B+ıq(u†ω−A+A0)dudq∫

Rn+1 e− 1

2tu†(1−tC)u−u†B+ıq(u†ω−A+A0)dudq

Let us introduce the partition function

Z[j] ≡∫

Rn+1e−

12tu†(1−tC)u−u†B+ıq(u†ω−A+A0)+j†(u−u∗)dudq

where u∗ is a n-dimensional vector that we specify below. By using theGaussian integration formula∫

RndXe−

12X†AX+j†X =

((2π)

n2

√detA

)e

12 j†A−1j

we obtain

χ[j] ≡ Z[j]Z[0]

= et2 j†Dtj+j

†(T−u∗)

with Dt a n × n-dimensional matrix and T a n-dimensional vector given re-spectively by (7.23) and (7.24). We choose u∗ = T . This gives

χ[j] = et2 j†Dtj

By expanding Φ(u) = Ci(t, u)Cj(t, u) at the second-order in u around u = u∗,we have

Φ(u) = Φ(u∗) +∇uΦ(u∗)(u− u∗) +12

(u− u∗)†∇2uΦ(u∗)(u− u∗)

Finally, we obtain

C(t, A)2 =n∑

i,j=1

ρijωiωj(Φ(u∗) +∇uΦ(u∗)†

∂χ[j]∂j

(j = 0) +12

Tr[∇2uΦ(u∗)

∂2χ[j]∂j2

(j = 0)])

=n∑

i,j=1

ρijωiωjCi(t, u∗)Cj(t, u∗)

(1 +

t

2Tr[∇2uC

iCj

CiCjDt]

)

An explicit computation shows that ∇2uC

iCj

CiCjis given by (7.22). At the first-

order, Dt can be replaced by D0.


REMARK 7.1 Correlation smile By analogy with the smile, thecorrelation smile is defined as the correlation (matrix) ρij(K,T ) that must beplugged in the Black-Scholes formula (7.14) in order to reproduce the marketprice of basket option with a strike K and a maturity T . We assume that eachasset is calibrated to its implied volatility surface. By analogy with the localvolatility, we can match the correlation smile by introducing a local correlationmatrix [121] which depends on the time t and the level of the basket At by

ρij(t, At) = ρ(t, At)ρhistij +

(1− ρ(t, At)

)ρhistij is the historical correlation matrix and ρ(t, At) ∈ [0, 1]. The matrix ρij

is still definite positive if ρhistij is so. From our previous computation, a good

guess for ρ(t, At) is

ρ(t, At) =Cmkt(t, A)2 −

∑ni,j=1 ωiωjC

imkt(t, f

∗i )Cjmkt(t, f

∗j )∑n

i,j=1

(ρhistij − 1

)ωiωjCimkt(t, f

∗i )Cjmkt(t, f

∗j )

where Cmkt (resp. Cimkt) is the local volatility derived from the basket (asset)implied volatility via the Dupire formula. Note that as the LV C(t, A) iscomputed by conditioning on A, our asymptotic LV computed using a constantcorrelation ρhist

ij is still valid.

In the next section, we present a classical approximation for the value of abasket option in the case when each asset has a constant volatility. We willcheck this approximation against our asymptotic implied volatility.

7.2.2 Second moment matching approximation

Let us assume that each forward f it follows a log-normal process (with avolatility σi) and denote the geometric average by

Gt =n∏i=1

(f it

)ωiBy using this new variable, we rewrite the fair value (7.13) as

C = P0T

n∑i=1

ωiSi0EPT [max(GT − (−AT + GT )− K, 0)|F0]

We approximate the expression above by

C ≈ P0T

n∑i=1

ωiSi0EPT [max(GT − K, 0)|F0]


where we have introduced the modified strike

K ≡ K + EPT [GT − AT |F0]

From the Jensen inequality which states that

E[Φ(ST )|F0] ≥ Φ (E[ST |F0])

for a convex function Φ, we deduce that our approximation gives a lowerbound.Gt is a log-normal process and GT is given by

GT = P−10T e

WT− c2T2

with WT =∑ni=1 ωiσiW

iT and c2 =

∑ni=1 ωiσ

2i . WT follows a normal law with

a zero mean and a variance v2 =∑ni,j=1 ρijωiωjσiσj . Therefore, the modified

strike K is

K = K + P−10T

(e

(v2−c2)T2 − 1

)(7.25)

Finally, we obtain the lower bound

PROPOSITION 7.2The approximate value of the price C of a basket call option with strike priceK and expiry date T equals

C = P0T

n∑i=1

ωiS0i CBS(K, T, σ|G0)

with σ2 =∑ki,j=1 ρijωiωjσ

iBSσ

jBS and G0 = P−1

0T .

By construction, for k = 1, the formula above reduces to the Black-Scholesformula. In the graphs 7.1, 7.2 and 7.3, we compare our asymptotic impliedvolatility against the second moment-matching formula and an exact Monte-Carlo pricing.

7.3 Collaterized Commodity Obligation

The payoff of a Collaterized Commodity Obligation (CCO) is given by

min(max(θT ,Kmin),Kmax)


25.0

25.5

26.0

26.5

27.0

27.5

28.0

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1Strike

Asymp

2nd

MC

FIGURE 7.1: Basket implied volatility with constant volatilities. ρij =e−0.3|i−j|, σi = 0.1 + 0.1× i, i, j = 1, 2, 3, T = 5 years.

with

θT =n∑i=1

1(Ki − f iT )

with f iT the forward of asset i at the maturity T . Kmin is a global floor andKmax a global cap. By using a static replication, we have

min(max(θT ,Kmin),Kmax) = Kmin + max(θT −Kmin, 0)−max(θT −Kmax, 0)

In order to price the CCO, we need to find the fair value for a European calloption with the payoff max(θT −K, 0).A CCO mimics a Collaterized Debt Obligation (CDO) whose payoff is linkedto a call option on the loss default of a basket

θT =n∑i=1

1(T − τi)


20.0

22.0

24.0

26.0

28.0

30.0

32.0

34.0

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1Strike

Asymp

2nd

MC

FIGURE 7.2: Basket implied volatility with CEV volatilities. ρij =e−0.3|i−j|, σi = 0.1 + 0.1× i, i, j = 1, 2, 3, βCEV = 0.5, T = 5 years.

where τi is the time when the asset i defaults.

Using the fact that θT can take only discrete values from 0 to n, we obtain

E[max(θT −K, 0)] =∫ n∏

i=1

df i max(θT −K, 0)p(T, f |f0)

=n∑i=1

max(i−K, 0)∫Hi

n∏i=1

df ip(T, f |f0)

where the hyperplane Hi is defined as

Hi = fkk=1,··· ,n|n∑j=1

1(Kj − f j) = i


5.0

10.0

15.0

20.0

25.0

30.0

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Strike

Asymp 2nd

MC

FIGURE 7.3: Basket implied volatility with LV volatilities (NIKKEI-SP500-EUROSTOCK, 24-04-2008). ρij = e−0.4|i−j|, T = 3 years.

It can be partitioned in the following way

Hi =⋃

cyclici

[0,K1]× · · · × [0,Ki]× [Ki+1,∞]× · · · × [Kn,∞]

where cyclici means that we sum over the configuration such that the numberof variables which take their value below their strike is i. For example for twoassets, we have

H1 = [0,K1]× [K2,∞] ∪ [K1,∞]× [0,K2]

Using this characterization of the hyperplane Hi, we have

E[max(θT −K, 0)] =n∑i=1

max(i−K, 0)∑

cyclici∫ K1

0

df1 · · ·∫ Ki

0

df i∫ ∞Ki+1

df i+1 · · ·∫ ∞Kn

dfnp(T, f |f0)


Using the fact that∫ Ki

0

p(T, f |f0)df i = p(T, f |f0)−∫ ∞Ki

p(T, f |f0)df i

with f ≡ (f1, · · · f i−1, f i+1, · · · fn), we obtain

E[max(θT −K, 0)] =n∑i=1

max(i−K, 0)i∑

k=0

Cn−in−k(−)k+i∑

cyclicn−k

Φn−k

with

Φi(K1, · · · ,Ki) =∫ ∞K1

df1 · · ·∫ ∞Ki

df ip(T, f1, · · · f i|f10 , · · · f i0) (7.26)

As a next step, we find an approximation of the functions Φi using the heatkernel expansion. Before moving on to the derivation, we consider the caseof zero correlation for which there is a model-independent exact solution de-pending on the fair value of digital European call-put options.

7.3.1 Zero correlation

Let us assume that the correlation is zero (i.e., ρij = δij). The probabilitydensity p(T, f |f0) becomes the product of the individual probability densitiesfor each asset

p(T, f |f0) =n∏i=1

pi(T, f i|f i0)

Moreover, we have

pi(T,K|f i0) =∂2Ci(K,T )

∂K2

with Ci(K,T ) a European option of strike Ki and maturity date T for asseti. Then∫Hi

n∏i=1

df ip(T, f |f0) =∑

cyclic

i∏j=1

(∫ Kj

0

dKj ∂2Cj(Kj , T )∂Kj 2

)n∏

j=i+1

(∫ ∞Kj

dKj ∂2Cj(Kj , T )∂Kj 2

)

=∑

cyclic

i∏j=1

(1 +

∂Cj(Kj , T )∂Kj

) n∏j=i+1

(−∂Cj(K

j , T )∂Kj

)Finally, we have


PROPOSITION 7.3In the zero correlation case, E[max(θT −K, 0)] is exactly given by

C =n∑i=1

max(i−K, 0)∑

cyclici

i∏j=1

(1−DCj)n∏

j=i+1

(DCj) (7.27)

with DCi = −∂C(Ki,T )∂Ki a digital call of strike Ki and maturity T for asset i.

Using the market-value for European call options, the above-stated formulagives an exact solution to the fair price of the CCO in zero correlation cases.For two assets, the formula (7.27) reduces to

Example 7.1 2 assets

C = max(1−K, 0)(DC1 +DC2 − 2DC1DC2)+ max(2−K, 0)(1−DC1)(1−DC2)

7.3.2 Non-zero correlation

In the following, we will assume that Ki < f i0 for all i as it is the case inpractice. The functions Φi (7.26) are computed using a saddle-point method.The saddle points f∗ are defined by

minfd2(u)

where d is the geodesic distance (7.4). The distance is minimized for

f∗ ≡ (f1∗ , · · · , f i∗) = (f1

0 , · · · , f i0)

which is a global minimum. The functions Φi can be computed as the con-ditional probability at the zero-order is a Gaussian distribution in the u co-ordinates. Therefore the saddle-point method consists in expanding the non-Gaussian part at the second-order around the point f∗ = f0. For this com-putation, we shall need the non-diagonal heat kernel coefficient a(0, u). Asusual, we will approximate the first heat kernel coefficient by its diagonal partgiven by

a(f, f0) = Q(fav)− G(fav)4

with

Q(f) = −18ρij∂iCi(0, f i)∂jCj(0, f j) +

14Ci(0, f i)∂2

i Ci(0, f i) (7.28)

G(f) = 2(Ci)−1(0, f i)Ci,t(0, fi) (7.29)


Zero correlation, Cst vol=20%, 3 Assets(FTSE, STO50XXE, SPX) 1 year

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

0.0% 50.0% 100.0% 150.0% 200.0% 250.0% 300.0%

ExactAsymptoticLVMC 15000

FIGURE 7.4: CCO. Zero-correlation case.

Finally, the asymptotic fair value of the CCO is

PROPOSITION 7.4

At the second-order, E[max(θT −K, 0)] is given by

E[max(θT −K, 0)] =n∑i=1

max(i−K, 0)i∑

k=0

∑cyclicn−k

Φn−k(−)k+iCn−in−k

with

Φn(K1, · · · ,Ks) = det[Aρ−1]Nn

ln( fj0Kj )

σjBS(Kj)√T− (AB)j

√T ,A

(1 + (Q− G

4+BTAB

2)T )


with

Nn(Xii, ρ) ≡ 1

(2π)n2

∫ X1

−∞· · ·∫ Xn

−∞

e−ρij xixj

2

det(ρ)

∏i

dxi

the n-th cumulative multi-Gaussian distribution N (ρ), and with

Bi =12ρij∂jC

j(0, f j0 )

Aij = ρij

(1− T

4(Cj(0, f j0 )∂2Cj(0, f j0 ) + Ci(0, f i0)∂2

i Ci(0, fi0)))

To build up our confidence in this asymptotic formula, we apply it in the caseof a simple digital call option and a constant volatility C(f) = σ0f . We derive

Example 7.2 Constant volatility

DCK = N

ln(f0

K

)σ0

√T− σ0

√T

2

This expression is the exact BS solution.

7.3.3 Implementation

We have tested the validity of this analytical approximation against a Monte-Carlo pricer. As per figures 7.5 and 7.6, the approximation is accurate.


Historical correlation, Cst vol=20%, 3 Assets,1 year

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

0.0% 50.0% 100.0% 150.0% 200.0% 250.0% 300.0%

AsymptoticLVMC 15000

FIGURE 7.5: Exact versus asymptotic prices. 3 Assets, 1 year.

Historical correlation, smile, 3 assets, 1 year

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

140.0%

0.0% 50.0% 100.0% 150.0% 200.0% 250.0% 300.0%

AsymptoticLVMC 15000

FIGURE 7.6: Exact versus asymptotic prices. 3 Assets, 5 years.

Chapter 8

Stochastic Volatility Libor MarketModels and Hyperbolic Geometry

Abstract In this chapter, we focus on the last generation of interest-ratemodels, the Libor Market Models (LMMs). After a quick review of LMMs, wediscuss the calibration of such models. In practice, this model is calibrated toa swaption matrix\cube. As a Monte-Carlo (MC) calibration routine is fairlytime-consuming and noisy, the calibration requires an analytical approxima-tion for the swaption implied volatility. For the BGM-LMM, some approxima-tions have been found which are based on the Hull-White-Rebonato freezingarguments.Using our geometrical framework, we give a justification for the freezing ar-guments and derive a more accurate asymptotic swaption implied volatilityat the first-order for general stochastic volatility LMMs. We apply this for-mula to a specific model where the forward rates are assumed to follow amulti-dimensional CEV process correlated to a unique SABR process. Thegeometry underlying this model is the hyperbolic manifold Hn+1 with n thenumber of Libor forward rates.Parts of this research were published by the author in Risk magazine [104].

8.1 Introduction

The BGM model [67, 111] has recently been the focus of much attention as itgives a theoretical justification for pricing caps-floors using the classical Black-Scholes formula. The BGM model is a special version of the LIBOR MarketModel (LMM), in which all forward rates are log-normal. While the LMM is avery flexible framework as it enables us to specify an individual dynamics foreach Libor, the calibration of swaption smiles is difficult as, even in the simplecase of log-normal forward rates, swaption prices are not known analyticallyand straight Monte Carlo calibration is too time-consuming.In such a model the instantaneous variance of a swap rate is a weighted sumof the covariances of all forward rates, the weights being dependent on theforward rates. A widely used approximation consists in approximating theseweights by their value at time t = 0 and deriving an effective log-normal

205


dynamics for the swap rate [138, 108].By freezing the weights, but without forcing the swap rate to be log-normal,one can get an autonomous dynamics for the swap rate by making the co-variances a function of the swap rate itself: this generates an effective localvolatility model for the swap rate. This technique has been used in models inwhich forward rates follow a constant elasticity of variance (CEV) dynamics[45].Note that, given a swaption smile, we know from Dupire work that there existsa unique effective local volatility (ELV) for the swap rate which is consistentwith the smile. The techniques mentioned above can be seen as heuristicapproximations to this ELV.While affording more flexibility than the BGM model, a CEV dynamics is stillnot able to calibrate to both caplet and swaption smiles: it was natural forpractitioners to consider stochastic volatility LMMs. The literature on thissubject is not particularly large. Andersen et al. [45, 46] introduced a LMMwhere each Libor follows a CEV process, with the same parameter β, all Liborsbeing coupled to an uncorrelated stochastic volatility of the Heston type.Piterbarg has recently extended this framework, allowing this single β to betime-dependent [136]. In both cases the method used for obtaining swaptionsmiles still relies on deriving an effective autonomous stochastic volatilitymodel for the swap rate.Our aim in this chapter is two-fold:

• to analyze a LMM model based on a SABR dynamics for forward rates.

• to pursue on the application of the heat kernel expansion on a Rieman-nian manifold endowed with an Abelian connection for deriving swaptionsmiles in mixed local/stochastic volatility LMMs, which is generic andequally applicable to all models mentioned above.

This chapter is organized as follows:We first review LMMs, in particular the calibration and pricing issues.Then, we present the generic framework that we make use of to computeapproximate swaption smiles, which relies on a geometric formulation of thedynamics that underlies the pricing equation. We explicitly carry out thederivation of swaption smiles for the SABR-LMM model. We prove that thegeometry underlying this model is the hyperbolic manifold Hn+1. Some im-portant properties of this space are then presented. Furthermore, we showthat the “freezing” argument is no longer valid when we try to price a swap-tion in/out the money: The Libors should in fact be frozen to the saddle-point (constrained on a particular hyperplane) which minimizes the geodesicdistance on Hn+1.Finally we illustrate the accuracy of our methodology by comparing our ap-proximate smiles with exact Monte Carlo estimates as well as other popularapproximations, such as those mentioned in this introduction.

Stochastic Volatility Libor Market Models and Hyperbolic Geometry 207

8.2 Libor market models

We adopt the same definition as in section 2.10.3.3 and set

Lk(t) ≡ L(t, Tk−1, Tk)

the forward rate resetting at Tk−1 with τk = Tk − Tk−1 the tenor. We recallthe link between the Libor Lk(t) and the bonds PtTk−1 and PtTk quoted at tand expiring respectively at Tk−1 and Tk

Lk(t) =1τk

(PtTk−1

PtTk− 1)

As the product of the bond PtTk with the forward rate Lk(t) is a difference oftwo bonds with respective maturities Tk−1 and Tk and therefore a traded asset,Lk is a (local) martingale under Pk, the (forward) measure associated withthe numeraire PtTk . Therefore, we assume the following driftless dynamicsunder Pk

dLk(t) = σk(t)Φk(a, Lk)dWk , ∀t ≤ Tk−1 , k = 1, · · · , ndWkdWl = ρkl(t)dt

with the initial conditions a(t = 0) = α and Lk(t = 0) = L0k. In order to

achieve some flexibility, we assume that the (normal) local volatility Φk(a, Lk)depends on an additional one-dimensional Ito diffusion process a (to be spec-ified later) representing a stochastic volatility. We therefore assume that allthe forward rates are coupled with the same stochastic volatility a.For the sake of comparison we list in Table 8.1 the specification of the modelspreviously mentioned - also in the forward measure Pk. Note that becausein such models the stochastic volatility a is uncorrelated with the forwardrates, its dynamics remains the same, regardless of the choice of measure, incontrast to our generic stochastic volatility LMM (SV-LMM).The BGM, (limited) CEV and shifted log-normal models correspond to localvolatility models (a = 1) and the others to stochastic volatility models witha unique stochastic volatility a driven by a Heston process.The model parameters in a generic SV-LMM are the volatility σi(t), the cor-relation ρij(t) and additional parameters coming from the eventual stochasticvolatility a, Φk(a, Lk). They are fitted to the prices of caplets and swaptions.In addition, they can also be fitted to the historical or terminal correlationsof forward rates.The calibration of market models has been one of the main focuses in recentresearch and we review the main idea in the following.


TABLE 8.1: Example of stochastic (or local) volatilityLibor market models.

LMM SDEBGM dLk = σk(t)LkdWk

CEV dLk = σk(t)LβkdWk

LCEV dFk = σk(t)Lk min(Lβ−1k , εβ−1)dWk

with ε a small positive numberShifted LN dXk = σk(t)XkdWk

with Lk = Xk + αkFL-SV dLk = σk(t)(βkLk + (1− βk)L0

k)√vdWk

dv = λ(v − v)dt+ ζ√vdZ ; dWkdZ = 0

FL-TSS dLk = σk(t)(βk(t)Lk + (1− βk(t))L0k)√vdWk

dv = λ(v − v)dt+ ζ√vdZ ; dWkdZ = 0

8.2.1 Calibration

From the definition (3.14), we rewrite the expression for a swap in the followingway

sαβ =β∑

i=α+1

ωi(L)Li

with the weight depending on the Libors given by

ωi(L) =τiPtTi∑β

k=α+1 τkPtTk

Approximating the weight by their initial value

ωi(L) ≈ ωi(L0i )

the swap becomes index-like and a swaption becomes similar to a basketoption. It is then clear that the swaption fair value depends on the volatilityof the Libors and the correlation matrix [ρ]ij . So, in theory, the swaptionscarry information about correlation between Libors.The swaption matrix can be used to calibrate both the Libor volatilities σi(t)and the correlations ρij(t) at the same time. This is the approach explained in[7]. However, calibration techniques in which correlation is an output typicallystruggle to simultaneously fit the swaptions as well as produce reasonableyield curve correlations [75]. The calibrated correlation matrix is quite noisyand relatively far from an historical correlation. This problem can not besolved by imposing smoothing constraints on the correlation functional form.Therefore, we disregard this methodology and decouple the calibration intotwo sub-problems: first, the correlation structure is chosen as an input andthen this correlation is used in the calibration of the Libor volatility to aswaption matrix [142].


Correlation input

In order to decrease the number of factors and simplify the Monte-Carlo pric-ing, we can use a piecewise constant reduced rank r correlation: Such a cor-relation can be written as

ρij(t) =r∑

k=1

bik(t)bjk(t)

where [b]ik is a n × r matrix. Furthermore, we assume that the coefficientsbik(t) have the following functional form ∀ t ∈ [Tl−1, Tl)

bip(t) =

∑rq=1 θqe

−kq(Ti−1−Tl−1)bqp√∑rq,s=1 θqθse

−kq(Ti−1−Tl−1)e−ks(Ti−1−Tl−1)ρsq(8.1)

depending on the parameters kp, θp and a r-dimensional correlation matrix[ρ ≡ bb†]pq (p, q = 1, · · · , r). kr set the time-dependence of the correlationand θr,[ρrp] control the shape of the correlation. Note that the matrix biq(t)has been normalized in order to have

ρii(t) =r∑

k=1

bik(t)bik(t) = 1

In practice, we use r = 2 or r = 3. Below, we have plotted a typical exampleof correlation shape between Libors 8.2.1.This low-rank parametrization of the correlation matrix is usually displayed bymean-reverting short rate models as in the Hull-White 2-factor model (HW2).

Correlation structure for the HW2 model

In the HW2 model, the dynamics of the instantaneous-short-rate process un-der the risk neutral measure P is given by

rt = xt + yt + ϕ(t) (8.2)

where xt and yt are Ornstein-Uhlenbeck processes

dxt = −axtdt+ σa(t)dWa

dy = −bytdt+ σb(t)dWb , dWadWb = ρdt

with a (resp. σa(t)) and b (resp. σb(t)) two constants (resp. two functions).With the HW2 model being an affine model (see exercise 3.2), the value of abond for the HW2 model quoted at t and expiring at T can be written as

PtT = A(t, T )e−B(a,t,T )xt−B(b,t,T )yt


1

6 11

16

21

26

31

1

611

16

21

26

31

36

0%

20%

40%

60%

80%

100%

2F- Correlation

FIGURE 8.1: Instantaneous correlation between Libors in a 2-factor LMM.k1 = 0.25, k2 = 0.04, θ1 = 100%, θ2 = 50% and ρ = −20%.


with

B(z, t, T ) =1− e−z(T−t)

z

A(t, T ) is left unspecified as we don’t need to know its expression. As

Li(t) =1τi

(PtTi−1

PtTi− 1)

(8.3)

we deduce that the Libor Li(t) has the following dynamics in the forwardmeasure Pi

dLi(t) = σa(t)(τiLi(t) + 1)((−e−a(Ti−t) + e−a(Ti−1−t)

aτi

)dWa+

σb(t)σa(t)

(−e−b(Ti−t) + e−b(Ti−1−t)

bτi

)dWb

)This SDE is easily obtained by applying Ito’s lemma on (8.3). We don’t needto care about the drift term as it must cancel at this end in the forwardmeasure Pi.Let us assume that the ratio σb(t)

σa(t) ≡ θb is a constant. If we choose new

parameters (kε, θεε=a,b) such as (θa ≡ 1)

θεe−kεTi−1ekεt = θεe

εt

(−e−εTi + e−εTi−1

ετi

);

equivalent to

kε = ε

θε = θε

(1− e−ετi

ετi

)≈ θε

HW2 can be rewritten as (σ(t) ≡ σa(t))

dLi(t) = σ(t)(τiLi(t) + 1)(θae−ka(Ti−1−t)dWa + θbe

−kb(Ti−1−t)dWb

)The correlation structure is identical to (8.1) with r = 2.Once the parameters kp, θp, ρpq have been chosen, we can calibrate the Liborvolatilities σi(t) to at-the-money swaptions.

Calibration to swaptions and Hull-White-Rebonato freezing argu-ments

The calibration uses an approximated formula for a swaption. Such an ap-proximation has been derived for the LMMs listed in Table 8.1. At this stage,it is useful to recall how this approximation is derived using the Rebonato,


Hull-White freezing argument [138], [108]. As in the LMMs presented in Table8.1, we assume in this section that the functional form Φk(a, Lk) is the productof a stochastic volatility a and a local volatility function C(Lk) independentof k

Φk(a, Lk) = aC(Lk)

and the process at is uncorrelated with the Libors Lk

ρka = 0 ,∀ k = 1, · · · , n

For a = 1 and C(x) = xβ , we have the Andersen-Andreasen CEV-LMM [44]and for C(Lk) = βk(t)Lk + (1 − βk(t))L0

k and a driven by an uncorrelatedHeston process, we have the FL-SV LMM [45].As the product of the swap rate sαβ(t),

sαβ(t) =PtTα − PtTβ∑βi=α+1 τiPtTi

with∑βi=α+1 τiPtTi is the difference between two bonds, PtTα − PtTβ , and

therefore a traded asset; the forward swap rate is a local martingale in theforward swap measure Pαβ associated to the numeraire

∑βi=α+1 τiPtTi . The

SDE followed by the swap rate is under Pαβ

dsαβ(t) =β∑

k=α+1

∂sαβ∂Lk

σk(t)aC(Lk)dZk (8.4)

Note that as we use a new measure, we have changed our notation for theBrownian motions from Wk to Zk. As a is not correlated with the Libors,its dynamics remains the same in the forward and forward swap measures.The “freezing” argument consists in assuming that the terms ∂sαβ

∂Lkand C(s)

C(Li)

are almost constant and therefore equal to their values at the spot. We notesαβ(t = 0) ≡ s0 below. Therefore, the SDE (8.4) can be approximated by

dsαβ 'β∑

k=α+1

a∂sαβ∂Lk

(L0)σk(t)C(L0

k)C(s0)

C(sαβ)dZk

= σαβ(t)aC(sαβ)dZt

with

σαβ(t)2 =β∑

i,j=α+1

ρij(t)σi(t)σj(t)(C(L0

i )C(s0)

)(C(L0

j )C(s0)

)∂sαβ∂Li

(L0)∂sαβ∂Lj

(L0)

The multi-dimensional process driving the Libors and the swap rate has beendegenerated into a two-dimensional (resp. one-dimensional) stochastic (resp.local) volatility model (resp. for a = 1). Using the asymptotic method in-troduced in chapters 5 and 6, we can derive an asymptotic swaption impliedvolatility. We explicitly do the computation for the CEV LMM as an example.


Swaption implied volatility for CEV LMM

By definition, we know that the square of the (Dupire) local volatility asso-ciated to the driftless swap process is equal to the mean value (in the swapmeasure Pαβ) of the square of the swap stochastic volatility conditional to thelevel of the swap

(σαβloc)2(t, s) = C(s)2σαβ(t)2Eαβ [a2|sαβ = s]

Taking a = 1 and C(x) = xβ , we obtain that the swap rate follows also aCEV process with the same parameter β

σαβloc(t, s) = σαβ(t)sβ

with

σαβ(t)2 =β∑

i,j=α+1

ρij(t)σi(t)σj(t)(L0i

s0

)β (L0i

s0

)β∂sαβ∂Li

(L0)∂sαβ∂Lj

(L0)

Doing a change of local time, the local volatility becomes time-independentand from example 5.6, we have that the swaption implied volatility is givenat the first-order by

σαβBS(Tα,K) =(1− β) ln

(Ks0αβ

)(K1−β − s01−β

αβ

)√∫ Tα

0σαβ(t)2dt√Tα

(8.5)

1 +(β − 1)2

24

(s+ s0

αβ

2

)2β−2 ∫ Tα

0

σαβ(t)2dt

Swaption implied volatility for FL-SV LMM

In the same way, assuming that

C(Lk) = βkLk + (1− βk)L0k

and a is driven by an uncorrelated Heston process, we have that the swaprate follows a Heston stochastic volatility model for which we have an exactsolution (modulo a Fourier integration) for a swaption.The “freezing argument” is crucial to obtain an analytical approximation forthe swaption implied volatility. However, its theoretical and numerical validityis unclear. Also if we generalize the functional form Φk(a, Lk) (8.2.1), it is notat all obvious how to use the freezing argument. In the following, we explainhow to calibrate the Libor volatility σk(t) using this analytical approximationto at-the-money swaption implied volatilities. Then, we justify in the nextsection the “freezing” argument and obtain more accurate approximationsusing the heat kernel expansion technique.


TABLE 8.2: Liborvolatility triangle.

1 σ10 0 0 02 σ20 σ21 0 03 σ30 σ31 σ32 04 σ40 σ41 σ42 σ43

Libor volatility triangle

The volatility parameters σk(t) are calibrated to the at-the-money swaptionimplied volatilities noted σmkt

αβ . In order to reduce the complexity of ageneral time-dependent function, we assume that the volatilities σk(t) areconstant piecewise functions on the tenor intervals:

σk(t) = σkq ∀ t ∈ [Tq−1, Tq)

As the Libor Lk(t) is fixed when t ≥ Tk, σk(t) cancels after this date. There-fore the volatilities [σkq] describe a triangle called Libor volatility triangle (seeTable 8.2). Without any additional specification, we have n(n+1)

2 parame-ters σkq with n the number of Libors. Using the approximation (8.5), anat-the-money swaption IV is given by

σαβBS(Tα, s0αβ)2Tα ' (s0

αβ)2(β−1)Σαβ

(1 + Σαβ(s0

αβ)2(β−1) (β − 1)2

24

)

with Σαβ =∫ Tα

0σαβ(t)2dt. By inverting this formula, we obtain

Σαβ = Σmktαβ

with

Σmktαβ =

−1 +√

1 + 16 (β − 1)2σαβBS(Tα, s0

αβ)2

(s0αβ)2(β−1) (β−1)2

12

The expression Σαβ is a quadratic function in the variables σiq. Smooth-ing constraints must be chosen in order to have a well-defined optimizationproblem. The objective function for the calibration is

min[σiq ]

∑α,β

((Σmktαβ

)2 − (Σαβ)2)

+ λP

raised by a penalty function P which can be [142, 137]

P =n∑i=1

n∑j=i

σij(−σij−1−σij+1−σi−1,j−σi+1,j +4σi,j)+ε

n∑i=1

n∑j=i

(σi,j−σ0ij)

2


with σ0ij a fixed Libor volatility triangle or

P = ε

n∑i=1

(σij − σi+1j+1)2

The last penalty function ensures that σk(t) stays close to a time-homogeneousLibor volatility.As n is usually large (n ≈ 40), we face a large optimization problem. In orderto decrease the number of parameters to be optimized, we define σkq with asmall number of parameters. In this context, two simple functional forms canbe used.

Volatility parametrization

The first one, and the simplest, is called Rebonato parametrization [138]. σkqdepends on n+ 3 parameters φk (k = 1, ..., n) and a, b, c:

σiq = φi

(1 + (a(Ti−1 − Tq−1) + d)e−b(Ti−1−Tq−1)

)In the second one [7], σiq depends on 2n parameters (φi, ϕi)i=1,··· ,n

σiq = φiϕq , q = 1, · · · , i

In practice, in both parametrizations, the parameters φi are calibrated exactlyto the caplets ATM which are given for the CEV LMM by

φ2i =

(Σmktii+1

)2∑iq=1 ϕ

2qτq

For example, in the Rebonato parametrization, we have

φ2i =

(Σmktii+1

)2∑iq=1

(1 + (a(Ti−1 − Tq−1) + d)e−b(Ti−1−Tq−1)

)2τq

The closer φi is to 1, the more [σiq] is time-homogeneous. Note that in thecase of a full swaption matrix, an exact calibration can be done which doesnot require any specific optimization subroutine [7] (see chapter 7). However,in order to get the full swaption matrix, an extrapolation must been doneon the market swaption surface. One can observe that some complex Liborvolatilities can be obtained if the extrapolation is not done properly. More-over, the calibration depends on this extrapolation scheme, a feature that itis not appropriate.In the following, we illustrate the difference between the short rate-model,HW2, and the BGM model with 2-factors as well (Fig. (8.2)). Using a time-dependent volatility, the HW2 model has been calibrated exactly to ATM


FIGURE 8.2: Comparison between HW2 and BGM models.

swaptions with a 10 year tenor. We have no guarantee on the error for theother swaption IVs (for example swaptions with tenor=2Y). The BGM modelhas been calibrated to ATM swaptions 1Y-2Y-10Y. Although the calibrationis not exact but rather based on a optimization routine, we can see that theerror is negligible.

Once the model is calibrated, the pricing of a derivative is performed with aMC pricer (see appendix B for a brief introduction to MC methods).

8.2.2 Pricing with a Libor market model

Monte-Carlo simulation

By choosing an appropriate measure such as the forward (terminal) measurePn where Tn coincides with the last coupon date of a derivative product orsuch as the spot Libor measure as defined below, the Libor dynamics acquiresa complicated drift. These drifts which are reminiscent of the non-Markoviannature of the HJM model are too time-consuming in a MC simulation.

In order to determine these drifts, it is necessary to specify a measure. Thereare usually two alternatives to choose from: the forward measure Pn associ-ated to the numeraire PtTn or the spot Libor measure Ps associated to the


numeraire

Ps(t) ≡PtTβ(t)−1∏β(t)−1

j=1 PTj−1Tj

= PtTβ(t)−1

β(t)−1∏j=1

(1 + τjLj(Tj))

The discontinuous index β(t) is equal to q if t ∈ [Tq−2, Tq−1).

Next, we compute the dynamics of Libors under the measure Pn and Ps. Aspreviously covered, the Libor Li(t) is driftless under the measure Pi (associ-ated to the bond PtTi). To pass from Pi to Pn (resp. Ps), we need to computethe volatility of the ratio PtTn

PtTi(resp. Ps(t)

PtTi) of numeraires. As the bond PtTk

is by definition

PtTk = PtTβ(t)−1

k∏j=β(t)

11 + τjLj(t)

we have1

PtTnPtTi

=n∏

j=i+1

11 + τjLj(t)

Ps(t)PtTi

=

∏β(t)−1j=i+1 (1 + τjLj(Tj))∏i

j=β(t)1

1+τjLj(t)

The volatility of the bond PtTn (resp. Ps(t)) minus the volatility of the bondPtTi is

[σPtTn − σPtTi ].dW = −n∑

j=i+1

τjσj(t)Φj(a, Lj)dWj

1 + τjLj(t)

[σPs(t) − σPtTi ].dW =i∑

j=β(t)

τjσj(t)Φj(a, Lj)dWj

1 + τjLj(t)

1We only explicit the computation for n > i.


From proposition 2.1, the Libor Li(t) in the forward measure Pn is

dLi(t) = −σi(t)Φi(a, Li)n∑

j=i+1

ρij(t)τjσj(t)Φj(a, Lj)1 + τjLj(t)

dt

+σi(t)Φi(a, Li)dZi , i < n (8.6)

dLi(t) = σi(t)Φi(a, Li)i∑

j=n+1


dt

+σi(t)Φi(a, Li)dZi , i > n (8.7)dLn(t) = σn(t)Φn(a, Ln)dZn (8.8)

Also the Libor dynamics in the spot Libor measure Ps is

dLi(t) = σi(t)Φi(a, Li)i∑

j=β(t)


dt

+σi(t)Φi(a, Li)dZi (8.9)

REMARK 8.1 Limit n → ∞ We take the continuous limit n → ∞in the equation (8.9). The Libors converge to the instantaneous forward ratecurve fTt = L(t, T,∆T ). Then we have the SDE

dfTt = σ(t, T )ΦT (at, fTt )∫ T

t

ρ(t, T, T ′)σ(t, T ′)ΦT ′(at, fT′

t )dT ′dt

+σT (t)ΦT (at, fTt )dZtT

This is the HJM dynamics. Note that in the limit n → ∞, the n Brown-ian motions converge to a Brownian sheet [33] which is a two-dimensionalcontinuous process parameterized by the variables t and T satisfying

dZtT dZtT ′ = ρ(T, T ′)dt

When dropped in the MC pricing, the processes (8.6, 8.7, 8.8, 8.9) should bediscretized.

Discretization: Log-Euler scheme

As the SDEs above are already fairly complex, it seems unreasonable to use ahigher-order discretization scheme such as the Milstein scheme (see appendixB ). We will therefore use a log-Euler scheme between two dates t and t+ ∆t


(∆ logLi(t) = logLi(t+ ∆t)− logLi(t))

∆ logLi(t) = σi(t)Φi(a, Li)Li(t)−1∆Zi , i < n

−

σi(t)Φi(a, Li)Li(t)−1n∑

j=i+1


+σi(t)2Φi(a, Li)2Li(t)−2

2

)∆t

∆ logLi(t) = σi(t)Φi(a, Li)Li(t)−1∆Zi , i > n

+

σi(t)Φi(a, Li)Li(t)−1i∑

j=n+1


−σi(t)2Φi(a, Li)2Li(t)−2

2

)∆t

∆ logLn(t) = −σn(t)2Φn(a, Ln)2Ln(t)−2

2∆t+ σn(t)Φn(a, Ln)Ln(t)−1∆Zn

Assuming that we simulate the n Libors under the forward measure Pn, thecomputation of the drift terms involves n(n−1)

2 operations which can be quitelarge. In the following, we explain how to reduce the number of operations toO(r × n) [113].As we use a low-rank correlation

ρij(t) =r∑

k=1

bik(t)bjk(t)

the drift term becomes

−n∑

j=i+1


= −r∑

k=1

bik(t)n∑

j=i+1

bjk(t)τjσj(t)Φj(a, Lj)1 + τjLj(t)

In particular, if we precompute the n terms xj = τjσj(t)Φj(Lj ,a)1+τjLj(t)

, we can define

ek,i ≡ −n∑

j=i+1

bjk(t)xj , i < n

ek,i ≡i∑

j=n+1

bjk(t)xj , i > n

ek,n ≡ 0

We deduce the recurrence equations

ek,i = ek,i+1 − xi+1bi+1,k(t) , i < n

ek,i = ek,i−1 + xibi,k(t) , i > n


and the SDEs for the Libors can be re-written as a function of the r×n terms[e]k,i

∆ log(Li(t)) = −12σi(t)2Φi(a, Li)2Li(t)−2∆t

+ σi(t)Φi(a, Li)Li(t)−1r∑

k=1

bik(ek,i∆t+ ∆Wk)

with ∆Wkk=1,··· ,r r-uncorrelated Brownians. The number of operations in-volved to compute the terms ek,i is r×n and the total number of operationsinvolved to compute the drift in this procedure is O(r × n). This algorithmconsiderably reduces the time involved in the computation of the drift.In order to decrease the discretization error, we can go one step further anduse a predictor-corrector scheme as explained below.

Predictor-corrector scheme

The predictor-corrector scheme consists of two steps:In a first step, we use the log-Euler scheme (with the drift algorithm above)to compute the Libors at the step t+∆t, noted Li(t+∆t). This Libor is thenused to average the drift by

xnewj =

τjσj(t)2

(Φj(a, Lj)

1 + τjLj(t)+

Φj(a, Lj)

1 + τjLj(t)

)

In the second step, using the same Brownian motion as before, we recomputethe Libors with this new drift

∆ log(Li(t)) = −12σi(t)2Φi(a, Li)2Li(t)−2∆t

+ σi(t)Φi(a, Li)Li(t)−1r∑

k=1

bik(enewk,i ∆t+ ∆Wk)

Note that there also exists a discretization consistent with the discretizationof the spot measure [94], where the bonds are arbitrage free in the discretizedLibor SDE.

8.3 Markovian realization and Frobenius theorem

The MC pricing for a LMM is too lengthly due to the complicated drifts whichare reminiscent of the non-Markovian nature of the HJM model. For exampleif we use the terminal forward measure Pn, we need to simulate n Libors. For


a derivative product with a maturity of 20 years and semi-annual Libors, n isequal to 40.As a consequence, a LMM requires to simulate a high number of Markovprocesses. This is a severe drawback which does not occur in the short-ratemodel framework where the whole yield curve is modeled by an instantaneousinterest rate.In order to overcome this difficulty, we can try to represent the Libors as func-tionals of low-dimensional Markov states (typically one or two). In general,such a map between Libors and a low-dimensional Markov process doesn’texist. For example, in exercise 8.1 we show that in the one-factor log-normalBGM model the Libors can not be written as a function of a one-dimensionalprocess.In this section, we give a necessary and sufficient condition to show that a n-dimensional Ito process (possibly infinite n =∞) can be written as a functionof a low-dimensional Ito process r < n. This criteria is based on the Frobeniustheorem.Let Xt be a n-dimensional Ito process following the Stratonovich SDE

dXt = V0dt+d∑i=1

VidW it , Xt=0 = X0 (8.10)

Without any loss of generality, we have assumed that we have a time-homogeneousSDE. A time-inhomogeneous SDE can be put into this normal form (8.10) byincluding an additional state Xn+1

t

dXn+1t = dt

DEFINITION 8.1 Markovian representation We say that the processXt admits a r-dimensional Markovian representation (r < n) if there exists asmooth function G : Rr → Rn such that X = G(z) with z a r-dimensional Itoprocess.

THEOREM 8.1 Frobenius theorem [64]If the free Lie algebra (see B.7 in appendix B) generated by (V0, V1, · · · , Vd)

has a constant dimension r (as a vector space) then the SDE (8.10) admitsa r-dimensional Markovian representation: X = G(z). We note f1, · · · , fr abasis for the free Lie algebra. The Markovian representation G(·) is given by

G(z1, · · · zr) = ez1f1ez2f2 · · · ezrfrX0

where ezifi is the flow (see the definition B.6 in appendix B) along the vectorfi at time zi.

As an example, we classify the one-factor LV LMMs which admits a two-dimensional Markovian representation.


Example 8.1 One-factor LV LMMsA one-factor LV LMM is defined by the following SDE for each Libor Lii=1,··· ,nin each forward measure Pi

dLi = σi(t)C(Li)dWt

In the terminal measure Pn, we have

dLi = σi(t)C(Li)n∑

j=i+1

σj(t)C(Lj)τj1 + τjLj

dt+ σi(t)C(Li)dZt

Using the Stratonovich calculus, we obtain the time-homogeneous SDE

dLiσi(u)C(Li)

=

−12σi(u)∂iC(Li) +

n∑j=i+1

σj(u)C(Lj)τj1 + τjLj

dt+ dZt

(8.11)du = dt (8.12)

for which we derive the vector fields

V0 =

−12σi(u)2C(Li)∂iC(Li) + σi(u)C(Li)

n∑j=i+1

σj(u)C(Lj)τj1 + τjLj

∂i + ∂u

V1 = σi(u)C(Li)∂i

where we have set ∂i ≡ ∂Li . We deduce that

[V0, V1] =(∂uσi(u)C(Li) +

σi(u)3

2C(Li)2∂2

i C(Li))∂i

− σj(u)C(Lj)σi(u)C(Li)n∑

j=i+1

σj(u)∂j

(C(Lj)τj1 + τjLj

)∂i

Therefore, a one-factor LV LMM admits a 2-dimensional Markovian represen-tation if and only if there exists a smooth function f : R×Rn → R such that[V0, V1] = f(u, L)V1. This gives the constraint(∂u lnσi(u) +

σi(u)2

2C(Li)∂2

i C(Li))− σj(u)C(Lj)

n∑j=i+1

σj(u)∂j

(C(Lj)τj1 + τjLj

)= f(u, L) ∀ i = 1, · · · , n

For example if we assume that τi = τ ∀ i, we cancel the sum over the index kby taking

C(Li) = (1 + τLi)


The constraint above reduces to

∂u lnσi(u) = f(u) ∀ i = 1, · · · , n

which shows that the volatility σi(·) should be a separable function in Ti andin time t

σi(t) = Φiν(t)

with ν(t) ≡ eR t0 f(u)du. This is the one-factor HW model

dLi = Φiν(t) (1 + τLi) dWt

From the theorem 8.1, the 2d-Markovian representation is given by the flow

L(t) = ez0V0ez1V1L0

The flow along the vector V1 corresponds to solve the following ODE at timet = z1

dLi(t)dt

= σi(u) (1 + τLi)

du

dt= 0

We obtain

ln(

1 + τLi(z1)1 + τLi(0)

)= τΦi

∫ z1

0

ν(s)ds

u(t) = u0

Similarly, the flow along the vector field V0 is

ln(

1 + τLi(z0)1 + τLi(0)

)= Φi

−12

Φi +n∑

j=i+1

Φj

τ2

∫ z0

0

ν(s)2ds

u(t) = z0

We get our final result thanks to the composition of the two flows describedabove

ln(

1 + τLi(z0, z1)1 + τLi(0)

)= Φi

−12

Φi +n∑

j=i+1

Φj

τ2

∫ z0

0

ν(s)2ds

+ τΦi∫ z1

0

ν(s)ds

u(t) = z0


By applying Stratonovich’s lemma on both sides of this equation and identi-fying with the SDEs (8.11) and (8.12), we get

dz0 = dt

dz1 =ν(t)ν(z1)

dWt

The first Markov state is the time t.

As seen above, a one-factor LV LMM admits a 2-dimensional Markovian rep-resentation if and only if the LV function is a displaced diffusion model

dL = (1 + τL)σ(t)dWt

Modulo several approximations, an (almost exact) Markovian representationcan be found for general LV LMMs. This is briefly discussed in the nextsubsection.

Markovian approximation

The simplest trick to account for these drifts is to freeze the Libor Li to thespot Libor L0

i inside the drifts. The BGM model becomes log-normal and canbe mapped to a low-rank Markov model if the volatility σi(t) is a separablefunction (see exercise 8.2)

σi(t) = Φiν(t)

This approximation is efficient for low volatility and small maturities butbecome quite off otherwise and it should not be used for MC pricing.After our short review of the calibration and pricing of the LMMs, we makethe connection with the main subject of this book and explain how to deducean accurate approximation for swaption implied volatilities for SV-LMMs.This is particularly useful for the calibration procedure as we saw previously.Although this technique is equally applicable to generic SV-LMMs, we applyit to a LMM based on a SABR dynamics for forward rates.

8.4 A generic SABR-LMM model

While keeping the idea of driving the Libors with a CEV dynamics we leteach Libor have its own elasticity parameter β. As the market standard forparameterizing swaption smiles is the SABR formula (see Fig. 8.3), we use alog-normal dynamics for the stochastic volatility - note that references [45, 46],[136] use the Heston dynamics.


Calibrated SABR vol (EUR 20x30 swaption)(alpha=0.53%,beta=0.13,rho=-13%, nu=24%)

5%

7%

9%

11%

13%

15%

17%

2.0% 4.0% 6.0% 8.0%strike

Marketvol

SABRvol

FIGURE 8.3: Calibration of a swaption with the SABR model.

Moreover, with the aim of enhancing the model’s ability to fit both capletand swaption smiles, we let the stochastic volatility process have non-zerocorrelation with the forward rates.In our case, because of the presence of stochastic volatility, we prefer to workin the spot measure Ps. In this measure, the dynamics of the forward ratesin the SABR-LMM model reads :

dLk = a2Bk(t, F )dt+ σk(t)aCk(Lk)dZkda = νadZn+1; , dZidZj = ρij(t)dt i, j = 1, · · · , n+ 1

with

Ck(Lk) = φkLβkk

Bk(t, F ) =k∑

j=γ(t)

τjρkjσk(t)σj(t)Ck(Lk)Cj(Lj)1 + τjLj

We introduce the constants φk for normalization purposes, so that σk(t =0) = 1, for all k.The forward rate dynamics under the forward measure Pk is much simpler


and is given by:

dLk(t) = σk(t)aCk(Lk)dWk

da = −νa2k∑

j=γ(t)

τjρjaσj(t)Cj(Lj)1 + τjLj

dt+νadWn+1, dWkdWn+1 = ρka(t)dt

with initial conditions a(t = 0) = a0 and Lk(t = 0) = L0k.

Note that as shown in chapter 6, the log-normal SABR model defines a mar-tingale as long as 0 ≤ βk < 1 or ρka ≤ 0 for βk = 1. The possibility of momentexplosions due to volatility being log-normally distributed in the case of theSABR-LMM model is an open question.

8.5 Asymptotic swaption smile

Our strategy for getting an approximate swaption smile involves the followingtwo main steps:

• Firstly we derive an approximation to the ELV for the swap rate athand.

• From the expression of this local volatility we derive an approximateexpression for the implied volatility.

8.5.1 First step: deriving the ELV

Let sαδ be the forward swap rate starting at Tα and expiring at Tδ. sαδsatisfies the following driftless dynamics in the forward swap measure Pαδ(associated to the numeraire Cαδ(t) =

∑δi=α+1 τiPtTi):

dsαδ =δ∑

k=α+1

∂sαδ∂Lk

σk(t)aCk(Lk)dZk

In order to be thorough, we give the dynamics of the Libors and the stochasticvolatility under the forward swap measure Qαδ:

dLk = a2bk(t, L)dt+ σk(t)aCk(Lk)dZkda = −νa2ba(t, L)dt+ νadZn+1; , dZidZj = ρij(t)dt i, j = 1, · · · , n+ 1


with the drifts

bk(t, L) =δ∑

j=α+1

(2.1(j≤k) − 1)τjPtTjCαδ(t)

(8.13)

max(k,j)∑i=min(k+1,j+1)

τiρkiσi(t)σk(t)Ci(Li)Ck(Lk)1 + τiLi

ba(t, L) =δ∑

j=α+1

τjPtTjCαδ(t)

j∑k=γ(t)

τkCk(Lk)ρkaσk(t)1 + τkLk

(8.14)

We know that there exists a unique local volatility function σαδloc – the ELV –which is consistent with sαδ’s smile. It is given by the expectation of the localvariance conditional on sαδ’s level. For convenience, we prefer to express thedynamics of sαδ in the Bachelier – rather than the log-normal – framework:

dsαδ ≡ σαδloc(t, sαδ)dWt

σαδloc(t, sαδ) is given by:

(σαδloc)2(t, s) ≡ Eαδ[δ∑

i,j=α+1

ρij(t)σi(t)σj(t)aCi(Li)aCj(Lj)∂sαδ∂Li

∂sαδ∂Lj|sαδ = s]

=δ∑

i,j=α+1

ρij(t)σi(t)σj(t)

∫B aCi(Li)aCj(Lj)

∂sαδ∂Li

∂sαδ∂Lj

p(t, a, L|L0, α)(da∏i dLi)∫

B p(t, a, L|L0, α)(da∏i dLi)

(8.15)

where the integration domain is restricted to the set B = Lii|sαδ = s andp(t, a, L|L0, α) is the joint density for the forward rates Li and the stochasticvolatility a in the forward swap measure Qαδ.There are two steps in evaluating this expression:

• We need an approximate expression for p(t, a, L|L0, α).

• We have to compute the integration over B.

The first step is achieved via the heat kernel expansion technique. It boilsdown to the calculation of the geodesic distance, the Van-Vleck-Morette de-terminant and the gauge transport parallel between any two given points, inthe metric defined by the SABR-LMM model as explained in chapter 4.

Hyperbolic geometry

While this is generally a non-trivial task, the geodesic distance is analyticallyknown for the special case of the geometry that the SABR-LMM model de-fines. It is also the case for most popular models (Heston, SABR, 3/2-SVM)which admit a large number of Killing vectors as seen in chapter 6.


By definition, the metric (4.78) (at t = 0) is given by

ds2 =2

ν2a2

n∑i,j=1

ρijνdLiCi(Li)

νdLjCj(Lj)

+ 2n∑i=1

ρiaνdLiCi(Li)

da+ ρaada2

Here ρij ≡ [ρ−1]ij , (i, j) = (1, · · · , n) and ρia ≡ [ρ−1]ia are the componentsof the inverse of the correlation matrix ρ. After some algebraic manipula-tions, we show that in the new coordinates [xk]k=1···n+1 (L is the Choleskydecomposition of the (reduced) correlation matrix: [ρ]i,j=1···n = [LL†]i,j=1···n)

xk =n∑i=1

νLki∫ Li

L0i

dL′iCi(L′i)

+n∑i=1

ρiaLika , k = 1, · · · , n

xn+1 = (ρaa −n∑i,j

ρiaρjaρij)12 a

the metric becomes

ds2 =2(ρaa −

∑ni,j ρ

iaρjaρij

)ν2

∑ni=1 dx

2i + dx2

n+1

x2n+1

Here ρij is the inverse of the reduced matrix [ρij ]i,j=1,...,n.

Geometry Hn+1

Written in the coordinates [xi], the metric is therefore the standard hyper-

bolic metric on Hn+1 modulo a constant factor2(ρaa−

Pni,j,k=1 ρ

iaρjaρij)

ν2 thatwe integrate out by scaling the time. By definition, the hyperbolic spaceis a (unique) simply connected n-dimensional Riemannian manifold with aconstant negative sectional curvature −1.The geodesic distance on Hn+1 is given by

THEOREM 8.2 Geodesic distance on Hn+1 [39]

The geodesic distance d(x, x′) on Hn+1 is given by

d(x, x0) = cosh−1

(1 +

∑ni=1(xi − x0

i )2

2xn+1x0n+1

)In particular for n = 1, we reproduce the geodesic distance (4.43) on thePoincare hyperbolic plane.

Using the geodesic distance on Hn+1 between the points x = (Lk, a) andthe initial point x0 = (L0k, α), the geodesic distance associated to the


SABR-LMM model is

d(x, x0) = cosh−1 (1+ (8.16)

ν2∑ni,j=1 ρ

ijqiqj + 2ν(a− a0)∑nj=1 ρ

jaqj + (a− a0)2ρaa

2(ρaa −∑ni,j=1 ρ

iaρjaρij)aa0

)

with

qi =∫ Li

L0i

dL′iCi(L′i)

ρµν are the components of the inverse correlation [(ρ−1)µν ]µ,ν=1,...,n+1.

An important property of the hyperbolic space Hn is that the heat kernelequation is solvable:

THEOREM 8.3 Heat kernel on Hn [95]

The heat kernel of the hyperbolic space Hn is given in even dimensions by

p2(m+1)(t, x|y) =(−12π

)m √2e−(2m+1)2t

4

(4πt)32

(1

sinh r∂

∂r

)m(∫ ∞

r

se−s24t

√cosh s− cosh r

ds

)

and in odd dimensions by

p2m(t, x|y) =(−12π

)me−m

2t

(4πt)12

(1

sinh r∂

∂r

)m (e−

r24t

)

Van-Vleck-Morette determinant

Using the explicit expression for the hyperbolic distance, the Van-Vleck-Morette determinant is

∆(a, L|α,L0) =(

d(a, L|α,L0)sinh(d(a, L|α,L0))

)n


8.5.2 Connection

The Abelian connection is given by (4.58)2

Ai =1

Ci(Li)

n∑j=1

ρij(bj(t, L)Cj(Lj)

− ∂jCj(Lj)2

)− νρiaba(t, L)

Aa =

1ν

n∑j=1

ρaj(bj(t, L)Cj(Lj)

− ∂jCj(Lj)2

)− νρaaba(t, L)

where we have used that

√g =

2n+1

2 det[ρ]−12

νa2∏ni=1 Ci(Li)

−νa2ba(t, L) (8.13) and a2bi(t, L) (8.14) are the drifts in the swap measure.Finally, the Abelian 1-form connection is

A =1ν

n∑j=1

(bj(t, L)Cj(Lj)

− ∂jCj(Lj)2

)(ν

n∑i=1

ρijdqi + ρajda

)

− ba(t, L)

(ν

n∑i=1

ρiadqi + ρaada

)In order to compute the log of the parallel gauge transport

ln(P)(a, q|α) = −∫CA

we need to know a parametrization of the geodesic curve on Hn+1. However,we can directly find ln(P)(a, q|α) if we approximate the drifts bk(t, L) andba(t, L) by their values at the Libor spots (and t = 0). A similar approxima-tion was done in the Hagan-al formula [100] as shown in remark 6.7. Modulothis approximation (see remark 4.3),

ln(P)(a, q|α) ' 1ν

n∑j=1

(bj(0, L0)Cj(L0

j )−∂jCj(L0

j )2

)(ν

n∑i=1

ρijqi + ρaj(a− α)

)

−ba(0, L0)

(ν

n∑i=1

ρiaqi + ρaa(a− α)

)

2

Ai =a2

2(bi −

1

2Ci∂iCi)

Aa = −ν2a2ba(L, t)

2


Saddle-point

At this stage, we need to perform the integration over B. This amounts tocomputing an integral of the following type:∫

f(L, a)e−1t φ(L,a)δ(g(L)) da

∏i

dLi (8.17)

where δ(g(L)) is introduced to restrict the integration to the hyperplane de-fined by g(x) = 0 – in our case the set of L such that sαβ(L) = s. φ(L, a)is related to the square of the geodesic distance. As we are interested in thelimit t→ 0, it is natural to use the saddle-point technique [5, 15] to computethis integral, introducing a Lagrange multiplier λ to enforce the constraintg(L) = 0. More precisely, the integral (8.17) is approximated in the limitt→ 0 by ∫

f(x)e−1t φ(x)dx ∼t→0 f(x∗)e−

1t φ(x∗) (1− t (8.18)(

−∂αβf2f

Aαβ +(∂αf

2f∂βγδφ+

18∂αβγδφ

)AαβAγδ

− 5∂αβγφ∂δµνφ

24AαβAγδAµν

))with Aαβ = [∂αβφ]−1, dx ≡

∏ni=1 dxi and x∗ the saddle-point (which mini-

mizes φ(x)). This expression can be obtained by developing φ(x) and f(x)in Taylor series around x∗. The quadratic part in φ(x) leads to a Gaussianintegration over x which can be performed. Details are left in appendix A.The saddle-point (a∗, L∗) is the point on the hyperplane sαβ = s which mini-mizes the geodesic distance that we have previously computed

(a∗, L∗i ) ≡ (a, Li) | mina,Li,λ

d2(x, x0) + λ(sαβ(L)− s)

The saddle-point (a∗, L∗) can then be computed efficiently. It is determinedby solving the following non-linear equations (q∗i = qi(L∗i ))

(ρia (a∗(s)−a0)ν +

∑nj=1 ρ

ijq∗j )d(a∗, q∗i )a∗(s) sinh(d(a∗, F ∗))

=λ

a0

∂sαδ∂qi|∗ (8.19)

with a∗(s) fixed by

a∗(s)2ρaa = (a0)2ρaa − 2νa0n∑i=1

ρiaq∗i + ν2n∑

i,j=1

ρijq∗i q∗j (8.20)

An approximation (which could be used as a guess solution in a numericaloptimization routine) can be obtained by linearizing these equations around


the spot Libor rates (i.e., qi = 0):

q∗i ≈∑nj=1 ρijωj(s− s0)∑np,q=1 ωpωqρpq

(8.21)

with ωi ≡ ∂sαδ∂qi

(qi = 0) and ρij = ρij − ρiaρja

ρaa . Note that when the strike isclose to the money, the saddle-point coordinates are close to the spot Liborsand a∗ = a0. In our experience, the solution of the linearized equations issufficiently accurate – this is what we have used to generate the swaptionsmiles shown in the following section.By plugging the asymptotic expression (4.67) for the density p(t, L, a|L0, α)into (8.15) and doing the integration over B using the saddle-point approxi-mation, we finally get the following expression for the effective local volatilityσαδloc at first order in time [104]: (∂n+1 ≡ ∂a, ∂i ≡ ∂Li i = 1, . . . , n)

(σαδloc)2(t, s) =n∑

i,j=1

ρijσi(t)σj(t)fij(a∗, L∗)(1 + 2t′n+1∑µ,ν=1

Aµν∂µνfij(a∗, L∗)

fij(a∗, L∗)

(8.22)

+2∂µfij(a∗, L∗)fij(a∗, L∗)

∂νψ(a∗, L∗)ψ(a∗, L∗)

−n+1∑γ,δ=1

Aγδ∂µfij(a∗, L∗)fij(a∗, L∗)

∂νγδd2(a∗, L∗))

with (a∗2(s), F ∗i i(s)) the saddle-point satisfying the equations (8.19,8.20)approximated by (8.21) and

fij(a, F ) = a2Ci(Li)Cj(Lj)∂sαδ∂Li

∂sαδ∂Lj

, ψ(a, F ) =√g∆P

Aαβ = [∂αβd2]−1

ln(P)(a, q|α) ' 1ν

n∑j=1

(bj(0, L0)Cj(L0

j )−∂jCj(L0

j )2

)(ν

n∑i=1


)

− ba(0, L0)

(ν

n∑i=1


)

∆(a, F ) =(

d(a, F )sinh d(a, F )

)n√g =

2n+1

2 det[ρ]−12

νa1+n∏ni=1 Ci(Li)

t′ =ν2

2(ρaa −∑ni,j=1 ρ

iaρjaρij)t; ρij ≡ [ρij ]−1

i,j=1,...,n

Here, by definition, a2bj(t, F ) and −va2ba(t, F ) are the drifts of the Libors(8.13) and the volatility (8.14) in the swap numeraire. The terms bj(0, F 0)


and ba(0, F 0) are small and can be neglected. As great an expression (8.22)may appear, it is analytical and the result of the straightforward applicationof methods which can be used in a similar fashion for all stochastic volatilitymodels listed in Table 8.1. It is important to point out that an additionalbenefit of using the techniques is that the expression (8.22) for the ELV isexact in the limit t→ 0.The effective local volatility for Libors Lk is given by plugging in a straight-forward manner δ = α+ 1 ≡ k in equation (8.22):

(σkloc)2(t, Lk) = σk(t)2fk(a∗, L∗k)(1 + 2ν2tA (8.23)(

−3 + 2a∗∂a lnP2(a∗, Lk)− a∗A∂3ad

2(a∗, Lk)))

with

fk(a, Lk) = a2Ck(Lk)2 , A = (∂aad2)−1(a∗, Lk) (8.24)

ln(P)(a, F ) = − ∂kCk(L0k)

2ν(1− ρ2)(νqk − ρ(a− a0))

d(a, Lk) = cosh−1[1 +ν2q2

k − 2ν(a− a0)ρqk + (a− a0)2

2(1− ρ2)aa0]

a∗2 = (a0)2 + 2νa0ρqk + ν2q2k

Note that as explained above, we have neglected the drift of the stochasticvolatility a.

8.5.3 Second step: deriving an implied volatility smile

Given the effective local volatility for sαδ we now need to compute its smile.The great benefit of using the ELV is that we have shrunk an (n + 1)-dimensional problem down to a one-dimensional problem. Solving numeri-cally the one-dimensional forward PDE is a natural method for obtaining thewhole smile. We prefer to derive an accurate analytical approximation, usingthe method explained in chapter 5:To start with, we re-write the swap rate dynamics under Pαδ as

dsαδ =σlocαδ (t, sαδ)σlocαδ (t, s0

αδ)σlocαδ (t, s0

αδ)dWt

Doing a change of time, we obtain

dsαδ = C(t′, sαδ)dWt′

with t′ =∫ t

0

(σlocαδ (u, s0

αδ))2du and C(t′, f) = σloc

αδ (t,sαδ)

σlocαδ (t,s0αδ)

. Finally, by applying

the formula (5.40), this yields the following asymptotic formula for σαδBS(K,Tα)


in the short-time limit:

σαδBS(K,Tα) =

√∫ Tα0

(σαδloc)2(u, sαδ0 )duTα

σαδBS(K)0 (8.25)(1 +

12

∫ Tα

0

(σαδloc)2(u, sαδ0 )duσαδBS(K)1

)

with

σαδBS(K)0 =ln(Ksαδ0

)∫Ksαβ0

dxC(x)

σαδBS(K)1 = − 1(∫Ksαδ0

dxC(x)

)2 ln(

(σαδBS(K)0)2Ksαδ0

C(K)

)

+1

(σαδloc)(0, sαβ0 )2

∂t

(σαδloc(0,fav)

σαδloc(0,sαβ0 )

)C(fav)

with C(f) ≡ σαδloc(0,K)

σαδloc(0,sαδ0 ), fav ≡ sαδ0 +K

2 and σαδloc(t, s) given by (8.22). Note that

this expression for σαδBS is exact when Tα → 0 and becomes identical to thefamiliar BBF formula (5.39) involving the harmonic average of local volatility.

8.5.4 Numerical tests and comments

We have tested our asymptotic swaption formula (8.25) in various scenarios.In the following graphs, an x × y swaption means an option maturity of xyears, a swap length of y years and a tenor of one year. The SABR-LMMmodels used semi-annual Libors. We have chosen a two factor correlationstructure as in [44]

dWk =2∑r=1

θre−kr(Tk−1−t)dZr; dZ1dZ2 = ρdt (8.26)

with θ1 = 1, θ2 = 0.5, k1 = 0.25, k2 = 0.04, ρ = −40%.

Firstly, by assuming that the CEV parameters have the same βk = βand the volatility of the volatility ν = 0, the model degenerates into the CEVLMM model: we can thus compare our smiles against the Andersen-Andreasen


asymptotic results [44]. The expressions above degenerate into

fij(L) = Ci(Li)Cj(Lj)∂sαδ∂Li

∂sαδ∂Lj

d(L) =

√√√√2n∑

i,j=1

ρijqiqj

ln(P)(q) =n∑j=1

(bj(L0, 0)Cj(L0

j )−∂jCj(L0

j )2

)n∑i=1

ρijqi

∆(L,L0) = 1

√g =

2n2 det[ρ]−

12∏n

i=1 Ci(Li)

with the saddle-points satisfying the non-linear equations (modulo the con-straint sαδ = s)

n∑j=1

ρijq∗j = −λ4∂sαδ∂qi|∗

The CEV model is calibrated (using an optimization routine) against ATMcaplets (with β = 0.2) and to all ATM swaptions x × 2 and x × 10 in orderto set the Libor volatility term structure (i.e., σi(t)). Calibration was doneusing the JPY curve on February 17th 2007. The two approaches yield veryclose results (see Fig. 8.4). Secondly, we take ν = 20% with a zero correlation between the volatilityand the Libors (i.e., ρia = 0). We calibrate the SABR-LMM model to allcaplet smiles (by adjusting φi for the ATM volatility and βi for the skew) andwe set the Libor volatility term structure (i.e., σi(t)) by matching (best fit)all ATM swaptions x × 10 and x × 2. Calibration was done using the EURcurve on February 17th 2007. Our formula is tested against a Monte-Carlosimulation with a time step ∆t = 0.02 and 217 paths3 (see Fig. 8.5). We alsoplot the calibrated values of beta (see Fig. 8.6). Thirdly, we choose ρia to be non-zero and run similar tests as above (seeFig. 8.7, 8.8). In the generic SABR-LMM model, once caplet smiles are calibrated, we canuse the parameters φi and βi to generate different swaption smiles using theparameters v and ρia. We illustrate this capability of the model we proposewith the example of the 10 × 10 swaption smile (caplets calibrated to theEUR curve-February 17th 2007) (see Fig. 8.9). Note that by construction,the 10× 10 ATM swaption is calibrated.

3We have used a predictor-corrector scheme with a Brownian bridge.


13%

15%

17%

19%

21%

23%

25%

27%

29%

31%

33%

50% 70% 90% 110% 130% 150% 170% 190%

Moneyness

Imp

lied

Vo

lati

lity

(%

)

Swaption10Y0.5Y, And

Swaption5Y15Y, And

Swaption10Y10Y, And

Swaption10Y0.5Y

Swaption5Y15Y

Swaption10Y10Y

FIGURE 8.4: The figure shows implied volatility smiles for swaption 10×0.5, 5 × 15 and 10 × 10 using our asymptotic formula and the Andersen-Andreasen expression. Here ν = 0 and β = 0.2.

Conclusion

We have investigated a LMM model coupled to a SABR stochastic volatilityprocess. Not only does this model allow each Libor to have its own value ofthe skew parameter β, but it also makes it possible to correlate the Liborswith the stochastic volatility process.By using the heat kernel expansion technique, we have derived accurate ex-pressions for swaption implied volatilities. We have tested the accuracy of ourswaption asymptotic formula against an exact (MC) pricing and other knownanalytical approximations.We have shown that the additional degree of freedom afforded by the corre-lation between the Libors and the stochastic volatility makes it possible toadjust swaption smiles once caplet smiles have been individually calibrated.This decoupling will assist in the joint calibration of caplet and swaptionsmiles.


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 6 11 16 21 26 31 36 41 46

Libor Number

Be

ta

EUR USD JPY

FIGURE 8.5: The figure shows implied volatility smiles for swaption 10×0.5, 5 × 15 and 10 × 10 using our asymptotic formula and a MC simulation.Here ν = 20% and ρia = 0%.

8.6 Extensions

The formula can be slightly extended to include a mean-reverting drift forthe stochastic volatility. Under the spot Libor measure, we assume that thevolatility follows the process

da = −νa2ψa(a)dt+ νadZn+1

with ψa(a) a general analytical function of a (the scaling a2 in front of ψa(a)has been put for convenience). After some algebraic computations, we derive


10.0%

10.5%

11.0%

11.5%

12.0%

12.5%

13.0%

13.5%

50% 70% 90% 110% 130% 150% 170% 190%

Moneyness

Imp

lied

Vo

lati

lity

(%

)

CMS10Y0.5Y

CMS5Y15Y

CMS10Y10Y

CMS10Y0.5Y, MC

CMS5Y15Y, MC

CMS10Y10Y, MC

FIGURE 8.6: CEV parameters βk calibrated to caplet smiles for EUR-JPY-USD curves. The x coordinate refers to the Libor index.

the new Abelian 1-form connection

A =1ν

n∑j=1

(bj(t, L)Cj(Lj)

− ∂jCj(Lj)2

)(ν

n∑i=1

ρijdqi + ρajda

)

− (ba(t, L) + ψ(a))

(ν

n∑i=1

ρiadqi + ρaada

)

Using a similar approximation as before, i.e., Cj(Lj) ∼ Cj(L0j ) and ψa(a) ∼

ψa(α), we obtain for the parallel gauge transport (see remark 4.3)

ln(P)(a, q|α) ∼ 1ν

n∑j=1

(bj(0, L0)Cj(L0

j )−∂jCj(L0

j )2

)(ν

n∑i=1


)

−(ba(0, L0)ψ(α)

)(ν

n∑i=1


)


Swaption 5*15

5%

7%

9%

11%

13%

15%

17%

19%

21%

23%

50% 70% 90% 110% 130% 150% 170% 190%

Moneyness

Imp

lied

Vo

lati

lity

(%

)

USD USD, MC

JPY JPY, MC

EUR EUR, MC

FIGURE 8.7: The figure shows implied volatility smiles for swaption 5×15(EUR, JPY, USD curves-February 17th 2007) using our asymptotic formulaand a MC simulation. Here ν = 20% and ρia = −30%.

Finally, the implied volatility is obtained using our general formula (8.25).Note that the metric and the geodesic equations remain unchanged when weonly modify drift terms.

8.7 Problems

Exercises 8.1 Markov LMMAs seen in this chapter, the MC simulation of a LMM is time-consuming.The main problem comes from the drifts which are reminiscent of the non-Markovian nature of the HJM model. To circumvent this difficulty, one can


Swaption 10*10

5%

7%

9%

11%

13%

15%

17%

19%

21%

23%

50% 70% 90% 110% 130% 150% 170% 190%

Moneyness

Imp

lie

d V

ola

tili

ty (

%)

JPY

JPY, MC

EUR

EUR, MC

FIGURE 8.8: The figure shows implied volatility smiles for swaption 10×10 (EUR, JPY-February 17th 2007) using our asymptotic formula and a MCsimulation. Here ν = 20% and ρia = −30%.

try to find a low-dimensional Markov representation of the LMM. By this, weassume that each Libor can be written as a functional of the time t and alow-dimensional Ito diffusion process Yt

Li ≡ Li(t, Yt) , i = 1, · · · , n

In this problem, we make the additional assumption that Yt is a onedimensional process.The process follows a general Ito SDE

dY = µ(t, Y )dt+ σ(t, Y )dWt

1. Prove that it is always possible to assume that the process Yt can be writtenas Yt = f(t,Xt) with Xt a driftless process. By redefining the functional Li,we have Li = Li(t,Xt).


9.00%

10.00%

11.00%

12.00%

13.00%

14.00%

15.00%

16.00%

17.00%

50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170% 180% 190% 200%

Moneyness

Imp

lie

d V

ola

tili

ty(%

)

Swaption10Y10Y, nu=0%

Swaption10Y10Y,nu=20%, rho=0

Swaption10Y10Y,nu=20%, rho=-30%

Swaption10Y10Y,nu=20%, rho=-40%

Market

FIGURE 8.9: The figure shows implied volatility smiles for swaption 10×10 (EUR-February 17th 2007) using different value of (ν, ρ). The LMM hasbeen calibrated to caplet smiles and the ATM swaptions x× 2, x× 10.

Note: The process Xt has no financial interpretation. However, for a shortrate model, Xt can be seen as the instantaneous interest rate rt.2. For an affine short-rate model such as the HW2 model (see 8.2), obtain theexplicit map Li(t, xt, yt).3. Prove that the Markovian property imposes from (8.6) the following con-straint

σi = σ(t,X)∂XLi (8.27)

−σ(t,X)2∂XLi

n∑j=i+1

τj∂XLj1 + τjLj

= ∂tLi +12σ(t,X)2∂2

XLi (8.28)

In the following, we explicitly check that the BGM model does not havea Markov representation. The Libors are driven by a log-normal processσi = ψi(t)Li.


The equations (8.27,8.28) reduce to

ψi(t) = σ(t,X)∂X lnLi (8.29)

−σ(t,X)2∂XLi

n∑j=i+1

τj∂XLj1 + τjLj

= ∂tLi +12σ(t,X)2∂2

XLi (8.30)

Without loss of generality, we take Ln = X.4. Prove that the first constraint (8.29) is equivalent to

Li = ξi(t)Lψn(t)ψi(t)n

5. Using this solution, prove that the second constraint (8.30) reduces

−ψ2nξi(t)

n∑j=i+1

τjξj(t)Lψn(t)ψj(t)n

1 + τjξj(t)Lψn(t)ψj(t)n

= ξi(t)′ + ξi(t)(ψn(t)ψi(t)

)′L−1n

+12ψn(t)2ξi(t)

(ψn(t)ψi(t)

)2

6. Specifying this equation for i = n − 1, show that this equation has only asolution in the degenerate case ξn−1(t) = 0.So, although every short-rate models admit a Markovian representation byconstruction, it is not the case for a particular LMM. To be able to do that, wemust assume that the volatility of each Libor is a general Dupire local volatilityσi(t, Li). In the following, we explain how to construct a MF introduced firstin [110].In order to be able to simulate exactly the process Xt, we take Xt as aGaussian process

dX = σ(t)dWt (8.31)

with the initial condition X0 = 0.We introduce the deflated bond

ζi(t) =PtTiPtTn

i = 1, · · ·n

We recall that a Libor Li(t) is given by

τiLi(t) =ζi−1(t)ζi(t)

− 1 (8.32)

7. Prove that ζi ≡ ζi(t,X) satisfies the PDE

∂tζi(t,X) +12σ(t)2∂2

Xζi(t,X) = 0


8. Doing a change of local time, prove that this is equivalent to the heat kernelon R and the fundamental solution is

ζi(t′, X) =∫ ∞−∞

ζi(T ′i , Y )pG(T ′i , Y |t′, X)dY (8.33)

where t′ < T ′i .Therefore the process ζi(t,X) is completely characterized if we specify itsterminal value at t = Ti. Its terminal value will be determined in order tocalibrate exactly the implied volatility of a unique swaption starting at Ti.9. Prove that the fair value of a digital swaption option with an underlyingswap sαβ is in the measure Pn

Dαβ(K) = −PtTnEPn [β∑

i=α+1

τiζi(Tα, Xα)1(sαβ(Tα, Xα) > K)|Ft]

(8.34)

10. Assuming that sαβ(Tα, X) is a monotone function in X, prove that thedigital fair value (8.34) becomes

Dαβ(K) = −PtTnβ∑

i=α+1

τi

∫ ∞X∗

ζi(Tα, X)pG(T ′α, X|0)dX (8.35)

11. In particular for α = n− 1, β = n, prove that (8.35) becomes

Dn−1,n(K) = −PtTnτn(1−N (X∗))

with N (X) the cumulative Gaussian distribution. Given the caplet impliedvolatility σBS

n−1,n(K), we can determine X∗ as a function of K by

Ln(Tn−1, X∗) = K∗

12. Now, let us assume that ζi(t,X) is known for ∀i > α. Then, identifying in(8.35) Dαβ(K) with the market fair value, we determine K∗ for a given X∗.We obtain

sαβ(Tα, X∗) = K∗

13. Prove the relation

ζα(Tα, X∗) = ζβ(Tα, X∗) +K

β∑i=α+1

τiζi(Tα, X∗) (8.36)

By integrating (8.33) with the terminal condition ζα(Tα, X∗) given by (8.36),we obtain ζα(t,X) for all t ≤ Tα and the Libors via (8.32).


14. Using a Taylor expansion at the first-order around the swap spot, provethat the terminal correlation of two swap rates is approximatively

< ln siβ(Ti), ln siδ(Tj) >≈

√√√√∫ Ti0σ(s)2ds∫ Tj

0σ(s)2ds

By taking σ(t) = eat where a is called the mean-reverting coefficient (seequestion 2.), we have in particular

< ln siβ(Ti), ln siδ(Tj) >≈

√e2aTi − 1e2aTj − 1

Exercises 8.2 Almost Markov LMMFor a generic LV LMM, the Libor dynamics in the spot Libor measure Ps is

dLi(t) = σi(t)Φi(Li)i∑

j=β(t)

ρk,j(t)τjσj(t)Φj(Lj)1 + τjLj(t)

dt

+σi(t)Φi(Li)dZi

1. To get rid of the local volatility Φi(Li), we define the new variables qi(t) =∫ LiL0i

dxΦi(x) . Show that in these new variables, the SDEs above become

dqi(t) = σi(t)(µi(t, L)dt− 1

2σi(t)Φ′i(Li)dt+ dZi

)The difficulty in making this SDE Markovian comes from the drift term. Thesimplest way to deal with this problem is to replace it with its value at thespot Libor

dqi(t) = σi(t)(µi(t, L0)dt− 1

2σi(t)Φ′i(L

0i )dt+ dZi

)(8.37)

This is equivalent in spirit to the Hull-White-Rebonato freezing argument.2. Integrate (8.37) to

qi(t) = di(t) +∫ t

0

σi(t)dZi

with di(t) a deterministic function.3. Deduce that a Libor Li can be written as a function of a time-changedBrownian motion

Li(t) = Li(t, Zi,t′)


4. Assuming the separability condition of the volatility σi(t)

σi(t) = νiσ(t)

prove that the Libors Li can be mapped to a unique time-changed Brownianmotion

Li(t) = Li(t,Wt′)

Chapter 9

Solvable Local and StochasticVolatility Models

Abstract In the previous chapters, we have been focusing on (geometric)approximation methods particularly useful for the calibration of local andstochastic volatility models. However, in the case of particular models, we canhave an exact solution for the Kolmogorov and the Black-Scholes equations. Inthis chapter we provide an extensive classification of one and two dimensionaldiffusion processes which admit an exact solution to the Kolmogorov (andhence Black-Scholes) equation (in terms of hypergeometric functions). Byidentifying the one-dimensional solvable processes with the class of integrablesuperpotentials introduced recently in supersymmetric Quantum Mechanics,we obtain new analytical solutions. In particular, by applying supersymmetrictransformations on a known solvable diffusion process (such as the Natanzonprocess for which the solution is given by a hypergeometric function), weobtain a hierarchy of new solutions. These solutions are given by a sumof hypergeometric functions. For two-dimensional processes, more preciselystochastic volatility models, the classification is achieved for a specific classcalled gauge-free models including the Heston model, the 3/2-model and thegeometric Brownian model. We then present a new exact stochastic volatilitymodel belonging to this class.In addition to our geometrical framework, we will use tools from functionalanalysis in particular the spectral decomposition of (unbounded) linear oper-ators.

9.1 Introduction

For most mathematical models of asset dynamics, an exact solution for thecorresponding Kolmogorov & Black-Scholes equation is usually not available;there are, however, a few notable exceptions. The known solutions for lo-cal volatility models are the constant elasticity of variance (CEV) [76] in-cluding the classical log-normal Black-Scholes process. For the instantaneousshort rate models, there are the CIR process [77] (Bessel process) and theVasicek-Hull-White process [108] (Ornstein-Uhlenbeck process). For stochas-

247


tic volatility models, the known exact solutions are for the Heston model [106],the 3/2-model [29] and the geometric Brownian model [29]. These analyticalsolutions can be used for calibrating a model quickly and efficiently or canserve as a benchmark for testing the implementation of more realistic modelsrequiring intensive numerical computations (Monte-Carlo, PDE). For exam-ple, the existence of a closed-form solution for the fair value of a Europeancall option in the Heston model allows us to quickly calibrate the model tothe implied volatilities observed on the market. The calibrated model canthen be used to value path-dependent exotic options using, for example, aMonte-Carlo methodology.In this chapter, we show how to obtain new analytic solutions to the Kol-mogorov & Black-Scholes equation, which we refer to as KBS throughoutthe rest of this chapter, for 1d & 2d diffusion processes. In order to get toour classification, we first present a general reduction method to simplify themulti-dimensional KBS equation. Rewriting the KBS equation as a heat ker-nel equation on a Riemannian manifold endowed with an Abelian connection,we show that this covariant equation can be simplified using both the groupof diffeomorphisms (i.e., change of variables) and the group of Abelian gaugetransformations. In particular for the models admitting a flat Abelian connec-tion, there always exists a gauge transformation that eliminates the Abelianconnection of the diffusion operator.In the second part we apply the reduction method, previously presented, toone-dimensional, time-homogeneous diffusion processes. Modulo a changeof variable, the metric becomes flat and the Abelian connection is an exactone-form for which a gauge transformation can always be applied. Usingthese two transformations, the resulting KBS equation becomes an EuclideanSchrodinger equation with a scalar potential. Extensive work has already beendone to classify the set of scalar potentials which admit an exact solution. Inparticular, using a supersymmetric formulation of the Schrodinger equationwhich consists in doubling the KBS equation with another equation, we showhow to generate a hierarchy of new solvable diffusion processes starting froma known solvable diffusion process (for, e.g., a Natanzon potential [129]). Inthis context, the local volatility function is identified with a superpotential.Applying supersymmetric transformations on the Natanzon potential (which isthe most general potential for which the Schrodinger equation can be reducedto either a hypergeometric or a confluent equation), we obtain a new classof solvable one-dimensional diffusion processes which are characterized by sixparameters.The classification of one-dimensional time-homogeneous solvable diffusion pro-cesses for which the solution to the KBS equation can be written as a hyper-geometric function has been achieved in [40, 41, 42, 120] using the well-knownNatanzon classification. The application of supersymmetric techniques to theclassification of solvable potentials for the Schrodinger equation has been re-viewed in [10] where a large number of references can be found. For theKolmogorov & Fokker-Planck equations, one can consult [118, 115].

Solvable Local and Stochastic Volatility Models 249

In the last part we pursue this classification for SVMs which admit a flatAbelian connection: we refer to these as gauge-free models. Surprisingly, thisclass includes all the well-known exact SVMs (i.e., the Heston model, the3/2-model and the geometric Brownian model). For these gauge-free models,we reduce the two-dimensional KBS equation to an Euclidean Schrodingerequation with a scalar potential. Then, we present a new exact SVM whichis a combination of the Heston and 3/2-models.

9.2 Reduction method

In this section, we explain how to simplify the KBS equation. This reductionmethod will be used in the next section to classify the solvable one and twodimensional time-homogeneous processes. This method is already well knownfor one-dimensional processes and is presented in [70, 30, 126] amongst others.However, the extension of this method to multi-dimensional diffusion processesrequires the introduction of differential geometric objects such as a metric andan Abelian connection on a Riemannian manifold, as we have already seen inchapter 4.Let us assume that our time-homogeneous multi-dimensional market modeldepends on n Ito processes xi which can either be traded assets or market-unobservable Ito processes (such as a stochastic volatility a or an instanta-neous short rate r). Let us denote x = (xi)i=1,··· ,n, with the initial conditionsα = (αi)i=1,··· ,n. These variables xi satisfy the following time-homogeneousSDE

dxit = bi(xt)dt+ σi(xt)dWi

dWidWj = ρijdt

with the initial condition xi(t = 0) = αi. The no-arbitrage condition impliesthat there exists an equivalent measure P such that the traded assets are(local) martingales under this measure. For P, the drifts bi are consequentlyzero for the traded assets (i.e., forwards). Note that the measure P is notunique as the market is not necessarily complete. Finally, the fair value of a(European) option, with payoff f(xi) at maturity T , is given by the discountedmean value of the payoff f conditional on the filtration Ft generated by theBrownian motions W i

s≤ti=1,··· ,n

C(α, t, T ) = EP[e−R Ttrsdsf |Ft]

with rs the instantaneous short rate. This mean-value depends on the prob-ability density p(T, x|α) which satisfies the backward Kolmogorov equation


(τ = T − t, ∂i = ∂∂αi

)

∂p

∂τ= bi∂ip+

12ρijσ

iσj∂ijp (9.1)

with the initial condition p(τ = 0) = δ(x − α). As usual we have used theEinstein convention meaning that two repeated indices are summed. Usingthe Feynman-Kac theorem 2.5, one can show that the fair value C of the optionsatisfies the Black-Scholes equation

∂C∂τ

= bi∂iC +12ρijσ

iσj∂ijC − r(α)C (9.2)

with the initial condition C(τ = 0, α) = f(α). In chapter 4, the PDE 9.1has been interpreted as a heat kernel on a smooth n-dimensional manifold Mendowed with a metric gij and an Abelian connection A

∂p(τ, x|α)∂τ

= Dp(τ, x|α) (9.3)

and D given by (4.55). Similarly, the Black-Scholes equation (9.2) can berewritten as

∂C(τ, α)∂τ

= (D − r)C(τ, α)

The heat kernel equation can now be simplified by applying the actions of thefollowing groups: The group of diffeomorphisms Diff(M) which acts on the metric gij andthe connection Ai by

(f∗g)ij = gpk∂ifp(x)∂jfk(x)

(f∗A)i = Ap∂ifp(x) , f ∈ Diff(M)

The group of gauge transformations G (see section 4.5 for details) whichacts on the conditional probability (and the fair value C) by

p′(τ, x|α) = eχ(τ,x)−χ(0,α)p(τ, x|α)C′(τ, α) = eχ(τ,α)C(τ, α)

Then p′ (C′) satisfies the same equation as p (C) (9.3) only with

A′i ≡ Ai − ∂iχQ′ ≡ Q+ ∂τχ

If the connection A is an exact form (meaning that there exists a smoothfunction Λ such that Ai = ∂iΛ), then by applying a gauge transformation, wecan eliminate the connection so that the heat kernel equation for p′ (or C′)


has a connection equal to zero. It can be shown that for a simply-connectedmanifold, the statement “A is exact” is equivalent to F = 0, where F is the2-form curvature given by

F = (∂iAj − ∂jAi)dαi ∧ dαj

It is straightforward to prove that if Ai = ∂iΛ, then F = 0 as ∂ijΛ = ∂jiΛfor a smooth function. In the following, we will restrict our classification tothose processes for which F = 0, meaning there exists a gauge transformationsuch that the transformed connection vanishes. The operator D reduces inthis case to the symmetric operator D = ∆ + Q for which we can use aneigenvector expansion. In this context, we review necessary materials fromthe (linear) operator theory and the spectral decomposition of (unbounded)self-adjoint linear operators on a (infinite-dimensional) Hilbert space. The Itogenerator of the Brownian D = ∂2

x on an interval I ⊂ R will be our guide toillustrate the notions we introduce in the next section.

9.3 Crash course in functional analysis

Classical references for this section are [35] and [36].Here H denotes a real separable Hilbert space with the scalar product (x, y).For completeness sake, a real (complex) Hilbert space is a real (complex)linear space equipped with an inner product (x, y) ∈ R (C) such that

(x, y) = (y, x)(λx1 + x2, y) = λ(x1, y) + (x2, y)(x, λy1 + y2) = λ(x, y1) + (x, y2)

where λ ∈ R (C) and · stands for complex conjugation.

(x, x) ≥ 0 , x ∈ H(x, x) = 0 if and only if x = 0

Such an inner product defines a norm ‖ x ‖≡√

(x, x) and by definition H iscomplete with respect to this norm. An Hilbert space is separable if it admitsa countable basis meaning that every vector x can be decomposed over anorthonormal basis eii=1,··· ,∞

1

x =∞∑i=1

xiei xi ∈ R (C)

1Note that we have necessarily ‖ x ‖2=P∞i=1 x

2i <∞.


9.3.1 Linear operator on Hilbert space

An operator D is a linear mapping D : Dom(D)→ H where Dom(D), calledthe domain of D, is a linear subspace of H.For any two operators D1, D2, their sum D1 +D2 and the product D1D2 aredefined as follows

(D1 +D2)(x) = D1x+D2x , ∀ x ∈ Dom(D1) ∩Dom(D2)(D1D2)(x) = D1(D2(x)) , ∀ x ∈ Dom(D1D2)

with Dom(D1D2) ≡ x ∈ Dom(D2) | D2x ∈ Dom(D1).

DEFINITION 9.1 Operator densely defined An operator D withdomain Dom(D) is said to be densely defined if the subset Dom(D) is densein H, i.e., for any x ∈ H one can find a sequence xn in Dom(D) whichconverges in norm to x.

We define the norm of the operator D by

DEFINITION 9.2 Norm

||D|| = supx∈H ; x 6=0

||Dx||||x||

(9.4)

If ||D|| <∞, then D is a bounded operator otherwise unbounded. The finite-ness of the norm ||D|| is equivalent to the continuity of D. Hence if D is abounded operator densely defined on H, it can be extended by continuity to thewhole Hilbert space H.

In the following, all the operators D are densely defined operatorson a (separable) Hilbert space H.

DEFINITION 9.3 Adjoint operator We call the subspace Dom†, thespace of vectors x such that the linear form x→ (y,Dx) is continuous for thenorm of H for all y ∈ H. Hence using Riesz’s theorem [36], there exists aunique x′ such that

(y,Dx) = (x′, x) (9.5)

By definition, we set D†y = x′. D† is called the adjoint operator of D and itsdomain is Dom(D†) ≡ Dom†.

Note that an adjoint operator D is only defined for an operator densely definedon H. Indeed, if Dom(D) is not dense in H, there exists z 6= 0 such that(z, y) = 0 ∀y ∈ H. If x′ and y are vectors satisfying (9.5), then x′ + λz for all


λ ∈ C satisfies (9.5) for the same y. The adjoint D† is not uniquely definedin this case.

DEFINITION 9.4 Symmetric operator An operator D is called sym-metric if for all x, y ∈ Dom(D), we have

(Dx, y) = (x,Dy)

It is equivalent to the condition

D† = D on Dom(D) ⊂ Dom(D†)

DEFINITION 9.5 Self-adjoint operator D is self-adjoint if addition-ally

Dom(D†) = Dom(D)

Before illustrating these notions on the Ito generator of a Brownian motion,we review the definition of a Sobolev space Hm and list few useful properties.

Sobolev space Hm

Let I be an interval of R. As a recall, a function g ∈ L2(I) is said to be theweak derivatives of f if∫

I

g(x)φ(x)dx = −∫I

f(x)φ′(x)dx

for all φ ∈ C∞0 (I) the space of C∞ function with compact support on I. Inthis case, we write g ≡ f ′. Similarly, we define the kth weak derivative f (k).For any integer m, the Sobolev space Hm(I) is

Hm(I) = f ∈ L2(I) : f (k) ∈ L2(I) , ∀ k = 1, · · · ,m

Endowed with the norm

||f ||Hm =

(m∑k=1

||fk||L2

) 12

Hm(I) is a Hilbert space. Below, we list some useful properties that charac-terize a function in Hm(I)

• f ∈ Hm(I) if and only if f ∈ L2(I)∩C(I) is continuously differentiableup to order m − 1, f (m) exists almost everywhere and f (m) ∈ L2(I).Moreover, all the functions f, f (1), · · · , f (m−1) are absolutely continu-ous.2

2A function F is said to be absolutely continuous on I if there exists a function f ∈ L2(I)such that F (x) = F (a) +

R xa f(y)dy , ∀x ∈ I.


• ∀f, g ∈ H1(I), the integration by parts formula holds∫ b

a

f(x)g′(x)dx = [fg]ba −∫ b

a

f ′(x)g(x)dx , ∀a, b ∈ I

and ∀f, g ∈ H2(I),∫ b

a

f(x)g′′(x)dx = [fg′]ba − [f ′g]ba +∫ b

a

f ′′(x)g(x)dx , ∀a, b ∈ I

(9.6)

• In the case of unbounded I, if f ∈ H1(I) then

lim|x|→∞

f(x) = 0 (9.7)

To illustrate the notions above, we look at the self-adjoint extensions of theIto generator of a one-dimensional Brownian motion D = ∂2

x on an interval I(not necessarily bounded). Here H = L2(I).

DEFINITION 9.6 Self-adjoint extension An operator D′ is a self-adjoint extension of an operator D if

D′ = D on Dom(D) ⊂ Dom(D†)

and

D′† = D′ on Dom(D) ⊂ Dom(D′) = Dom(D′†) ⊂ Dom(D†)

Self-adjoint extensions: Examples

Using the integration by parts formula (9.6), we have

(Df, g) = (f,Dg) ∀f, g ∈ H2(I)

with D = ∂2x and I = (a, b) if and only if

[fg′]ba = [f ′g]ba (9.8)

Example 9.1 I = RWe set Dom(D) = C∞0 (R). As the space C∞0 (R) is dense in H, the adjoint canbe defined. From the condition (9.8) and the property (9.7), we obtain thatthe adjoint is D† = ∂2

x with the domain Dom(D†) = H2(R). (D,Dom(D))is therefore symmetric but not self-adjoint. It admits a unique self-adjointextension: (D,H2(R)) is self-adjoint.


Example 9.2 I = R+

(D,C∞0 (R+)) is symmetric but not self-adjoint. Note that because of theproperty (9.7), the condition (9.8) reduces to

f ′(0)f(0)

=g′(0)g(0)

We deduce that D admits self-adjoint extensions parameterized by the groupU(1) = eiθ , θ ∈ [0, 2π):

Dom(Dθ) = f ∈ H2(R+) , (f(0)− ıf ′(0)) = eiθ(f(0) + ıf ′(0)) , θ ∈ [0, 2π)

which is equivalent to

Dom(Dθ) = f ∈ H2(R+) , f(0) = λf ′(0) , λ ∈ R ∪ ∞

with λ = − tan( θ2 ).The Neumann (resp. Dirichlet) boundary f ′(0) = 0 (resp. f(0) = 0) corre-sponds to λ =∞ (resp. λ = 0).

Example 9.3 I = [0, 1](D,C∞0 (I)) is symmetric but not self-adjoint. It admits self-adjoint extensionsparameterized by the group G of two-dimensional unitary matrices U(2) sat-isfying the condition:

G = U ∈ U(2) , det(id− UU) = 0 (9.9)

where · is the complex conjugation. The domain of all of these extensions is

Dom(DG) = f ∈ H2(I) ,(f ′(0)− ıf(0)f ′(1) + ıf(1)

)= U

(f ′(0) + ıf(0)f ′(1)− ıf(1)

), U ∈ G

The reality of the function f implies the condition (9.9).

9.3.2 Spectrum

For any operator on a complex Hilbert spaceH, we say that a complex numberλ is a regular value of D if the operator D − λid has a bounded inverse, e.g.,

ker(D − λid) = 0 , ran(D − λid) = H , ||(D − λid)(−1)|| <∞

Any λ ∈ C which is not a regular value is called a singular value. The setof all regular values is called the resolvent of D and the set of all singularvalues is called the spectrum of D and is denoted spec(D). For example, if λis an eigenvalue of D, that is Dx = λx for some x 6= 0, then λ belongs to thespectrum of D.


If D is an operator in a real Hilbert space H then we define first complexifiedspace HC = H + ıH and extend D by linearity to an operator DC in HC.We set spec(D) = spec(DC). The spectrum of a self-adjoint operator can becharacterized as

PROPOSITION 9.1The spectrum of a self-adjoint operator D is a non-empty closed subset of R.Also, we have

||D|| = supλ∈spec(D)

|λ|

9.3.3 Spectral decomposition

Let us assume that (D,Dom(D)) is self-adjoint. Before presenting the spectraldecomposition in the general case of a self-adjoint unbounded linear operatoron an (infinite-dimensional) Hilbert space, we recall the finite-dimensionalcase in a form generalizable to the infinite-dimensional case. For a self-adjointoperator D on a finite-dimensional Hilbert space, we can find a basis wherethe operator D is diagonalizable. If we denote (λi)i=1,··· ,m the set of differenteigenvalues and Pi the projector operator on the vector subspace generated bythe eigenvectors associated to the eigenvalue λi, then D can be decomposedas

D =m∑i=1

λiPi (9.10)

We recall that the projectors Pi satisfy the algebra

PiPj = Piδij , P†i = Pi ,

m∑i=1

Pi = id

Introducing the operator E(λ) =∑i:λi≤λ Pi, we prove that E(λ) satisfies the

following properties

E(λ)E(λ′) = E(min(λ, λ′))limλ→λ−i

E(λ) = E(λi)

limλ→−∞

E(λ) = 0

limλ→∞

E(λ) = 1

with 1 the identity map on H. The sign lim should be understood as thestrong convergence meaning that En → E strongly if

limn→∞

||En − E|| = 0


The operators E(λ) allow to define a specific measure where the sign sum in theformula (9.10) can be written as a Lebesgue-Stieltjes integral. This spectraldecomposition formula is generalizable to unbounded self-adjoint operators.Beforehand, let us recall briefly the construction of the Lebesgue-Stieltjesintegral.

DEFINITION 9.7 Stieltjes function Let F (λ) be a monotone, increas-ing, left-continuous function satisfying limλ→−∞ F (λ) = 0 and limλ→∞ F (λ) <∞.

From the properties satisfied by E(λ), ||E(λ)x|| is an example of Stieltjesfunctions.From this function, we can define a measure µF on the Borel σ-algebra on R:On the semi-open interval [a, b), we set

µF ([a, b)) = F (b)− F (a)

As µF is a σ-additive function on the semi-ring formed by all semi-open in-tervals, it can be extended to all Borel sets by the Caratheodory extensiontheorem (2.1).Hence we can integrate a measurable function φ against the measure µF :∫

Rφ(λ)dµF (λ)

This integral is called the Lebesgue-Stieltjes integral. If a function F canbe decomposed as the difference of two Stieltjes functions F1 and F2 (F =F1 − F2), we set∫

Rφ(λ)dµF (λ) ≡

∫Rφ(λ)dµF1(λ)−

∫Rφ(λ)dµF2(λ) (9.11)

Note that for any two vectors f, g ∈ H, we have

(E(λ)f, g) = (E(λ)f,E(λ)g) =14(||E(λ)(f + g)||2 − ||E(λ)(f − g)||2

)Hence, (E(λ)f, g) is a difference of two Stieltjes functions and therefore definesa measure according to (9.11). In this context, the sum in the formula (9.10)can be rewritten as a Lebesgue-Stieltjes integral:

(g,Df) =∫

Rλd(g,E(λ)f) (9.12)

Here (g,E(λ)f) is a discrete measure with support the eigenvalues λii=1,··· ,m.The formula (9.12) can be written formally as

D =∫

RλdE(λ) (9.13)


For a self-adjoint operator in an infinite-dimensional Hilbert space, a similarspectral decomposition is still valid. However, the spectrum is not necessarilydiscrete and can have a continuous part. The spectrum can belong to a Borelsubset of R, and E(λ) defines a resolution of the identity:

DEFINITION 9.8 Resolution of the identity Let H be a separablereal Hilbert space and let B(R) be the Borel σ-algebra on R. A family ofbounded linear operators E(B) : B ∈ B(R) in H such that

• Each E(B) is an orthogonal projection (i.e., E(B)2 = E(B) and E(B)† =E(B)).

• E(∅) = 0, E(R) = 1 (1 is the identity operator in H).

• If B = ∪∞n=1Bn with Bn ∩Bm = 0 if n 6= m, then E(B) =∑∞n=1 E(Bn)

(where the limit involved in the infinite series is taken in the strongconvergence topology).

• E(B1)E(B2) = E(B1 ∩B2)

is called a resolution of the identity.

The definition above ensures that (f,E(B)f) is a Stieltjes function.For a self-adjoint operator, one can show that there exists a resolution of theidentity and the operator can be decomposed as (9.13). More precisely, wehave

THEOREM 9.1 Spectral decompositionLet D be a self-adjoint operator in a real Hilbert space H. Then there exists

a unique spectral resolution E(B) : B ∈ B(R) in H such that the followingspectral decomposition holds

(g,Df) =∫

Rλd(g,E(λ)f)

Moreover, the domain of D is given by

Dom(D) = f ∈ H|∫

Rλd(f,E(λ)f) <∞

This formula is written formally as (9.13).

The spectral decomposition is valid only if the symmetric operator D is an(unbounded) self-adjoint operator or admits a self-adjoint extension. This willdepend strongly on the boundary conditions as seen in the examples above.In order to show that D is self-adjoint or admits self-adjoint extensions, we


can use the deficiency indices technique introduced by Von Neumann [36] (see[66] for a pedagogical introduction). The deficiency indices are defined by

n± = dim ker(D† ∓ ıid)

THEOREM 9.2For an operator with deficiency indices (n−, n+), there are three possibilities:

1. If n− = n+ = 0, then D is self-adjoint (necessary and sufficient condi-tion).

2. If n+ = n− = n ≥ 1, then D has infinitely many self-adjoint extensions,parameterized by the unitary group U(n) satisfying the real condition(9.9).

3. If n− 6= n+, then D has no self-adjoint extension.

REMARK 9.1 Note that if D is a linear differential operator of order nand the deficiency indices are (n, n) then the spectrum is purely discrete [35].

If D admits a self-adjoint extension, the resulting conditional probability isnot unique but depends on the boundary conditions which are parameterizedby the unitary group U(n). We look back at the self-adjoint extensions of theIto generator of a one-dimensional Brownian motion D = ∂2

x on an intervalI using the Von-Neumann criteria. Here D† = ∂2

x. The eigenvectors for theeigenvalues ±ı are

φ±(x) = e±√±ıx

We examine if φ± belong to L2(I).

Example 9.4 I = R(n−, n+) = (0, 0). D is self-adjoint.

Example 9.5 I = R+

(n−, n+) = (1, 1). D admits a self-adjoint extension parameterized by U(1).

Example 9.6 I = [0, 1](n−, n+) = (2, 2). D admits a self-adjoint extension parameterized by U(2).Moreover, the spectrum is discrete from the remark 9.1.


Having proved that −D ≡ ∆ + Q − r is a self-adjoint operator or admitsa self-adjoint extension, the gauge-transform option value C′ satisfying theequation

∂C′(τ, α)∂τ

= (∆ +Q− r)C′(τ, α)

admits a unique solution in H given by

C′(τ, α) =∫

spec(D)

e−λτdE(λ)f (9.14)

We have applied the spectral decomposition to D and not −D only for thepurpose of convenience. As we will see in the next section, the spectrum of−D is included in R+.

DEFINITION 9.9 Semi-group In the equation (9.14), the familyDτ (τ ≥ 0) defined as

Dτ ≡ e−τD =∫

spec(D)

e−λτdE(λ)

is called the heat semi-group associated to D and satisfies the following prop-erties:

1. Dt+s = DtDs t, s > 0 and D0 = 1. (semi-group axiom)

2. ||Dt|| ≤ 1 (contraction property)

3. The mapping t→ Dt is strongly continuous on [0,∞). That is, for anyt ≥ 0 and f ∈ H,

lims→t

Dsf = Dtf

where the limit is understood in the norm of H. In particular, for anyf ∈ H,

limt→0

Dtf = f

4. For all f ∈ H and t > 0, we have Dtf ∈ Dom(H) and

d

dt(Dtf) = −D(Dtf)

where ddt is the Frechet derivative.


How can we construct a resolution of identity in practice?

We consider the generalized eigenvectors

Dφλ = λφλ

By “generalized,” we mean that it is not required that φλ belongs to H andtherefore λ does not necessarily belong to the discrete spectrum. We normalizethe function φλ ( φλ is a distribution) such that∫

spec(D)

φλ(s)φλ(s0)dλ = δ(s− s0) (9.15)

Then, the resolution of identity of D is

(g,E(λ)f) =∫ λ

−∞(g, φλ)(φλ, f)dλ

and the unique solution to the gauge-transform Kolmogorov equation is

p′(t, s|s0) =∫

spec(D)

e−λtφλ(s)φλ(s0)dλ (9.16)

=∑k

e−λktφλk(s)φλk(s0) +∫

spec/specd(D)

e−λtφλ(s)φλ(s0)dλ

where we have split the spectrum into a discrete specd(D) and continuousparts spec/specd(D).

Example 9.7 D = ∂2s , I = R

The eigenvector φλ(s) associated to the eigenvalue −λ2 satisfies the ODE

Dφλ(s) = −λ2φλ(s)

for which the solutions are

φλ(s) = e±iλs , λ ∈ R

The (real) eigenvector which satisfies the completude relation (9.15) is

φλ(s) =1√2π

(cos (λs)− sin (λs))

From (9.16), we get the normal distribution

p′(t, s|s0) =∫ ∞−∞

e−λ2tφλ(s)φλ(s0)dλ

=1√4πt

e−(s−s0)2

4t


9.4 1D time-homogeneous diffusion models

In the next section, we apply the general reduction method to the one-dimensional KBS equation. Though similar reductions to a Schrodinger equa-tion with a scalar potential (without any references to differential geometryhowever) can be found in [70, 30, 126], we rederive this classic result usingour general reduction method. This method will prove to be useful when wediscuss the classification of SVMs. What is more, we find the supersymmetricpartner of this Schrodinger equation and show how to generate new exactsolutions for European call options.

9.4.1 Reduction method

Let us consider a one-dimensional, time-homogeneous diffusion process withdrift3

dft = µ(ft)dt+ σ(ft)dWt

This process has been used as a basis for various mathematical models infinance. If f is a traded asset (i.e., a forward), the drift vanishes in the forwardmeasure and we end up with a local volatility model where we assume that thevolatility is only a function of f . This has been explored in detail in chapter 5.The one-dimensional process is not necessarily driftless as the random variableis not always a traded asset as it is the case for an instantaneous short ratemodel, or a model of stochastic volatility.In our framework, this process corresponds to a (one-dimensional) real curveendowed with the metric gff = 2

σ(f)2 . For the new coordinate

s(f) =√

2∫ f

·

df ′

σ(f ′)

the metric is flat: gss = 1. It follows that the Laplace-Beltrami operator(4.57) becomes M= ∂2

s . Using the definition (4.58), (4.59), we find that theAbelian connection A and the function Q are given by

A =(−1

2∂f lnσ(f) +

µ(f)σ2(f)

)df (9.17)

Q =14

(σ(f)σ

′′(f)− 1

2σ′(f)2

)− µ′(f)

2+µ(f)σ′(f)σ(f)

− µ2(f)2σ2(f)

(9.18)

3The time-dependent process dft = µ(ft)A2(t)dt+σ(ft)A(t)dWt is equivalent to this process

under the change of local time t′ =R t0 A(s)2ds.


A is an exact form (it is always the case in one dimension) with

Λ(f) = −12

ln(σ(f)σ(f0)

)+∫ f

f0

µ(f ′)σ2(f ′)

df ′

By applying a gauge transformation on the conditional probability p(t, f |f0)

P (t, s) =σ(f0)√

2eΛ(f)p(t, f |f0) (9.19)

the connection vanishes and P satisfies the equation (in the s flat coordinate)

∂τP (τ, s) =(∂2s +Q(s)

)P (τ, s) (9.20)

This is known as an Euclidean Schrodinger equation with a scalar potentialQ(s). This equation is considerably simpler than the Quantum MechanicsSchrodinger equation seeing that all the terms are real-valued.The solution P has been scaled by the (constant) factor σ(f0)√

2in order to

obtain the initial condition

limτ→0

P (τ, s) = δ(s)

Moreover, Q is given in the s coordinate by

Q =12

(lnσ)′′(s)− 1

4((lnσ)

′(s))2 − µ′(s)√

2σ(s)+√

2µ(s)σ′(s)σ(s)2

− µ(s)2

2σ(s)2

(9.21)

where the prime ′ indicates a derivative according to s.

Example 9.8 Quadratic volatility processLet us assume that f satisfies a driftless process (i.e., µ(f) = 0). The Black-Scholes equation reduces to the heat kernel on R if Q(s) = cst (i.e., Q(s) iszero modulo a time-dependent gauge transformation) which is equivalent to

σ(f)σ′′(f)− 12σ′(f)2 = 4cst

By differentiating with respect to f , we get

σ(f)σ′′′(f) = 0

and finally

σ(f) = αf2 + βf + γ


(i.e., the quadratic volatility model studied in detail in [145]) with cst =αγ2 −

β2

8 . The solution P (t, s|s0) to the Schrodinger equation is

P (t, s|s0) =1√4πt

e−(s−s0)2

4t +cstt

By using the explicit map (9.19) between p(t, f |f0) and its gauge transformP (t, s|s0) with s =

√2∫ f df ′

σ(f ′) , we obtain (see remark 4.3) the solution to theKolmogorov equation

p(t, f |f0) =1

σ(f)√

2πtσ(f0)σ(f)

e−(R ff0

df′σ(f′) )2

2t +cstt

Example 9.9 CEV processFor the CEV process dft = fβt dWt, the potential is

Q(s) =β(β − 2)

4(1− β)2s2, 0 < β < 1

Q(s) = −18, β = 1

Note that in order to ensure that ft is a true martingale, we have assumedthat β ≤ 1. The Schrodinger equation is

Dφλ(s) = λφλ(s)

with D = −∂2s − gs−2 and g = β(β−2)

4(1−β)2 .

(Generalized) eigenvectors

We will now attempt to find the (generalized) eigenvectors φλ(s). Let us takethe ansatz

φλ(s) =√sΦλ(s)

We obtain

s2Φ′′λ(s) + sΦ′λ(s) +(−1

4+ g + λs2

)Φλ(s) = 0

Thanks to the scaling u =√λs ;λ 6= 0, the PDE becomes a Bessel differential

equation4 [1]

u2Φ′′λ(u) + uΦ′λ(u) +(−1

4+ g + u2

)Φλ(u) = 0

4Bessel Iα(x): x2I′′α(x) + xI′α(x) + (x2 − α2)Iα(x) = 0.


The eigenvectors are

φλ(s) =√s(Iν(√λs) + aI−ν(

√λs)), λ 6= 0

with

ν ≡ 12|(1− β)|

For the zero eigenvalue state, we take the ansatz

φ(s) = sα

This implies that g = −α(α − 1). The solution to this equation is α± =12 ±

√1−4g2 . This gives two independent solutions and by completeness all

the solutions. These solutions do not lead to normalizable states since theydiverge at the origin or at infinity. As a result, there is no normalizable λ = 0solution and the (continuous) spectrum is (0,∞).

Deficiency indices

In order to determine the deficiency indices, we search the square integrablesolutions of

D†φ±(s) = ±ıφ±(s)

The solutions are given by

φ+ =√s(Iν(sei

π4 ) + aI−ν(sei

π4 ))

φ− =√s(Iν(se3iπ4 ) +AI−ν(se3iπ4 )

)with a,A two constants. Note that Iν(z) and I−ν(z) (z ∈ C) are linearlyindependent except when ν is an integer.When ν is fixed and |z| → ∞ (| arg z| < π)

Iν(z) =

√2πz

(cos(z − νπ

2− π

4) + e|=z|O(|z|−1)

)We deduce that φ+ and φ− are renormalizable at s = ∞ if and only if a =A = −eiνπ. By using that Iν(ze−πi) = e−πiIν(z), we get

φ+ =√s(Iν(sei

π4 )− I−ν(se−i

3π4 ))

φ− =√s(Iν(sei

3π4 )− I−ν(se−i

π4 ))

When ν is fixed and |z| → 0

Iν(z) =(z

2

)ν ( 1Γ(ν + 1)

+ o(z2))


We deduce that φ+ and φ− are not square integrable at s = 0 if and only ifν ≥ 1 equivalent to β ∈ [ 1

2 , 1). In this case D has deficiency indices (0, 0) andis (essentially) self-adjoint with the domain

Dom(D) =(φ ∈ L2(R+) : φ(0) = φ′(0) = 0

)(9.22)

Both φ± are square integrable when ν < 1 equivalent to β ∈ (0, 12 ). In these

ranges, D has deficiency indices (1, 1). D is not self-adjoint on the domainDom but admits self-adjoint extensions parameterized by θ ∈ U(1). Thedomain is

Dom(Dθ) =(φ ∈ L2(R+) : φ(0) = − tan(

θ

2)φ′(0)

)(9.23)

We have reproduced properties 2 and 5 from lemma 5.1.

Conditional probability

By assuming the absorbing condition Φλ(0) = 0, we obtain

Φλ(s) =√s

Iν(√λs)√

2λ14

The scale factor 1√

2λ14

has been included in order to satisfy the completude

relation (9.15) as we have the identity∫ ∞0

Iν(λs)Iν(λs0) =1s0δ(s− s0)

From (9.16), the conditional probability is

p′(t, s|s0) =√ss0

∫ ∞0

e−λ2tIν(λs)Iν(λs0)dλ

=√ss0√2te−

s2+s202t Iν

(ss0

t

)By using the explicit map (9.19) between p(t, f |f0) and its gauge transformp′(t, s|s0) with s = f1−β

1−β , we obtain (see remark 4.3)

p(t, f |f0) =f

12−2β

(1− β)t

√f0e− f

2(1−β)+f2(1−β)0

2(1−β)2t I 12(1−β)

((ff0)1−β

(1− β)2t

)

Boundary conditions If the operatorD = (−∂2s−Q) is self-adjoint or admits

self-adjoint extensions, the spectral decomposition can be used and P (τ, s) isdecomposed as (9.16). For a one-dimensional diffusion process, one doesn’t


TABLE 9.1: Feller boundary classification forone-dimensional Ito processes.

Boundary type S(e, d) M(e, d) Σ(e) N(e)Regular <∞ <∞ <∞ <∞

Exit <∞ =∞ <∞ <∞Entrance =∞ <∞ <∞ <∞Natural <∞ =∞ =∞ =∞

=∞ <∞ =∞ =∞=∞ =∞ =∞ =∞

need to use the deficiency indices technique as the complete classification ofboundary conditions is given by the Feller classification [27]. More precisely,for a 1D time-homogeneous diffusion process, the boundary falls into one ofthe four following types: regular, entrance, exit or natural. For entrance, exitor natural, no boundary conditions are needed. As for a regular boundary,the conditional probability is not unique but depends on the boundary condi-tions. It corresponds to the case when the deficiency indices are (n, n). Theboundary classification depends on the behavior of the following functions

S(c, d) =∫ d

c

s(f)df

M(c, d) =∫ d

c

m(f)df

Σ(e) = limc→e

∫ d

c

S(c, f)m(f)df

N(e) = limc→e

∫ d

c

S(x, d)m(x)dx

with s(f) = e−2R f µ(x)dx

σ(x)2 and m(f) = 1σ2(f)s(f) (see Table 1 below).

Example 9.10 European call option with deterministic interest rateWe have that the forward f satisfies a driftless process (i.e., µ(f) = 0)

dft = σ(ft)dWt

The value at t of a European call option (with strike K and expiry date T ) isgiven by (τ = T − t)

C(K, τ |ft) = PtT

∫ ∞K

max(f −K, 0)p(τ, f |ft)df

Doing an integration by parts, or equivalently, applying the Ito-Tanaka for-mula on the payoff max(ft −K, 0) (see equation (5.29)), the option C can be


rewritten as

C(K, τ |ft)PtT

= max(ft −K, 0) +σ(K)2

2

∫ τ

0

dt′p(t′,K|ft)

Using the relation between the conditional probability p(t′, f |ft) and its gauge-transform P (t′, s(f)|st) (9.19), we obtain (see remark 4.3)

C(K, τ |ft)PtT

= max(ft −K, 0) +

√σ(K)σ(ft)√

2

∫ τ

0

P (t′, s(K)|st)dt′ (9.24)

Plugging the expression for P (t′, s|st) (9.16) into (9.24) and doing the inte-gration over the time t, we obtain

C(K, τ |ft)PtT

=

√σ(K)σ(ft)√

2

∫spec(D)

φλ(s(K))φλ(s(ft))(1− e−λτ )

λdλ

+ max(ft −K, 0) (9.25)

Example 9.11 Quadratic model, σ(f) = αf2 + βf + γ

C(τ,K|f0)P0T

= max(f0 −K, 0) +

√σ(K)σ(f0)√

2

∫ τ

0

1√4πt′

e−s2

2t′+Qt′dt′

with s = |∫Kf0

dxσ(x) | and Q = αγ

2 −β2

8 . By doing the integration over the time,we obtain

C(τ,K|f0)P0T

= max(f0 −K, 0) +

√σ(K)σ(f0)2√−2Q(

es√−QN

(− s√

2τ−√−2Qτ

)− e−s

√−QN

(− s√

2τ+√−2Qτ

))

A specific local volatility model admits an exact solution for a European calloption if we can find the eigenvalues and eigenvectors for the correspond-ing Euclidean Schrodinger equation with a scalar potential. As examples ofsolvable potentials, one can cite the harmonic oscillator, Coulomb, Morse,Poschl-Teller I&II, Eckart and Manning-Rosen potentials. The classificationof solvable scalar potentials was initiated by Natanzon [129]. This work pro-vides the most general potential for which the Schrodinger equation can bereduced to either a hypergeometric or confluent equation. We show that theSchrodinger equation can be doubled into a set of two independent Schrodingerequations with two different scalar potentials which transform into each other


under a supersymmetric transformation. Moreover, if one scalar potential issolvable, the other one is. Applying this technique to the Natanzon potentialwhich depends on 6 parameters, we obtain a new class of solvable potentialscorresponding to a new class of solvable diffusion processes. For these mod-els, the solution to the KBS equation is given by a sum of hypergeometricfunctions.

9.4.2 Solvable (super)potentials

In this part, we show that the Schrodinger equation can be formulated usingsupersymmetric techniques (see [10] for a nice review). In particular, thelocal volatility is identified as a superpotential. Using this formalism, weshow how to generate a hierarchy of solvable diffusion models starting from aknown solvable superpotential, for example, the hypergeometric or confluenthypergeometric Natanzon superpotential.

Superpotential and local volatility

We have seen that the Kolmogorov equation can be cast into (9.16) whereφλ(s) are formal eigenvectors of the Schrodinger equation

−(∂2s +Q(s)

)φλ(s) = λφλ(s) (9.26)

The eigenvectors φλ(s) must satisfy the identity (9.15). Let us write (9.26) as

λφ(1)λ = A†1A1φ

(1)λ (9.27)

where we have introduced the first-order operator A1 and its adjoint A†1 (calledsupercharge operators)

A1 = ∂s +W (1)(s) , A†1 = −∂s +W (1)(s)

W (1) is called a superpotential which satisfies the Riccati equation

Q(1)(s) = ∂sW(1)(s)−W (1)(s)2 (9.28)

Surprisingly, this equation is trivially solved for our specific expression forQ(1) (9.21) (even with a drift µ(f)!)

W (1)(s) =12d lnσ(1)(s)

ds− µ(1)(s)√

2σ(1)(s)(9.29)

Example 9.12 Coulomb superpotential and CEV processThe CEV process corresponding to σ(f) = fβ and µ(f) = 0 has the Coulombsuperpotential

W (s) =β

2(1− β)s


Equation (9.27) implies that λ ∈ R+ as

λ||φ(1)λ ||

2 = ||A1φ(1)λ ||

2

For a zero drift, the local volatility function is directly related to the super-potential by

σ(s) = e2R s· W (z)dz

A similar correspondence between superpotentials and driftless diffusion pro-cesses has been found in [118, 115]. In addition, if we have a family of solvablesuperpotentials W (1)

solv(s), we can always find an analytic solution to the Kol-mogorov equation for any diffusion term σ(1)(s) by adjusting the drift withthe relation (9.29)

µ(1)(s) =σ(1)′(s)√

2−√

2σ(1)(s)W (1)solv(s)

Note that D admits a zero eigenvalue if and only if the Kolmogorov equa-tion admits a stationary distribution. By observing that A†1A1φ

(1)0 = 0 is

equivalent to A1φ(1)0 = 0 as

(φ(1)0 , A†1A1φ

(1)0 ) =‖ A1φ

(1)0 ‖2

we obtain the stationary distribution

φ(1)0 (s) = Ce−

R sW (1)(z)dz (9.30)

with C a normalization constant. Therefore, the stationary distribution existsif the superpotential is normalisable (i.e., φ(1)

0 (s) ∈ L2).Next, we define the Scholes-Black equation by intervening the operator A1

and A†1

λφ(2)λ (s) = A1A

†1φ

(2)λ (s)

= −(∂2s +Q(2)(s)

)φ

(2)λ (s) (9.31)

This corresponds to a new Schrodinger equation with the partner potential

Q(2)(s) = −∂sW (1) − (W (1))2 (9.32)

By plugging our expression for the superpotential (9.29) in (9.32), we have

Q(2) = −12

(lnσ(1))′′(s)− 1

4(lnσ(1))

′(s)2 +

µ(1)′(s)√2σ(1)(s)

− µ(1)(s)2

2σ(1)(s)2(9.33)


In the same way as before, H2 admits a zero eigenvector (i.e., stationarydistribution) if

φ(2)0 (s) = Ce

R sW (1)(z)dz

is normalisable.

REMARK 9.2 In physics, the supersymmetry (SUSY) is said to bebroken if at least one of the eigenvectors φ(1,2)

0 (s) exists.

Now we want to show that provided that we can solve the equation (9.27),then we have automatically a solution to (9.31) and vice versa. The SUSY-partner Hamiltonians H1 = A†1A1 and H2 = A1A

†1 obey the relations A†1H2 =

H1A†1 and H2A1 = A1H1. As a consequence H1 and H2 are isospectral. It

means that the strictly positive eigenvalues all coincide and the correspondingeigenvectors are related by the supercharge operators A1 and A†1: If H1 admits a zero eigenvalue (i.e., broken supersymmetry), we have therelation

φ(1)0 (s) = Ce−

R sW (1)(z)dz , λ(1) = 0

φ(2)λ (s) = (λ)−

12A1φ

(1)λ (s) , λ(2) = λ(1) = λ 6= 0 (9.34)

φ(1)λ (s) = (λ)−

12A†1φ

(2)λ (s)

If H1 (and H2) doesn’t admit a zero eigenvalue (i.e., unbroken supersym-metry)

λ(2) = λ(1) = λ 6= 0

φ(2)λ (s) = (λ)−

12A1φ

(1)λ (s)

φ(1)λ (s) = (λ)−

12A†1φ

(2)λ (s)

The eigenvectors have been scaled with the factor (λ)−12 to get

‖ φ(1)λ ‖=‖ φ

(2)λ ‖

In the unbroken SUSY case, there are no zero modes and consequently thespectrum of H1 and H2 is the same. One can obtain the solution to theScholes-Black (resp. Black-Scholes) equation if the eigenvalues/eigenvectorsof the Black-Scholes (resp. Scholes-Black) are known. We clarify this cor-respondence by studying a specific example: the CEV process df = fβdW .In particular, we show that for β = 2

3 , the partner superpotential vanishes.It is therefore simpler to solve the Scholes-Black equation as Scholes-Black(rather than Black-Scholes) reduces to the heat kernel on R+. Applying asupersymmetric transformation on the Scholes-Black equation, we can then


derive the solution to the Black-Scholes equation. For example, in the case ofunbroken SUSY, we have

p(t, s|s0) =∫

spec(H(2))

e−tλ(λ)−1(A†1φ

(2)λ

)(s)(A†1φ

(2)λ

)(s0)dλ

where the generalized eigenvectors φ(2)λ (s) satisfy

−(∂2s +Q(2)(s))φ(2)

λ (s) = λφ(2)λ (s)

Example 9.13 CEV with β = 2/3 and Bachelier processWe saw previously that the superpotential associated with the CEV processis given by

W (1)(s) =β

2s(1− β)

with the flat coordinate s =√

2f1−β

(1−β) ∈ [0,∞) . The potential (9.28) is

Q(1)(s) =β(β − 2)

4(1− β)2s2

from which we deduce the partner potential (9.32)

Q(2)(s) =β(2− 3β)

4(1− β)2s2

This partner potential corresponds to the potential of a CEV process df =fBdW with B depending on β by

B(B − 2)(1−B)2

=β(2− 3β)(1− β)2

Surprisingly, we observe that for β = 23 , Q(2)(s) cancels and the corresponding

partner local volatility model is the Bachelier model df = dW for whichthe heat kernel is given by the normal distribution. The eigenvectors of thesupersymmetric Hamiltonian partner H2 = −∂2

s to H1 are given by (with theabsorbing boundary condition φλ(0) = 0)

φ(2)λ (s) =

sin(√λs)√

4πλ14

with a continuous spectrum R+. Applying the supersymmetric transforma-tion (9.34), we obtain the eigenvectors for the Hamiltonian H1 = −∂2

s + 2s2

corresponding to the CEV process with β = 23

φ(1)λ (s) = λ−

12

(−∂s +

1s

)φ

(2)λ (s)

=1√

4πλ34

(−√λ cos(

√λs) +

sin(√λs)

s

)


By plugging this expression in (9.25), we obtain the fair value for a Europeancall option

C(τ, ft,K)PtT

= max(ft −K, 0) +(ftK)

13

√2

∫ ∞0

dλ(1− e−λτ )

λφ

(1)λ (s)φ(1)

λ (s0)

This expression can be integrated out and written in terms of the cumulativedistribution [30]. The fact that the CEV model with β = 2

3 depends on thecumulative normal distribution and is therefore related to the heat kernelon R+ has been observed empirically by [30]. Here we have seen that itcorresponds to the fact that the supersymmetric partner potential vanishesfor this particular value of β.

9.4.3 Hierarchy of solvable diffusion processes

In the previous section we saw that the operators A1 and A†1 can be used tofactorize the Hamiltonian H1. These operators depend on the superpotentialW (1) which is determined once we know the first eigenvector φ(1)

0 (s) of H1

(see eq. (9.30)):

W (1)(s) = −d lnφ(1)0 (s)ds

We have assumed that H1 admits a zero eigenvalue. By shifting the eigenvalueλ(1) it is always possible to achieve this condition. The partner Schrodingerequation (9.31) can then be recast into a Schrodinger equation with a zeroeigenvalue

H(2) = A1A†1 = A†2A2 + E

(1)1

where A2 ≡ ∂s +W2(s) and A†2 ≡ −∂s +W2(s),

W (2)(s) ≡ 12d lnσ(2)(s)

ds− µ(2)(s)√

2σ(2)(s)(9.35)

We have introduced the notation E(m)n where n denotes the first eigenvalue and

(m) refers to the mth HamiltonianHm. By construction, this new HamiltonianH2 = A†2A2 + E

(1)1 is solvable as A1A

†1 is so and the associated diffusion

process with volatility σ(2) and drift µ(2) satisfying (9.35) is solvable. Thesuperpotential W (2)(s) is determined by the first eigenvector of H2, φ(2)

0 (s)associated to the eigenvalue E(1)

1 ,

W (2)(s) = −d ln(φ(2)0 )(s)ds


At this stage, we can apply a supersymmetric transformation on H2. The newHamiltonian H3 can be refactorised exactly in the same way as was done forH2. Finally, it turns out that if H1 admits p (normalisable) eigenvectors, thenone can generate a family of solvable Hamiltonians Hm (with a zero-eigenvalueby construction)

Hm = A†mAm + E(1)m−1 = −∂2

s +Qm(s)

where Am = ∂s + Wm(s). This corresponds to the solvable diffusion processwith a drift and a volatility such that

Wm(s) = −d lnφ(m)0

ds=

12d lnσ(m)(s)

ds− µ(m)(s)√

2σ(m)(s)

The eigenvalues/eigenvectors of Hm are related to those of H1 by

E(m)n = E

(m−1)n+1 = · · · = E

(1)n+m−1

φ(m)n = (E(1)

n+m−1 − E(1)m−2)−

12 · · · (E(1)

n+m−1 − E(1)0 )−

12Am−1 · · ·A1φ

(1)n+m−1

In particular, the superpotential of Hm is determined by the (m− 1)th eigen-vector of H1, φ(1)

m−1(s),

Wm(s) = −d ln(Am−1 · · ·A1φ

(1)m−1(s))

ds

=12d lnσ(m)(s)

ds− µ(m)(s)√

2σ(m)(s)

Consequently, if we know all the m discrete eigenvalues and eigenvectors ofH1, we immediately know all the energy eigenvalues and eigenfunctions of thehierarchy of m−1 Hamiltonians. In the following we apply this procedure witha known solvable superpotential, the Natanzon superpotential, as a startingpoint.

9.4.4 Natanzon (super)potentials

The Natanzon potential [129] is the most general potential which allows us toreduce the Schrodinger equation (9.20) to a Gauss or confluent hypergeometricequation (GHE or CHE).

Gauss hypergeometric potential

The potential is given by

Q(s) =S(z)− 1R(z)

−(r1 − 2(r2 + r1)z

z(1− z)− 5

4(r2

1 − 4r1r0)R(z)

+ r2

)z2(1− z2)R(z)2


with R(z) = r2z2 + r1z + r0 > 0 and S(z) = s2z

2 + s1z + s0 (two secondorder polynomials). The z coordinate, lying in the interval [0, 1], is implicitlydefined in terms of s by the differential equation

dz(s)ds

=2z(1− z)√

R(z)

Example 9.14The hypergeometric Natanzon potential includes as special cases the Poschl-Teller potential II

Q(s) = A+Bsech

(2s√r1

)2

+ Ccsch

(2s√r1

)2

for r0 = r2 = 0 and the Rosen-Morse potential

Q(s) = A+B tanh(2s√r0

) + Csech(2s√r0

)2

r1 = r2 = 0.

By construction, the solution to the Schrodinger equation with a GHE poten-tial is given in terms of the Gauss hypergeometric function F (α, β, γ, z)

z(s)′ ≡ φλ(s) = (z′)−12 z

γ2 (1− z)

−γ+α+β+12 F (α, β, γ, z)

where F (α, β, γ, z) satisfies the differential equation [1]

z(1− z)d2F

dz2+ (γ − (α+ β + 1)z)

dF

dz− αβz = 0 (9.36)

The most general solution to this equation (9.36) is generated by a two-dimensional vector space

F (α, β, γ, z) = c12 F1(α, β, γ, z) + c2z1−γ

2 F1(α− γ + 1, β − γ + 1, 2− γ, z)

with 2 F1(α, β, γ, z) satisfying 2 F1(α, β, γ, 0) = 1 and c1 and c2 two arbitrarycoefficients. The parameters α, β, γ depend explicitly on the eigenvalue λ by

1− (α− β)2 = r2λ+ s2

2γ(α+ β − 1)− 4αβ = r1λ+ s1

γ(2− γ) = r0λ+ s0

From these equations, one can show that λ satisfies a fourth-order polynomialand find λ as an explicit function of α, β, γ [97]. By imposing the conditionthat the eigenvectors are normalisable (i.e., belong to L2([0, 1])) we obtain


TABLE 9.2: Condition at z = 0.r0 6= 0 c1 = 0, γ < 1 or c2 = 0, γ > 1

r0 = 0, r1 6= 0 c1 = 0, γ < 2 or c2 = 0, γ > 0r0 = 0, r1 = 0 c1 = 0, γ < 3 or c2 = 0, γ > −1

TABLE 9.3: Condition at z = 1.c1 = 0 α− 1 ∈ N∗, α+ β − γ < 0 or −1− α+ γ ∈ N∗ , α+ β − γ > 0c2 = 0 −α ∈ N∗, α+ β − γ > 0 or α− γ ∈ N∗ , α+ β − γ < 0

the discrete spectrum λn and can determine the coefficients c1 and c2. Weimpose conditions on α, β, γ, c1 and c2 such that∫ ∞

−∞dsφλ(s)2 =

∫ 1

0

dzR(z)

4zγ−2(1− z)−γ+α+β−1F (α, β, γ, z)2 <∞

Looking at the asymptotic behavior of 2F1(α, β, γ, z) near z = 0 and z = 1[1]5, we obtain the following conditions (see table 9.2, 9.3)We have the discrete eigenvalues (αn = −n ;n ∈ N∗)

2n+ 1 = −√

1− r0λn − s0 +√

1− r2λn − s2

−√

1− (r0 + r1 + r2)λn − (s0 + s1 + s2)

γn = 1 +√

1− r0λn − s0

αn − βn = −√

1− r2λn − s2

αn + βn = γn +√

1− (r0 + r1 + r2)λn − (s0 + s1 + s2)

and the (normalised) eigenvectors

φn(s) = Bn(z′)−12 z

γn2 (1− z)

−γn−n+βn+12 P (γn−1,−n+βn−γn+1)

n (1− 2z)

with

Bn =((

R(1)α+ β − γ

+r0

γ − 1− r2

β − α

)Γ(γ + n+ 1)Γ(α+ β − γ)

n!Γ(β − α− n)

)− 12

and P(γ−1,α+β−γ)n (2z − 1) the Jacobi polynomials [1].

Confluent hypergeometric potential

A similar construction can be achieved for the class of scaled confluent hyper-geometric functions. The potential is given by

Q(s) =S(z)− 1R(z)

−(r1

z− 5

4(r2

1 − 4r2r0)R(z)

− r2

)z2

R(z)2

52F1(α, β, γ, z) ∼z→1 Γ(γ)

“(−1 + z)−α−β+γ +

Γ(−α−β+γ)Γ(−α+γ)Γ(−β+γ)

”.


with R(z) = r2z2 +r1z+r0 > 0 and S(z) = s2z

2 +s1z+s0. The z coordinate,lying in the interval [0,∞), is defined implicitly in terms of s by the differentialequation

dz(s)ds

=2z√R(z)

Example 9.15The confluent Natanzon potential reduces to the Morse potential

Q(s) =−1 + s0 + s1e

2s√r0 + s2e

4s√r0

r0

for r1 = r2 = 0, to the 3D oscillator

Q(s) =− 3

4 + s0

s2+s1

r1+s2s

2

r21

for r0 = r2 = 0 and to the Coulomb potential

Q(r) =−r2s0 − 2s

√r2s1 − 4s2s2

4r2s2

for r0 = r1 = 0.

By construction, the eigenvector solutions to the CHE potential are given interms of the confluent hypergeometric function F (α, β, γ, z)

φλ(s) = z(s)γ2 e−

ωz(s)2 (z′(s))−

12F (α, β, γ, ωz(s))

By definition, φ(z) ≡ F (α, β, γ, ωz(s)) satisfies the differential equation [1]

zφ′′(z) + (γ − ωz)φ′(z)− ωαφ(z) = 0 (9.37)

The parameters ω, γ, α depend explicitly on the eigenvalue λ by

ω2 = −r2λ− s2

2ω(γ − 2α) = r1λ+ s1

γ(2− γ) = r0λ+ s0

The most general solution to (9.37) is generated by a two-dimensional vectorspace

F (α, γ, ωz) = c1M(α, γ, ωz) + c2(ωz)1−γM(1 + α− γ, 2− γ, ωz)

with M(α, γ, ωz) the M-Whittaker function [1] and c1 and c2 two arbitrarycoefficients. By imposing that the eigenvectors are normalisable (i.e., belong


TABLE 9.4: Condition atz =∞.α > 2 no conditionα ≤ 2 c1 = 0,−1− α+ γ ∈ N∗

or c2 = 0,−α ∈ N∗

TABLE 9.5: Example of solvable superpotentials and local volatilitymodels.

Superpotential W (s) σ(s)σ0

Shifted oscillator as+ b eas2+2bs

Coulomb a+ bs sbe2as

Morse a+ be−αs e2(−( bαeαs )+as)

Eckart a coth(αs) + b e2(bs+ a log(sinh(αs))α )

Rosen-Morse a tanh(αs) + b e2bs+ 2aα ln(cosh(α))

3D oscillator as+ bs eas

2+2b ln(s)

P-T I α > 2b a tan(αs) + bcotg(αs) e2(−( a log(cos(αs))α )+

b log(sin(αs))α )

P-T II α > 2b a tanh(αs) + bcoth(αs) e2( a log(cosh(αs))α +

b log(sinh(αs))α )

to L2(R+)), we obtain the following conditions (see Table 9.4) which give thediscrete spectrum λ and the coefficients c1 and c2.6 We have the discreteeigenvalues αn = −n ;n ∈ N

γn = 1 +√

1− r0λn − s0

ωn =√−r2λn − s2

2n+ 1 =r1λn + s1

2√−r2λn − s2

−√

1− r0λn − s0

and the (normalized) eigenvectors

φn(s) =n!

(γn)nz(s)

γn2 e−

ωnz(s)2 (z′(s))−

12Lγn−1

n (ωnz(s))

with Lγn−1n (z) the (generalized) Laguerre polynomial [1]. In tables 9.5, 9.6, we

have listed classical solvable superpotentials and the corresponding solvablelocal volatility models and solvable instantaneous short-rate models.

Natanzon hierarchy and new solvable processes

We know that the Natanzon superpotential is related to the zero-eigenvector

Wnat = −∂s lnφ0(s)

6M(α, β, z) ∼z→∞ Γ(β)Γ(α)

ezzα−β(1 +O(|z|−1)).


TABLE 9.6: Example of solvable one-factor short-rate models.one-factor short-rate model SDE Superpot.

Vasicek-Hull-White dr = k(θ − r)dt+ σdW Shifted Osc.CIR dr = k(θ − r)dt+ σ

√rdW 3D Osc.

Dooleans dr = krdt+ σrdW ConstantEV-BK dr = r(η − α ln(r))dt+ σrdW Shifted Osc.

and the corresponding supercharge A is

A = ∂s +Wnat(s)

=2z(1− z)√

R

(∂z −

γ0

2z− (1 + α0 + β0 − γ0)

2(z − 1)

−α0β0F (1 + α0, 1 + β0, 1 + γ0, z)γ0F (α0, β0, γ0, z)

+z′−

32 z′′(s)2

)

With a zero drift, the Natanzon superpotential corresponds to the diffusionprocess (9.29)

σ(1)(s) =σ

(1)0

φ(1)0 (s)2

with σ(1)0 a constant of integration. Applying the results of section 9.4.3, we

obtain that the driftless diffusion processes

σ(m)(s) =σ

(m)0

Am−1 · · ·A1φ(1)m−1(s)2

are solvable. Using the fact that the eigenvector φ(1)m−1(s) associated to a

discrete eigenvalue is an hypergeometric function and that the derivative ofa hypergeometric function is a new hypergeometric function,7 the action ofAm−1 · · ·A1 on φ(1)

m1(s) results in a sum of (m− 1) hypergeometric functions,thus generalizing the solution found in [40].

9.5 Gauge-free stochastic volatility models

In this section, pursuing our application of the reduction method, we try toidentify the class of time-homogeneous stochastic volatility models which leads

72F ′1(α, β, γ, z) = αβ

γ 2F1(α+ 1, β + 1, γ + 1, z) and M ′(α, β, z) = αβM(α+ 1, β + 1, z).


to an exact solution of the KBS equation. We assume that the forward f andthe volatility a are driven by two correlated Brownian motions in the forwardmeasure PT

dft = atC(ft)dWt

dat = b(at, ft)dt+ σ(at, ft)dZt ; dWtdZt = ρdt

with the initial conditions a0 = a and f0 = f .Using the definition for the Abelian connection (4.58), we obtain8

Af = − 12(1− ρ2)

∂f ln(C

σ

)− ρ

(1− ρ2)

(b

aCσ− 1

2C∂aσ

a

)Aa =

1(1− ρ2)

(b

σ2− 1

2∂a ln

(σa

))+

ρ

2(1− ρ2)a∂f

(C

σ

)Then, the field strength is

F = (∂aAf − ∂fAa) da ∧ df

=1

(1− ρ2)

((∂af ln(σ)− ∂f

b

σ2

)−ρ(

1C∂a

b

aσ− 1

2C∂2a

σ

a+a

2∂2f

C

σ

))da ∧ df

We assume that the connection is flat,

Faf = 0

meaning that the connection can be eliminated modulo a gauge transforma-tion. In this case, the stochastic volatility model is called a gauge-free model.This condition is satisfied for every correlation ρ ∈ (−1, 1) if and only if

∂af lnσ = ∂fb

σ2

∂ab

aσ− 1

2∂2a

σ

a+aC

2∂2f

C

σ= 0

Moreover, if we assume that σa(a) is only a function of a (this hypothesis isequivalent to assuming that the metric admits a Killing vector), the model isgauge-free if and only if

b

σ=a

2∂aσ

a+ aφ(f)− aC(f)

2∂2fC(f)

∫a′da′

σ(a′)

8

Af = −a2Cσ

4∂fC

σ

Aa =1

2

“b−

aσ

2∂aσ

a

”


TABLE 9.7: Example of Gauge free stochastic volatility models withdf = δ(µ+ νf)

√vdW ′1.

name σ(a) SDE

Heston σ(a) = η dv =√δ(2vγ)dt+ 2η

√δ√vdZt

Geometric Brownian σ(a) = ηa dv =√δ(2ηγv

32 + η2v)dt+ 2

√δηvdZt

3/2-model σ(a) = ηa2 dv = 2√δη(η + γ)v2dt+ 2

√δηv

32 dZt

with φ(f) satisfying

∂fφ(f) =∂f (C∂2

fC)2

∫a′da′

σ(a′)

This last equation is equivalent to C(f)∂2fC(f) = β with β a constant and

φ = γ a constant function. For β = 0, the last equation above simplifies andwe obtain

C(f) = µf + ν

b(a) = aσ(a)(γ +

12∂aσ(a)a

)with µ, ν , γ three integration constants. The gauge-free condition has there-fore imposed the functional form of the drift term:

dv = 2vσ(v) (γ + ∂vσ(v)) dt+ 2√vσ(v)dZt

with v ≡ a2. When the volatility function is fixed respectively to a constant(Heston model), a linear function (geometric Brownian model) and a quadraticfunction (3/2-model) in the volatility, one obtains the correct (mean-reverting)drift9 (see table 9.7). Note that the gauge-free Heston model has a zero-meanreverting volatility (this corrects a typo in [105]).The gauge transformation eliminating the connection is then

Λ(f, a) =1

1− ρ2

(−1

2lnC(f)− γρ

∫ f df ′

C(f ′)+(γ +

ρ

2∂fC

)∫ a a′da′

σ(a′)

)

Finally, by plugging the expression for C(f) and b(a) into (4.59), we find that

9In order to obtain the correct number of parameters, one needs to apply a change of localtime dt = δdt′, dW1,2 =

√δdW ′1,2.


the function Q is10

Q = Aa2 +Bσ2∂a

( aσ

)The Black-Scholes equation for a European call option C(τ = T−t, a, f) (withstrike K and maturity T ) satisfied by the gauge transformed function

C′(τ, a, f) = eΛ(f,a)C(τ, a, f)

is

∂τC′(τ, a, f) = ∆C′(a, f, τ) +Q(a)C′(τ, a, f)

with the initial condition C′(τ = 0, a, f) = eΛ(f,a) max(f − K, 0). In thecoordinates q(f) =

∫ f·

df ′

C(f ′) and a, the Laplace-Beltrami operator is given by

∆ =aσ

2(a

σ∂2q + 2ρ∂aq + ∂a

σ

a∂a)

Applying a Fourier transformation according to q,

C′(τ, k, a) = FC′(τ, q, a)

≡ 1√2π

∫ ∞−∞

e−ikqC′(τ, q, a)dq

we obtain

∂τ C′(τ, k, a) =aσ

2

(−k2 a

σ+ 2ikρ∂a + ∂a

σ

a∂a

)C′(τ, k, a) +Q(a)C′(τ, k, a)


C′(τ = 0, k, a) = F [eΛ(f,a) max(f −K, 0)]

This is a Schrodinger-type equation. Using the spectral decomposition

C′(τ, k, a) =∑n

φnk(a)e−Enkτ

10A and B are two constants given by

A =1

2(−

1

2ρff∂fC + γρaf )2 +

1

2ρffC2∂2

f ln(C)

+ ρ(−1

2ρff∂fC + γρaf )(γρff −

ρaf

2∂fC) +

1

2(ρffγ −

ρaf

2∂fC)2

B = −1

2(ρffγ − ρaf

∂fC

2)


the eigenvectors φnk(a) satisfy the equation

−Enkφnk(a) =aσ

2

(−k2 a

σ+ 2ikρ∂a + ∂a

σ

a∂a

)φnk(a)

+ Qφnk(a) (9.38)

This equation (9.38) can be further simplified by applying a Liouville trans-formation consisting in a gauge transformation and a change of variable [128]

ψnk(s) =(σa

) 12eikρ

R a a′σ(a′)da

′φnk(a)

ds

da=√

2σ(a)

ψnk(s) satisfies a Schrodinger equation

ψ′′nk(s) + (Enk − J(s))ψnk(s) = 0

with the scalar potential

J(s) = Q(a)− k2a2

2+

12a, s

+4 a4 k2 ρ2 − 3σ(a)2 + a2 σ′(a)2 + 2 a σ(a) (σ′(a)− a σ′′(a))

8 a2

and where the curly bracket denotes the Schwarzian derivative of a with re-spect to s

a, s =(a′′(s)a′(s)

)′− 1

2

(a′′(s)a′(s)

)2

The two-dimensional PDE corresponding to our original KBS equation forour stochastic volatility model has therefore been reduced via a change ofcoordinates and two gauge transformations to a Schrodinger equation with ascalar potential J(s). The stochastic volatility model is therefore solvable interms of hypergeometric functions if the potential J(s) belongs to the Natan-zon class. Finally, the solution is given (in terms of the eigenvectors ψnk withthe appropriate boundary condition at τ = 0) by

C(τ, a, f) = e−Λ(a,f)F−1[(σa

)− 12e−ikρ

R a a′σ(a′)da

′∑n

ψnk(s(a))e−Enkτ ]

Let us examine classical examples of solvable stochastic volatility models andshow that the potentials J(s) correspond to the Natanzon class (see table9.8).We also present a new example of solvable stochastic volatility model whichcorresponds to the Posh-Teller I potential.


TABLE 9.8: Stochastic volatility models and potential J(s).name potential J(s)

Heston 3D osc. J(s) =−3+4Bs2η+s4η2(2A+k2(−1+ρ2))

4s2

Geometric Brownian Morse J(s) =−η2+4 e

√2s η (2A+k2 (−1+ρ2))

8

3/2-model Coulombian J(s) =8A+η (−8B+η)+4 k2 (−1+ρ2)

4 s2 η2

Example 9.16 Posh-Teller IFor a volatility function given by σ(a) = α+ ηa2, the potential is given by aPosh-Teller I potential11

J(s) =α

8η(4(−2A+ k2 + 4Bη − k2ρ2

)− 3η2csc

(s√α√η

√2

)2

+

(8A+ η (−8B + η) + 4k2

(−1 + ρ2

))sec(s√α√η

√2

)2

)

9.6 Laplacian heat kernel and Schrodinger equations

In chapter 6, we have seen that the Kolmogorov PDE for the log-normal SABRmodel can be mapped to the Laplacian heat kernel on H3. In this section,we generalize this result and exhibit a strong relation between a Schrodingerequation on an a n-dimensional Riemannian manifold M with a negativepotential and a pure Laplacian heat kernel equation on a (n+ 1)-dimensionalRiemannian manifold M with a Killing vector.

Reduction

M can be considered as a R-principal bundle over the manifold M . In short,a G-principal bundle is a vector bundle with a Lie group G acting transitivelyon the fibers. In our case, the Lie algebra of G is generated by the Killingvector and is isomorphic to R. In such a space M, the data of a metric areequivalent to the data of

• a metric on the base M

• an Abelian connection A

11csc(z) ≡ 1sin(z)

and sec(z) ≡ 1cos(z)

.


• a scalar function φ (i.e., a scalar potential)

Indeed, an Abelian connection on M corresponds to the orthogonal decom-position of the tangent space TpM to M at a point p into a vertical partTpV (i.e., vectors co-linear to the Killing vector) and an horizontal part TpH.The one-form on M which cancels on these horizontal vectors is an Abelianconnection.Using our metric g on M, we can define an horizontal vector space as theorthogonal complement to TpV. This gives an Abelian connection A. More-over, the metric restricted to TpV gives rise to a positive function eφ and themetric pullback to the base gives a metric on M .Finally, the metric gµ,ν with µ, ν = 1, · · · , n+ 1 can be written locally as

ds2 = gµνdxµdxν

= gijdxidxj + eφ(x)(dθ +Aidxi)(dθ +Aidxi) (9.39)

with the Latin indices i, j = 1, · · · , n.In the following, to get our connection with a Schrodinger equation on M , weassume that A = 0. M becomes a so-called wrapped product, noted M ./ R,of M with R. We consider the (Laplacian) heat kernel equation on M:

∂tp(t, x, θ|x′, θ′) = 4Mp(t, x, θ|x′, θ′)

with 4M = g−12 ∂igijg

12 ∂j the Beltrami-Laplace operator and with the initial

condition limt→0 p(t, x, θ|x′, θ′) = δ(x − x′)δ(θ − θ′). From (9.39), we obtainby direct computation

4M = 4M +12

(∂iφ)gij∂j + e−φ∂2θ

By assuming that p(t, x, ·|x′, θ′) is in L2(R), we can apply a Fourier transformover the variable θ

p(t, x, θ|x′, θ′) ≡ 12π

∫Rpk(t, x|x′, θ′)eıkθdk

or equivalently

pk(t, x|x′, θ′) =∫

Rp(t, x, θ|x′, θ′)e−ıkθdθ (9.40)

We take θ′ = 0 on order to impose the initial condition

limt→0

p(t, x, k|x′, θ′ = 0) = δ(x− x′)

pk(t, x|x′, θ′) satisfies then a heat kernel equation on M :

∂tpk(t, x|x′, θ′ = 0) = Dpk(t, x|x′, θ′ = 0)


where D = g−12 (∂i +Ai) g

12 gij (∂j +Aj) +Q with

Ai =14∂iφ

Q = −k2e−φ

Then by doing a gauge transform

p′k(t, x|x′) = e14 (φ(x)−φ(x′))pk(t, x, k|x′, θ′ = 0)

= e14 (φ(x)−φ(x′))

∫R

p(t, x, θ|x′, θ′ = 0)e−ıkθdθ (9.41)

we obtain that p′k(t, x|x′) satisfies a Schrodinger equation on M with a nega-tive potential

∂tp′k(t, x|x′) = ∆Mp

′k(t, x|x′)− k2e−φ(x)p′k(t, x|x′)

Analytical solution for gauge-free SVMs

The Kolmogorov equation reduces to a Schrodinger equation with a nega-tive potential on a non-compact Riemann surface for the so-called gauge-freeSVMs. In this chapter, modulo a Fourier transform, we have shown that this2d PDE can be converted into a 1d-Schrodinger equation for which we canapply a spectral decomposition if the potential belongs to the Natanzon class.From the previous section, this is equivalent to solve a pure heat kernel equa-tion on a three-dimensional (non-compact) manifold. Once we have a solution,we only need to apply a Fourier transform without relying on spectral expan-sion. To our knowledge, the known manifolds for which we have an analyticalheat kernel are the spaces of constant sectional curvature. As M has at leasttwo Killing vectors, ∂y and ∂θ, M can be H3, H2×R and R3. The log-normal(resp. normal) SABR model corresponds to H3 (resp. H2 × R) (see chapter6).

Conclusion

From all this, we have shown how to use supersymmetric methods to generatenew solutions to the Kolmogorov & Black-Scholes equation (KBS) for one-dimensional diffusion processes. In particular, by applying a supersymmetrictransformation on the Natanzon potential, we have generated a hierarchyof new solvable processes. Then, we have classified the stochastic volatilitymodels which admit a flat Abelian connection (with one Killing vector). Thetwo-dimensional KBS equation has been converted into a Schrodinger equa-tion with a scalar potential. The models for which the scalar potential belongs


to the Natanzon class are solvable in terms of hypergeometric functions. Thisis the case for the Heston model, the geometric Brownian model and the 3/2-model. A new solution with a volatility of the volatility σ(a) = α + ηa2,corresponding to the Posh-Teller I, has been presented.

9.7 Problems

Exercises 9.1 Symmetry

1. For a LVM defined by dft = C(ft)dWt, prove that the intrinsic fair valueof a European call option defined as

G(τ, f0,K) ≡ C(τ,K)−max(f0 −K, 0)

is

G(τ, f0,K) =

√C(K)C(f0)√

2

∫ τ

0

P (t′, s(K))dt′

where the fundamental solution P (t′, s(K)) satisfies an Euclidean Schro-dinger equation.

2. Deduce the symmetry

G(τ, f0,K) = G(τ,K, f0)

Chapter 10

Schrodinger Semigroups Estimatesand Implied Volatility Wings

Abstract We study the small/large strike behavior of the implied volatil-ity for local and stochastic volatility models. The derivation uses two-sidedestimates for Schrodinger equations on Riemannian manifolds with scalar po-tentials belonging to different classes.

10.1 Introduction

Since Black-Scholes, several alternative models have emerged such as localvolatility models (LVMs) and stochastic volatility models (SVMs) which havebeen reviewed in chapters 5 and 6. Although all these models are able tofit reasonably the market implied volatilities across a (liquid) range of strikesand maturities, the overall properties of each model are quite different.Notable differences come from the dynamics of the implied volatility. Moreprecisely, although matching the initial market implied volatility surface, twomodels, belonging to different model classes, can give different prices whenpricing exotic options such as forward-starting options.Similarly, the different (illiquid) large-strike behaviors, not quoted on themarket, produce different prices for volatility derivatives, such as varianceswaps which strongly depend on the implied volatility wings.In order to choose the best model for pricing and capturing the risk, it istherefore important to understand the general properties of these models.Below is a list of common properties that can characterize a model:

• Is the model exploding in a finite time? A study for some particularstochastic volatility models was achieved in [47, 29] and reviewed inchapter 6 using the Feller criteria.

• What is the short-time behavior of the implied volatility? Can we usethis short-time asymptotics to calibrate the model if no analytical so-lutions are available? A general short-time asymptotics for LVMs andSVMs at the first-order in the maturity is proposed in chapters 5 and 6

289


using the heat kernel expansion on a Riemann manifold endowed withan Abelian connection.

• Is it solvable? In particular, can we obtain analytical prices for liquidoptions and calibrate efficiently the model? An extensive classificationof solvable LVMs and SVMs is achieved in chapter 9.

• What is the large-strike behavior of the implied volatility? A partialanswer to this question was given by the Lee moment formula [122]which translates the behaviors of the wings in the existence of highermoments for the forward. Recently, Benaim and Friz gave a tail-wingformula [53] sharpening the moment formula which relates the tails ofthe risk neutral returns with the wings behavior of the implied volatility.

• What is the large-time behavior of the implied volatility? A study of thelarge-time behavior of the implied volatility for analytical SVMs, mainlyHeston, 3/2 and geometric Brownian models was achieved in [29].

• What is the forward implied volatility and skew implied by a model?An asymptotic analytical answer was given in [59] for the Heston modelfor small/long maturities.

In this chapter, we study the large-strike behavior of the implied volatility forLVMs and SVMs. The motivation is to obtain elegant expressions to describethe left/right-tail behavior in the regime where numerical PDE or Monte Carlomethods break down.In this context, we use sharp bounds for heat kernel equations. In order toobtain accurate estimates, we restrain ourselves to SVMs for which the Itogenerator can be transformed into a self-adjoint unbounded operator. Then,the Kolmogorov equation reduces to an (Euclidean) Schrodinger equation ona Riemannian manifold with a time-dependent scalar potential as explainedin detail in chapter 9. For LVMs, this reduction is always possible. Althoughthe reading of chapter 9 is recommended, we have included a brief reminder.

10.2 Wings asymptotics

For the sake of completeness, we recall in this section a few basic definitionsand results. The implied volatility is the (unique) value of the volatilityσBS(τ, k) that, when put in the Black-Scholes formula, reproduces the marketprice for a European call option with log-strike k = ln

(Kf0

)and maturity τ

P−10τ C(τ, k, σ) = f0N(d+)−KN(d−)

Schrodinger Semigroups Estimates and Implied Volatility Wings 291

with d± = − kσ√τ± σ

√τ

2 . Below, we denote the dimensionless parameterVBS(τ, k) ≡ σBS(τ, k)

√τ and let the strictly decreasing function Ψ : [0,∞]→

[0, 2] be

Ψ(x) = 2− 4(√x2 + x− x)

A first step in the study of the implied volatility wings is achieved by theLee moment formula [122] which translates the behavior of the wings in theexistence of higher moments for the forward. This formula states that theimplied volatility is at-most linear in the moneyness k as |k| → ±∞ with aslope depending on the existence of higher moments. More precisely, we have

THEOREM 10.1 Lee’s theorem [122]Let q∗ = supq : E[f−qT ] <∞. Then

lim supk→∞

VBS(τ,−k)2

|k|= Ψ(q∗)

Also let p∗ = supp : E[f1+pT ] <∞. Then

lim supk→∞

VBS(τ, k)2

k= Ψ(p∗)

REMARK 10.1 Small-strike asymptotics from large-strike asymp-totics We can deduce the small-strike asymptotics from the large-strikeasymptotics using the put-call duality (see section 6.2 for details) which meansthat

EPf [max(XT −K, 0)|Ft] =K

f0EP[max(K−1 − fT , 0)Ft] (10.1)

with XT ≡ f−1T . Here Pf is the measure associated to the forward fT defined

by the Radon-Nikodym derivative

dPf

dP|FT = fT

From (10.1) and the definition of the implied volatility, we have

lim supk→∞

V XBS(τ, k)2

|k|= lim sup

k→∞

V fBS(τ,−k)2

|k|

By using the large-strike asymptotics for the process XT , we have

lim supk→∞

V XBS(τ, k)2

k= Ψ(p∗X)


with p∗X = supp : EPf [Xp+1T ] <∞. As

EPf [Xp+1T ] = EP[

dPf

dPf−p−1T ] = EP[f−pT ]

we deduce the small-strike asymptotics formula.

For example, when all the moments exist, we have

lim supk→±∞

VBS(τ, k)2

|k|= 0

If we apply this result to the Black-Scholes log-normal process, the lim sup isrough as the implied volatility is flat. Moreover, the Lee moment formula isquite hard to use as we generally don’t have at hand an analytical conditionalprobability, hence the difficulty to examine the existence of higher moments.The moment formula was recently sharpened by Benaim and Friz [53] whoshow how the tail asymptotics of the log stock price translate directly tothe large-strike behavior of the implied volatility. Their tail-wing formulareliably informs us when the lim sup in Lee moment formula [122] can bestrengthened to a true limit. More specifically, it links the tail asymptotics ofthe distribution function of the log return to the implied volatility behaviorat extreme strikes. Moreover, Benaim and Friz [54] develop criteria based onTauberian theorems to establish when the lim sup in the Lee moment formulacan be replaced by a true limit.We define the function class Rα of regularly varying function of index α by

DEFINITION 10.1 Class Rα A positive real-valued measurable functiong is regularly varying with index α, in symbols g ∈ Rα, if

limx→∞

g(λx)g(x)

= λα

Assumption (IR): We assume the integrability condition on the right taildenoted by (IR)

∃ ε > 0 : E[f (1+ε)τ ] <∞

Then we have the following theorem [53]

THEOREM 10.2 [53]Assume condition (IR). Then if − ln p(τ, k) ∈ Rα>0 with p(τ, k) the probabil-

ity of the log-forward, we have1

VBS(τ, k)k

2

∼ Ψ(−1− ln p(τ, k)

k

)1f ∼ g means limk→∞

f(k)g(k)

= 1.


In the following, we try to derive the wings asymptotics directly from thebehavior of the local or stochastic volatility function. Our technique is ap-plicable when neither the moment generating function nor the distributionfunction is known.Note that for commonly used LVMs such as the Constant Elasticity of Vari-ance model (CEV), the probability density (5.23) is not a regularly varyingfunction and the theorem above is not applicable. Our study will rely on thetheorem 10.3 which relates the asymptotics of the wings to the behavior ofthe call option price. We note C(τ, k) the fair value of a European call optionwith log-strike k and maturity τ and we assumeAssumption (IR’): The condition (IR) is satisfied and − ln C(τ, k) is a reg-ularly varying function in the moneyness k (or in the strike K) with a strictlypositive index.

THEOREM 10.3 [55]Let us assume (IR’). Then

VBS(τ, k)2

k∼ Ψ

(− ln C(τ, k)

k

)(10.2)

In particular, if − ln(C(τ,k))k goes to infinity as k → ∞, the expression above

becomesVBS(τ, k)2

k∼ k

−2 ln C(τ, k)Note that a similar result can be obtained for the left wing using the put-callduality (see remark 10.1). This is why we only focus on the right wing in thenext sections.Let us now explain how to find Gaussian estimates for the conditional prob-ability for LVMs. From these Gaussian estimates, we obtain lower and upperestimates of the wings using the theorem 10.3. In order to find these Gaussianestimates, we first reduce the Kolmogorov equation associated to a LVM to an(Euclidean) Schrodinger equation with a (time-dependent) scalar potential.

10.3 Local volatility model and Schrodinger equation

10.3.1 Separable local volatility model

Before moving on to the general case, we recall briefly the reduction methodexplained in chapter 9 with a simpler example, namely a separable LV model:


We assume that under P, the forward f follows a one-dimensional, time-homogeneous regular diffusion on an interval I ⊂ R+

dft = A(t)C(ft)dWt

with ft=0 = f0. The so-called local volatility C(f) is a positive continuousfunction on I and A(t) a strictly positive continuous function on R+. The Itoinfinitesimal generator H associated to the backward Kolmogorov equationfor the conditional probability p(t, f |f0) is

Hp(t, f |f0) =12A(t)2C(f0)2∂2

f0p(t, f |f0)

We now convert the backward Kolmogorov equation into a simpler form. Forthis purpose, we assume that C ′(f) and C ′′(f) exist and are continuous on Iin order to perform our (Liouville) transformation. By introducing the newcoordinate s =

√2∫ f0

·df ′

C(f ′) which can be interpreted as the geodesic distance

and the new time t′

=∫ t

0A(s)2ds, one can show that the new function P (t′, s)

defined by

p(t, f |f0) = P (t′, s)

√2C(f0)

C(f)32

(10.3)

satisfies a (Euclidean) one-dimensional Schrodinger equation

(∂2s +Q(s))P (t′, s) = ∂t′P (t′, s) (10.4)

The time-homogeneous potential is2

Q(s) =12

(lnC)′′(s)− 1

4((lnC)

′(s))2 (10.5)

where the prime ′ indicates a derivative according to s. Table 10.1 is a list ofexamples of potentials for a few particular time-homogeneous LV models.By applying Ito-Tanaka’s formula on the payoff max(ft − K, 0) (5.29), weobtain that a European call option C(τ, k) with maturity τ and log-strike kcan be rewritten as a local time (without loss of generality, we assume a zerointerest rate)

C(τ, k) = max(f0 −K, 0) +∫ τ

0

C(K)2

2p(t′,K|f0)dt′

Using the relation (10.3) between the conditional probability p(t, f |f0) and itsgauge-transform P (t, s), we obtain

C(τ, k) = max(f0 −K, 0) +

√C(K)C(f0)√

2

∫ τ

0

P (t′, s(K))dt′ (10.6)

2In the f -coordinate, we have Q(f) = 182C(f)C′′(f)− C′(f)2.


TABLE 10.1: Example of potentials associated to LV models.LV Model C(f) Potential

Black-Scholes f Q(s) = − 18

Quadratic af2 + bf + c Q(s) = − 18 (b2 − 4ac)

CEV fβ , 0 ≤ β < 1 QCEV (s) = β(β−2)4(1−β)2s2

LCEV f min(fβ−1, εβ−1

)sε ≡

√2ε1−β

(1−β)

with ε > 0 QLCEV (s) = QCEV (s) ∀ s ≥ sεQLCEV (s) = − 1

8ε2(β−1) ∀ s < sε

10.3.2 General local volatility model

In the Dupire LV model [81], it is assumed that under P the forward followsa one-dimensional regular diffusion on an interval I ⊂ R+

dft = C(t, ft)dWt

The so-called Dupire local volatility C(t, f) is a strictly positive continuousfunction on R+ × I. The Ito infinitesimal generator H is

Hp(t, f [f0) =12C(t, f0)2∂2

f0p(t, f |f0)

Throughout this chapter, we adopt the reduced variable s = ln ff0

, σ(t, s) =C(t,f0e

s)f0es

and p(t, s) = p(t, f |f0). The transformed forward Kolmogorov equa-tion satisfies

∂tp(t, s) =12∂s(σ(t, s)2∂sp(t, s)

)+

12∂s(σ(t, s)2p(t, s)

)We assume alsoAssumption 1: σ(t, s), ∂sσ(t, s), are uniformly continuous on R+ × I.Note that these conditions are not at all restrictive and are satisfied for (rea-sonable) market conditions. The new function P (t, s) defined by

p(t, s) = e−s2P (t, s)

satisfies a one-dimensional (Euclidean) Schrodinger equation

12∂s(σ(t, s)2∂s)P (t, s) +Q(t, s)P (t, s) = ∂tP (t, s) (10.7)

with a time-dependent scalar potential given by

Q(t, s) = −18σ(t, s)2 +

12σ(t, s)∂sσ(t, s) (10.8)

Proceeding as in the previous section, the fair value of a European call optionC(τ, k) can be written as

C(τ, k) = max(f0 −K, 0) +

√f0

K

∫ τ

0

C(t′,K)2

2P (t′, k)dt′ (10.9)


Still proceeding as previously discussed, we convert the PDE (10.7) into a moreconventional Schrodinger equation on R by doing a time-dependent change ofcoordinates st =

√2∫ ff0

df ′

C(t,f ′) . Thus, the new function P (t, s) defined by

p(t, f |f0) =√

2C(t, f)

√C(t, f0)C(t, f)

P (t, s)eR ff0

df′C(t,f′)∂t

R f′f0

df′′

C(t,f′′ ) (10.10)

satisfies a one-dimensional (Euclidean) Schrodinger equation

(∂2s +Q(t, s))P = ∂tP (10.11)

with a time-dependent scalar potential given by

Q(t, s) = −∂sµ(t, s)− µ(t, s)2 −∫ s

0

∂tµ(t, s′)ds′ (10.12)

with µ(t, s) = ∂t(∫ f(s)

f0

df ′

C(t,f ′) )− 12∂s ln(C(t, s)). Note that for a time-homogeneous

local volatility C(t, f) ≡ C(f), the potential Q(t, s) reduces to (10.5).After the reduction of the Kolmogorov equation into a symmetric semigroup,we explain how to find Gaussian lower and upper bounds.

10.4 Gaussian estimates of Schrodinger semigroups

The history of Gaussian estimates of parabolic PDEs is quite rich starting withthe works of Nash [130] and Aronsov [48] on Gaussian estimates for Laplacianheat kernel equation. By Gaussian estimates, we mean in short that thefundamental solution P (t, s) can be bounded by two Gaussian distributions

c1pG(c2t, s|s0) ≤ P (t, s) ≤ C1pG(C2t, s|s0)

with pG(t, y|x) = 1

(4πt)12

exp(− (y−x)2

4t ) the Gaussian heat kernel and Ci, cii=1,2

some constants.

10.4.1 Time-homogenous scalar potential

The fundamental solution of (10.4) satisfies Gaussian bounds provided thatthe potential Q(s) belongs to the Kato class:

DEFINITION 10.2 Autonomous Kato class We say that Q(·) is inthe Kato class K if

limδ→0

supx∈R

∫ δ

0

dt

∫Rdy|Q(y)|pG(t, y|x) = 0


Moreover, we say that Q ∈ Kloc if ∀N ≥ 1, Q(y)1N (y) ∈ K.3

Additional properties for potentials belonging to the Kato class can be foundin [37]. Then, we have

THEOREM 10.4 [139]Let Q+ ≡ max(Q, 0) ∈ K and Q− = max(−Q, 0) ∈ Kloc. Then we have an

upper bound

P (t, y|x) ≤ C1eC2tpG(t, y|x), t > 0, x, y ∈ R

with two constants C1, C2. Note that the constant C2 = 0 if Q+ = 0. Byassuming that Q+ and Q− are both in the Kato class K, we have also a lowerbound

c1ec2tpG(t, y|x) ≤ P (t, x|y), t > 0, x, y ∈ R

with two constants c1 and c2.

One can check that the models listed in (Table 10.1) except the CEV model(due to the singularity at f = 0) belong to the Kato class.

Example 10.1For the short-rate Vasicek model, Q(y) is the harmonic potential Q(y) = y2.This potential does not belong to the Kato class.

REMARK 10.2 Lower bound from upper bound It is a classicalfact that for Schrodinger operators H = ∆ + Q on L2(Rd), Gaussian lowerbounds follow from upper bounds as reviewed in [133]. The derivation relieson the Feynman-Kac formula.Let us denote Wt the d-dimensional Brownian motion and ∆ the Laplacianon Rd. We have for every non-negative f ∈ C∞c (Rd)

e−t∆f(x) = EPx[f(Wt)]

≤√

EPx[f(Wt)e−

R t0 Q(Ws)ds]

√EPx[f(Wt)e

R t0 Q(Ws)ds]

=√e−t(∆−Q)f(x)

√e−t(∆+Q)f(x)

where we have used the Cauchy-Schwartz inequality. Then, we have

e−t∆f(x) ≤ 1εe−t(∆+Q)f(x) + εe−t(∆−Q)f(x)

31N (y) = 1 if y ≤ N , zero otherwise.


This inequality holds for all ε > 0. We deduce

pG(t, x|x0) ≤ 1εp∆+Q(t, x|x0) + εp∆−Q(t, x|x0)

where p∆±Q(t, x|x0) is the conditional probability associated to the Ito gen-erator H = ∆ ± Q. By assuming that Q− is in the Kato class, we have anupper bound for p∆−Q

εpG(t, x|x0) ≤ ε2C1eC2tpG(t, x|x0) + p∆+Q(t, x|x0)

and

p∆+Q(t, x|x0) ≥ pG(t, x|x0)(ε− ε2C1e

C2t)

We optimize over ε and we obtain the Gaussian lower bound

p∆+Q(t, x|x0) ≥ 14C1

e−C2tpG(t, x|x0)

10.4.2 Time-dependent scalar potential

A similar result [127] can be obtained for Schrodinger equation (10.11) witha time-dependent scalar potential Q(t, s) provided that the Kato class K isextended to the non-autonomous Kato class K defined by

DEFINITION 10.3 Non-autonomous Kato class We say that Q(·, ·)is in the non-autonomous Kato class K if

limδ→0

N±δ (Q) = 0

where

N±δ (Q) = sups,x∈R

∫ δ

0

dt

∫Rdy|Q(s± t, y)|pG(t, x|y)

Unfortunately, even for simple models such as C(t, f) = a(t)f + b(t), thepotential Q(t, s) (10.12) does not belong to the non-autonomous Kato class.Therefore, as explained by [87], we assume a stronger boundness conditionon the potential Q defined by (10.8) so that we can apply a sharp Gaussianestimate found by Norris and Stroock [131] for the PDE (10.7).We define the control distance

d(t, s)2 = infγ∈Γ(t,s)

∫ t

0

2σ(u, γ(u))2

(dγ(u)du

)2

du


where Γ(t, s) = γ ∈ C1([0, t],R) : γ0 = 0, γt = s,∫ t

0

(dγ(u)du

)2

du <∞. Notethat for time homogeneous LVMs, the control distance reduces to the classicalgeodesic distance.Then, we have (as reported in [87])

THEOREM 10.5Assume the condition Assumption 1 and that there are constants λ ∈ [1,∞)

and Λ ∈ [0,∞) such that, uniformly on R+ × I,

λ−1 ≤ 12σ(t, s)2 ≤ λ and |Q(t, s)| ≤ Λ

Let α ∈ ( 12 , 1) satisfying α2

2α−1 > λ2 be given. Then for all λ ∈ [1,∞),Λ ∈[0,∞) and all T ∈ (0,∞), we have the following asymptotic result for thefundamental solution P (t, s) associated with (10.7)

limM→∞

inf0<t≤T, s∈,R,E≥M

logP (t, s) + E

E1

4α−1= 0

lim supM→∞

sup0<t≤T, s∈,R,E≥M

logP (t, s) + E

logE≤ N

4α− 2(10.13)

where we have set E ≡ d(t,s)2

4t .

REMARK 10.3 Norris & Stroock [131, 87] also mention that we havethe estimate

|s|2

8λt− 1

4Λt < E <

λ|s|2

2t+

12

Λt (10.14)

in terms of the coefficient bounds λ and Λ.

For fixed t, s, by (10.14), we see that E can be made arbitrarily large if s issufficiently large. In particular for all ε > 0, there exists a s such that

e−εE1

4α−1e−E < P (t, s) < E( N

4α−2 +ε) e−E

Thus, the theorem (10.5) implies the following corollary of the Norris-Stroockresult [87]

COROLLARY 10.1

− lnP (t, s) ∼ d(t, s)2

4t(|s| → ∞)


Note that in [87], this corollary was obtained in terms of the energy functionalwhich is more difficult to compute than the control distance.In the subsequent pages, we apply the theorems (10.2, 10.3) to obtain thelarge-strike behavior of the implied volatility.

10.5 Implied volatility at extreme strikes

10.5.1 Separable local volatility model

Assuming that the scalar potential (10.5) associated to a separable localvolatility function belongs to the Kato class, we have the Gaussian bounds onthe function P (t, s) from the theorem (10.4)

c1ec2tpG(t, s) ≤ P (t, s) ≤ C1e

C2tpG(t, s)

This inequality directly translates on an estimation of the fundamental solu-tion p(t, f |f0) using the relation between p(t, f |f0) and its gauge-transformP (t, s) (10.3)

c1ec2tpG(t, s) ≤ C(f)

32√

2C(f0)p(t, f |f0) ≤ C1e

C2tpG(t, s) (10.15)

In the following example, we will check the validity of these Gaussian estimatesfor the CEV model for which we know analytically the conditional probability.

Example 10.2 CEV modelThe CEV model is a LV model for which the local volatility function is apower of the forward: C(f) = fβ , with 0 < β < 1. A closed-form expressionfor the risk-neutral conditional probability (5.23) is given by

p(t, f |f0) =f

12−2β

(1− β)t

√f0e− f

2(1−β)+f2(1−β)0

2(1−β)2t I 12(1−β)

((ff0)1−β

(1− β)2t

)where I is the modified Bessel function of the first kind. As

I 12(1−β)

(x) ∼x→∞1√2πx

ex

we deduce that the large forward limit exhibits a Gaussian behavior√2C(f0)

C(f)32p(t, f |f0) ∼ 1√

4πte−

s(f)2

4t (10.16)


with s(f) =∫ f· x−βdx. The potential associated to the CEV model is

Q(s) =β(β − 2)

4(1− β)2s2

Unfortunately, the potential doesn’t belong to Kato class K due to the sin-gularity at s = 0. Therefore, we modify the CEV model by the limitedCEV model (LCEV) 5.5 for which the potential belongs to K and we have aGaussian estimate for the heat kernel. From the LCEV definition, the large-forward behavior of the CEV and LCEV conditional probabilities are the sameas shown in [44]. In the limit f → ∞, from (10.16), the inequalities (10.15)are trivially satisfied for the LCEV model. Note that the lower and upperbound are exact in the limit f →∞ for C1 = 1, c1 = 1 , C2 = 0 and c2 = 0.

By plugging our Gaussian estimates for the conditional probability (10.15)into the expression (10.6) and by doing the integration over the time t, weobtain lower and upper bounds for the fair value of a European call option.Then, assuming that the condition (IR’) is fulfilled and using theorem (10.3),we can directly translate the Gaussian bounds on the call option into boundson the implied volatility. For a time-homogeneous local volatility model, weobtain

THEOREM 10.6By assuming that the quantity − 1

2 ln(C(K)) + s(K)2

4τ belongs to Rα>0 (in kor K) and that the potential (10.5), associated to a time-homogeneous LVM,belongs to the Kato class K, the large strike behavior of the implied volatilityis given by

VBS(τ, k)2

k∼k→∞ Ψ

(− 1

2 ln(C(K)) + s(K)2

4τ

k

)

with s(K) =√

2∫Kf0

df ′

C(f ′) . Moreover, if s(K) is the leading term, we havethat the large-strike behavior of the implied volatility involves the harmonicaverage of the local volatility function

σBS(τ, k) ∼k→∞√

2ks(K)

REMARK 10.4 Note that this limit is the BBF formula (5.39) and wasobtained for the short-time limit of the implied volatility in chapter 5: In thelimit τ → 0, the implied volatility is the harmonic mean of the local volatility,namely

limτ→0

σBS(τ, k) =√

2ks(K)


Therefore assuming that Q belongs to the Kato class, the large-strike andshort-time behaviors coincide.

As an example, we obtain the following tail estimates for the implied volatilityin the CEV model

Example 10.3 CEV modelFor 0 ≤ β < 1,we have

σBS(k, τ) ∼k→∞k(1− β)K1−β (10.17)

and for β = 1, we have σBS(τ, k) ∼k→∞ 1.

This relationship is derived in [87, 55] using the Freidlin-Wentzell theory.This result should be compared with the result obtained using the Lee’s mo-ment formula

REMARK 10.5 By applying the Lee’s moment formula, we obtain for0 ≤ β ≤ 1

lim supk→∞

VBS(k, τ)2

k= 0

as all the moments exist.

10.5.2 Local volatility model

For the Dupire LV model, we proceed as in the previous section by pluggingthe Gaussian estimates for P (t, f |f0) given by corollary (10.1) into (10.9). Weobtain

THEOREM 10.7

By assuming that the control distance d(t, s) ∈ Rα>0 (in k or in K) and thatthe local volatility C(t, f) and the potential (10.8), associated to a LV model,satisfies the assumption in the theorem (10.5), the large strike behavior of theimplied volatility is given by

VBS(k, τ)2

k∼k→∞ Ψ

(−1

2+d(τ, k)2

4kτ

)


10.6 Gauge-free stochastic volatility models

We assume that the forward f and the volatility a are driven by two correlatedBrownian motions in the forward measure PT

dft = atC(ft)dWt (10.18)dat = b(at)dt+ σ(at)dZt (10.19)dWtdZt = ρdt

with the initial conditions a = α and f = f0. We assume that the diffu-sion process is non-explosive. The resulting Ito generator is generally not asymmetric operator. However, by doing a gauge transform on the conditionalprobability p(τ, f, a|f0, α) with a specific time-homogeneous function Λ(f, a):

P (τ, f, a|f0, α) = eΛ(f,a)p(τ, f, a|f0, α)

the Ito generator can be reduced to a symmetric operator for certain SVMs,called gauge free stochastic volatility models (GFSVM) in section 9.5. By con-struction for these models, P (τ, f, a|f0, α) satisfies a self-adjoint heat kernelequation (x ≡ (f, a), x0 ≡ (f0, α))

∂τP (τ, x|x0) = 4P (x, τ |x0) +Q(x)P (τ, x|x0) (10.20)

where 4 is the Laplace-Beltrami operator on a (non-compact) Riemann sur-face. Here Q is an (unspecified) time-homogeneous potential. In section 9.5,these (GFSVM) models have been completely characterized and are definedby

C(f) = µf + ν

b(a) = aσ(a)(γ +

12∂aσ(a)a

)with µ, ν, γ three constants. The symmetric condition has therefore im-posed the functional form of the drift term b(a) and the volatility C(f) of theforward.The gauge transformation Λ(a, f) which reduces the backward Kolmogorovequation to (10.20) is

(1− ρ2)Λ(f, a) = −12

ln(C(f)C(f0)

)− γρ

∫ f

f0

df ′

C(f ′)+ (γ +

ρ

2∂fC(f))

∫ a

α

a′da′

σ(a′)

The derivation of Gaussian estimates for a Schrodinger equation on a Rie-mannian manifold (10.20) is a difficult problem. A review of approaches andresults can be found in [96]. Note that it is not possible to use the Norris-Stroock result as the metric gij is not uniformly elliptic and the potential is


not bounded (it does not even belong to the Kato class in general). For ex-ample for the log-normal SABR model, gij is the metric on the non-compactcomplete hyperbolic plane and the potential Q is unbounded: Q = − a2

8(1−ρ2) .If the potential Q cancels and the Ricci curvature is bounded for below, thereexists Gaussian estimates found by Li-Yau [125] and improved in [78, 141]:

THEOREM 10.8Let Σ be a complete (non-compact) Riemannian manifold without boundary.If the Ricci curvature of M is bounded from below by −K, i.e.,

Rµν(x)XµXν ≥ −(n− 1)κXµXν (∀x ∈M,∀X ∈ TxM)

with a constant κ ≥ 0, then the heat kernel P (τ, x|x0) satisfies an upper bound[78]: For any ε > 0, there exists a constant C1 = C1(κ, d, ε) such that for allt > 0 and x, y ∈M :

p(t, x|y) ≤ C1m− 1

2 (B√t(x))m−12 (B√t(y))e

−(1−ε)d2(x,y)4t e(ε−λ(M))t

d(x, y) is the geodesic distance on Σ between the points x and y. λ(M) ≥ 0is the bottom of the L2-spectrum of the operator − 1

2∆ on M . B√t(x) is thegeodesic volume of a ball of radius

√t and center x.

We also have the lower bound [141]

p(t, x|y) ≥ (2πt)−d2 e−(1−ε) d

2(x,y)4t −(n−1)2−

32√κd(x,y)e−λ

κ,nt

with λκ,n = (n−1)2

8 κ.

Note that the geodesic distance associated to the SVM defined by (10.18,10.19)has been computed in chapter 6 and is given by (6.17).Warning: In the following, we will assume that our Schrodingerequation (10.20) satisfies Gaussian bounds:

14πt

c1e−d2(x,y)

4t ≤ p(t, x|y) ≤ 14πt

C1e−d2(x,y)

4t (10.21)

From (10.21), we derive an upper (lower) bound for the value of a call Vanillaoption with strike K and maturity τ

C(τ, k) ≤ max(f0 −K, 0) +C1C(K)

4π√

1− ρ2

∫ ∞0

ada

σ(a)eΛ(K,a)E1

(d(K, a)2

4τ

)Note the sign =+ in front of Λ as we have switched f0 (resp. α) with K (resp.a) (see remark 4.3). E1(x) =

∫∞x

e−u

u du is the exponential integral function.By assuming that limK→∞ d(K, a) =∞ and as

E1(x) ∼x→∞e−x

x

(1− 1

x+

2x2

)


we obtain from the lower and upper bounds

ln(C(τ, k)) ∼k→∞ ln(C(K)) + ln

(∫ ∞0

ada

σ(a)e−

d(K,a)2

4τ +Λ(K,a)

d(K, a)2

)

∼k→∞ ln(C(K))− d(K, amin)2

4τ+ Λ(K, amin)

where amin = amin(K) is the saddle-point which minimizes the function

−d(K, a)2

4τ+ Λ(K, a)

From the theorem 10.3, we deduce

VBS(τ, k)2

k∼k→∞ Ψ

(− ln(C(K)) + d(K,amin)2

4τ − Λ(K, amin)k

)Specifying the function Λ, we have our final result

VBS(τ, k)2

k∼k→∞ Ψ

(− lnC(K)

k+d(K, amin)2

4kτ(10.22)

+1

2(1− ρ2)k

(12

lnC(K)C(f0)

+ γρ

∫ K

f0

df ′

C(f ′)−(γ +

ρ

2∂fC

)∫ amin

α

a′da′

σ(a′)

))

Example 10.4 Normal SABR ModelFor the normal SABR model (i.e., C(f) = 1), the volatility is σ(a) = νaand γ = 0. The metric associated to this model corresponds to the two-dimensional hyperbolic manifold H2. Besides, the gauge function Λ and thepotential cancel: Λ = 0, Q = 0. In this case, the Schrodinger equation reducesto the Laplacian heat kernel on H2 for which the Li-Yau-Davies Gaussianestimate is valid.We found that the effective distance is explicitly given by

d(amin) =

√2ν2

cosh−1

(−qνρ− αρ2 + amin

α(1− ρ2)

)with q = K − f0 and a2

min = α2 + 2ανρq + ν2q2. We obtain

d(amin) ∼√

2ν2

ln q

From (10.22), we get

σBS(τ, k) ∼k→∞ ν


Example 10.5 Log-normal SABR ModelFor the log-normal SABR model, the potential is unbounded Q(a) = − a2

8(1−ρ2)

and we note that at the saddle-point it diverges when k →∞

Q(amin) ∼ − ν2k2

8(1− ρ2)

We suspect that our Gaussian estimates (10.21) are not correct. Indeed, from(10.22), we get

VBS(τ, k)2

k∼k→∞ Ψ

(− 1 + 2ρ

2(1 + ρ)

)This result is incorrect as the argument for Ψ should be positive. In [55], theLee’s moment formula has been applied

lim supk→∞

VBS(τ, k)2

k= Ψ

(ρ2

1− ρ2

)This result should also be compared with the (uncorrect) result obtained usingthe short-time asymptotics of the implied volatility

limk→∞

σBS(τ, k)2

k= limk→∞

ν2k

ln2(k)=∞!

This illustrates the fact that the limits τ → 0 and k →∞ do not commute ingeneral.

We conclude this section with two conjectures.

Some conjectures

The potential for the SABR model with β < 1 evaluated at the saddle-pointis constant

Q(amin(f)) =ν2

4(1− β)2

(β(β − 1)− β2

2(1− ρ2)

)In this case, we conjecture that the Gaussian estimate (10.21) is valid and wegetConjecture 1:For the SABR model with β < 1, we have

σBS ∼ν

1− β

This conjecture is proved in [55] in the case ρ = 0.Conjecture 2:If the potential Q(amin) is bounded, the relation (10.22) is valid.


10.7 Problems

Exercises 10.1 SVI parametrizationGatheral [89] presents the following “Stochastic Volatility Inspired” (SVI)parametrization of the implied volatility

σBS(τ, k)2τ = a+ b(ρ(k −m) +

√(k −m)2 + σ2

)where the five parameters a, b, ρ, σ and m depend on the maturity τ . Thisparametrization agrees with the Lee moment formula, i.e., the large-strikebehavior is at most linear in the moneyness k.

1. Prove that the no-arbitrage condition ∂2C(τ,K)∂K2 > 0 is preserved when

k >> 1 if and only

|b(1 + ρ)| ≤ 2

Chapter 11

Analysis on Wiener Space withApplications

Abstract We review the main notions and results in the stochastic calcu-lus of variations, commonly called Malliavin calculus and developed by Malli-avin in 1976 in order to give a probabilistic proof of the “sums of squares”Hormander theorem. In the last section, we focus on various applications:The convexity adjustment of CMS option, the probabilistic representation ofsensitivities and the calibration of stochastic volatility models.

11.1 Introduction

Since the works of Fournie et al. [85, 86] on the probabilistic representationof sensitivities of derivatives products (i.e., Greeks), various lecture notes onMalliavin calculus have appeared. Let us cite [31, 33, 132, 52, 88]. From ourperspective, all these lectures are highly technical and do not give a motivationto get familiar with such a method. Moreover, the mathematical proofs of themain theorems are hidden by various technical probability results that areoften out of reach of the quantitative analysts.In this chapter, we explain the main notions and sketch the proofs in theMalliavin calculus, highlighting the link with the formal (simpler) functionalintegration invented by the physicists in the fifties to find a Lagrangian for-mulation of a Quantum Field Theory (QFT). In particular, we connect theMalliavin derivative to the functional derivative, the Skorohod integral to theWick product and the Wiener chaos decomposition to the Fock space. Inspirit, our presentation follows [73].We consider three distinct applications:Firstly, we compute the convexity adjustment for CMS options in the case ofshort-rate affine models, mainly the Hull-White model. Our tool will be theWick calculus and the Wiener chaos decomposition.Secondly, we obtain a probabilistic representation of sensitivities of derivativeproducts allowing an accurate computation via Monte-Carlo simulation.Finally, we obtain a probabilistic representation of conditional expectationssuch as the instantaneous variance of a stock process conditional to its forward.

309


TABLE 11.1: A dictionary from Malliavin calculusto QFT.

Malliavin Derivative Functional DerivativeSkorohod integration Wick product

Ornstein-Uhlenbeck semigroup Harmonic oscillatorWiener chaos Fock space

This gives the local volatility associated to a SVM. This representation pavesthe way for the calibration of a mixed local and stochastic volatility model.

11.2 Functional integration

11.2.1 Functional space

Let W be the path space of pointed continuous paths ω on the interval [0, 1]

W = ω : [0, 1]→ R : ω(0) = 0

In QFT, ω is a (0 + 1)-field. As usual, we endow this space with the Wienermeasure (Ω,F ,P). For further use, we introduce the Cameron-Martin spaceH:

H = h ∈ L2([0, 1]) : h(·) =∫ ·

0

h(s)ds ∈W

The derivative of h, h(s)′ = h(s) (where the ′ indicates a derivative accordingto s) exists ds-almost surely. We note below < ·, · > the scalar product onL2([0, 1]).On this space, we define R-valued F-measurable functions F (·) : W → R.From the functional integration point of view, F (·) is called a (Wiener) func-tional.We introduce integration and derivation operations on this space of function-als. These operations are first defined on some simple functions F (·) calledcylindrical functions and then extended to a large class using an integrationby parts formula.

11.2.2 Cylindrical functions

DEFINITION 11.1 Cylindrical functions A functional F : W → Ris called a cylindrical function if it has the form

F = f (ω(t1), · · · , ω(tn))

Analysis on Wiener Space with Applications 311

where 0 < t1 < · · · < tn ≤ 1 and f is a smooth function on Rn such that all itsderivatives have at most polynomial growth. The set of cylindrical functionsis denoted C1 and forms an algebra.

REMARK 11.1 Note that we have C1 ⊂ Lp(Ω) for all p ≥ 1 as thepolynomial growth of f ensures the existence of all the moments.We can also define the algebra C2 of functionals of the form

F = f (W (h1), · · · ,W (hn))

where W (hi) =∫ 1

0hi(s)dWs with hi ∈ L2([0, 1]).

In particular, replacing each W (hi) by the Riemann sums

N∑j=1

h j−1N

(ω jN− ω j−1

N

)the resulting function belongs to C1. We have therefore that C1 is dense in C2.

In the following, we introduce the Wiener measure P using the Feynman pathintegral.

11.2.3 Feynman path integral

We endow the space W with the Wiener measure dγ(ω) (i.e., Feynman quan-tification procedure) which is characterized by its generating function [73]∫

Wdγ(ω)e−ı<ω

′,ω> = e−12

R 10

R 10 dω

′(t)dω′(t′) min(t,t′) (11.1)

where ω′ is an element of the topological dual1 W′ of W, i.e., a boundedmeasure on the semi-open interval (0, 1) where

< ω′, ω >=∫ 1

0

dω′(t)ω(t)

Formally, we would like to represent the measure dγ(ω) as

dγ(ω) = D(ω)e−12

R 10 ( dωdt )2dt (11.2)

This equation is formal as the term D cannot be defined as a measure onW and moreover the derivative according to the time dω(t)

dt does not exist.

1The set of all continuous linear functionals, i.e., continuous linear maps from W into R. Atopology on the dual can be defined to be the coarsest topology such that the dual pairingW′ ×W→ R is continuous.


This derivative could be put on a rigorous ground if we introduced the Hidadistribution [132] but we do not go down this route as we prefer to use formalcomputations. We call the (formal) derivative dω(t)

dt a white noise.Although D is not a measure on W, we try to give a meaning to (11.2) andthe following expression, called a functional integration

EP[F ] ≡∫D(ω)F (ω)e−

12

R 10 ( dωdt )2

dt (11.3)

Firstly, the “Riemann” integral is replaced by its discrete sum∫ 1

0

(dω

dt

)2

dt =1

∆t

n∑i=1

(ω(ti)− ω(ti−1))2

where the interval [0, 1] is subdivided into 0 < t1 < · · · < ti < · · · < tn = 1with length ∆t. Then, we view the functional integral (11.3) as the limitn→∞ of a (finite) n-dimensional Gaussian integration:

DEFINITION 11.2 Functional integration

EP[F ] ≡ limn→∞

∫Rn

(n∏i=1

dωi√2π∆t

)F (ω1, · · · , ωn)

(n∏i=1

e−1

2∆t (ωi−ωi−1)2

)with ω0 = 0.

Using this definition, we will to reproduce our previous characterization (11.1)of the Wiener measure dγ. To start with, using an integration by parts, wewrite the characteristic function

I ≡∫

Wdγ(ω)e−ı

R 10 ω′(t)ω(t)dt

as

I =∫

WD(ω)e−

12

R 10 dt

R 10 dt

′ω(t′)K(t,t′)ω(t)e−ıR 10 ω′(t)ω(t)dt

with the kernel operator K(t, t′) defined as

K(t, t′) = −δ(t′ − t) d2

dt2

Using the definition 11.2, the expression above becomes

limn→∞

∫Rn

(n∏i=1

dωi√2π∆t

)e−

(∆t)2

2

Pni,j=1 ωiKijωi−ı

Pni=1(ω′i−ω

′i−1)ωi

= limn→∞

1(2π)

n2e− 1

2(∆t)2Pni,j=1(ω′i−ω

′i−1)iGij(ω

′j−ω

′j−1)


with G the inverse of the n-dimensional matrix K:

KikGkj = δij

In the limit n→∞, we obtain

I = e−12

R 10 dt

R 10 dt

′ω′(t′)G(t′,t)ω′(t)

with G(t, t′), the so-called Green function, solution of∫ 1

0

dsK(t, s)G(s, t′) = − d2

dt2G(t, t′) = δ(t− t′)

The solution is given by G(t, t′) = min(t, t′) and we reproduce (11.1).Via the definition 11.2, it is clear that the functional integration is linear

EP[F +G] = EP[F ] + EP[G]

We obtain as an exercise

EP[ω(t1)ω(t2)] = min(t1, t2)

This can be generalized for any product of ω. It is called the Wick identity

THEOREM 11.1 Wick identity

EP[ω(t1)ω(t2) · · ·ω(t2n)] = min(t1, t2) · · ·min(t2n−1, t2n) + cyclic perm.EP[ω(t1)ω(t2) · · ·ω(t2n−1)] = 0

Below, we note

(F,G) ≡∫

WF (ω)G(ω)dγ(ω)

11.3 Functional-Malliavin derivative

Having defined how to integrate functionals on W, we introduce a derivation,called in the mathematical literature a Malliavin derivative and in the physicsliterature a functional derivation. The derivative is first defined for cylindricalfunctions belonging to C1 (or equivalently C2) . Then using an integration byparts formula, we show how to extend its domain ⊂ L2(Ω).


DEFINITION 11.3 Malliavin derivative 1 For F ∈ C1, we definethe Malliavin derivative DhF along the direction h ∈ H by

DhF (ω) =d

dεF

(ω + ε

∫ ·0

h(s)ds)|ε=0

= limε→0

f(ω(t1) + ε

∫ t10h(s)ds, · · ·

)− f (ω(t1), · · · )

ε

The limit is under L2(Ω).

Formally, the Malliavin derivative corresponds to the bumping of the pathω(·) by the function

∫ ·0h(s)ds. This definition can be simplified and we have

DEFINITION 11.4 Malliavin derivative 2 For F ∈ C1, the Malliavinderivative DhF along the direction h ∈ H is defined by

DhF (ω) =n∑i=1

∂if(ω)∫ ti

0

h(s)ds (11.4)

Below are the main properties of the Malliavin derivative derived from theformula above for all F,G ∈ C1:

1. Linearity:

Dh (F (ω) +G(ω)) = DhF (ω) +DhG(ω)

2. Leibnitz rule:

Dh (F (ω)G(ω)) = DhF (ω)G(ω) + F (ω)DhG(ω)

For a fixed ω, the map h→ DhF (ω) is a linear bounded functional on L2([0, 1])(see 11.4). By the Riesz representation theorem, there exists a unique elementDF (ω) ∈ L2([0, 1]) such that

DhF (ω) =< DF (ω), h >≡∫ 1

0

(DF (ω))(s)h(s)ds

Below, we note DF (ω)(s) ≡ DsF (ω). DtF can be considered as a L2([0, 1])-valued stochastic process.The assumption that the functional f is a smooth function on Rn such thatall its derivatives have at most polynomial growth ensures that for all p ≥ 1,we have DF ∈ Lp(Ω, L2([0, 1])) (we take p = 2):

D : C1 → L2(Ω, L2([0, 1])) ' L2(Ω× [0, 1])


From (11.4), we obtain

DsF (ω) =n∑i=1

∂if(ω)1(s ∈ [0, ti])

The Malliavin derivative can be formally obtained with the following definition

DEFINITION 11.5 Functional derivative

DsF (ω) = limε→0

F (ω(·) + εδ(· − s))− F (ω(·))ε

with F ∈ C1

and

DhF (ω) =∫ 1

0

h(s)DsF (ω)ds

where h ∈ L2([0, 1]). In order to differentiate the use of the functional deriva-tive from the Malliavin derivative, we note below the functional derivative asDs ≡ δ

δω(s) .

Applying this definition, we reproduce the result

δω(t)δω(s)

=δ∫ t

0ω(u)duδω(s)

=∫ t

0

δ(u− s)du

= 1(s ∈ [0, t]) (11.5)

Since C1 is dense in L2(Ω), we could extend the domain of the Malliavinderivative in L2(Ω) by means of taking limits. For this purpose, we define theclass D1,2.

DEFINITION 11.6 D1,2 We define D1,2 to be the closure of the familyC1 with respect to the norm || · ||1,2:

||F ||1,2 = ||F ||L2(Ω) + ||DtF ||L2([0,1]×Ω)

It means that D1,2 consists of all F ∈ L2(Ω) such that there exists Fn ∈ C1with the property that Fn → F in L2(Ω) as n → ∞ and DFnn=1,··· ,∞ isconvergent in L2([0, 1]× Ω).

Then, we could extend the Malliavin derivative for F ∈ L2(Ω) by

DF ≡ limn→∞

DFn

However, there is no guarantee that if Fn and Fn are two sequences in C1 bothconverging to F under L2, then DFn and DFn converge to the same limit.


When this property is satisfied, D is called a closable operator on D1,2 withcore C1. In the following, having introduced the integration by parts formulaon C1, we show that D is closable.

LEMMA 11.1 Integration by parts formulaSuppose F,G ∈ C1 and h ∈ H. Then

(DhF,G) = (F,D∗hG) (11.6)

with

D∗h = −Dh +∫ 1

0

h(s)dω(s)

PROOF By the means of the Girsanov theorem, we have

EP[F (ω(·) + ε

∫ ·0

h(s)ds)G(ω)] = EP[e−12 ε

2 R 10 h(s)2ds+ε

R 10 h(s)dω(s)

F (ω(·))G(ω(·)− ε

∫ ·0

h(s)ds)

]

Note that the Girsanov transform is well defined as∫ 1

0h(s)2ds < ∞ and the

Novikov condition is satisfied. Differentiating both sides with respect to ε atε = 0, we obtain

EP[DhF (ω)G(ω)] = E[∫ 1

0

h(s)dω(s)F (ω)G(ω)]− E[FDhG]

We give another proof using the formal calculus provided by the Feynmanpath integral and the functional derivative. The proof is based on the followingidentity ∫

Dω δ

δω(s)F (ω) = 0 (11.7)

Then, we have(δF

δω(s), G

)=∫Dω

(δ

δω(s)[F (ω)G(ω)e−

12

R 10 ω(t)2dt]

+(−F (ω)

δG(ω)δω(s)

+ ω(s)F (ω)G(ω))e−

12

R 10 ω(t)2dt

)=∫Dω

(−F (ω)

δG(ω)δω(s)

+ ω(s)F (ω)G(ω))e−

12

R 10 ω(t)2dt

where we have used

δ

δω(s)e−

12

R 10 ω(t)2dt = −ω(s)e−

12

R 10 ω(t)2dt


For the smooth derivative, Dh(F ) ≡∫ 1

0h(s) δF (ω)

δω(s) ds, we obtain

(DhF,G) = (F,AhG)− (F,DhG)

with

Ah =∫ 1

0

h(s)ω(s)ds =∫ 1

0

h(s)dω(s)

THEOREM 11.2D is closable.

PROOF Considering the differenceHn = Fn−Fn, we see thatD is closableif and only if Hn → 0 under L2(Ω), then DHn → 0 under L2(Ω× [0, 1]). Bythe lemma 11.1, we get

(DhHn, G) = (Hn, D∗hG)→n→∞ 0

with G ∈ C. Since DhHn converges by definition and C is dense in L2(Ω), weconclude that DhHn →n→∞ 0 in L2(Ω× [0, 1]).

Example 11.1

Ds

∫ T

0

f(u)dω(u) = f(s)1(s ∈ [0, T ])

Dseω(t0) = eω(t0)1(s ∈ [0, t0])

11.4 Skorohod integral and Wick product

11.4.1 Skorohod integral

The Malliavin derivative is a closed and unbounded operator valued in L2(Ω×[0, 1]) and defined on a dense subset D1,2 of L2(Ω):

D : D1,2 ⊂ L2(Ω)→ L2(Ω× [0, 1])

One can then define an adjoint operator

∂ : Dom(∂) ⊂ L2(Ω× [0, 1])→ L2(Ω)


The domain of ∂, denoted Dom(∂), is the set of L2([0, 1])-valued square in-tegrable r.v. u ∈ L2(Ω × [0, 1]) for which there exists an unique u∗ ∈ L2(Ω)such that

EP[< DF, u >] = EP[Fu∗]

As ∂ is densely defined, we set u∗ = ∂(u) and we have

EP[< DF, u >] = EP[F∂(u)] (11.8)

PROPOSITION 11.1For F ∈ C1 and h ∈ L2([0, 1]), the following formula holds

∂(Fh) = F

∫ 1

0

h(s)dω(s)−DhF

PROOF For F,G ∈ C1 and h ∈ L2([0, 1]), we have

EP[∂(Fh)G] = EP[< Fh,DG >] using (11.8)= EP[F < h,DG >] ≡ (F,DhG)

= EP[−GDhF ] + EP[FG∫ 1

0

h(s)dω(s)] using (11.6)

REMARK 11.2 Skorohod reduces to an Ito for adapted processIf u(s, ω) is a Fs-adapted process, then the Skorohod integral reduces to anIto integral:

∂(u) =∫ T

0

u(ω, s)dω(s)

LEMMA 11.2For F ∈ D1,2, u ∈ Dom(∂) such that Fu ∈ L2(Ω× [0, 1]). Then F belongs tothe domain of ∂ and the following equality is true

∂(Fu) = F∂(u)− < DF, u > (11.9)

PROOF

EP[< Fu,DG >] = EP[< u,FDG >] = EP[< u,D(FG)−GDF >]= EP[∂(u)FG]− EP[< DF, u > G]


In the following, we give a second definition of the Skorohod integral using theso-called Wick calculus. The Wick product (also called normal ordering) wasintroduced as a first step to renormalize a QFT, in particular to eliminate theinfinite energy of the vacuum. The Wick calculus will lead us to the Wienerchaos expansion.

11.4.2 Wick product

This product can be defined using the Hermite polynomials Hn(x)n∈N oforder n

H0(x) = 1

Hn(x) =(−)n

n!e−

x22dn

dxne−

x22 , n ≥ 1

These polynomials are the coefficients of the expansion in powers of t of thefunction

F (t, x) = etx−t22 =

∞∑n=0

tnHn(x)

We scale these functions by

Hen(x) = 2−n2 Hn

(x√2

)In particular, we have

He1(q) = q

He2(q) = q2 − 1He3(q) = q3 − 3qHe4(q) = q4 − 6q2 + 3

DEFINITION 11.7 Wick product 1 The Wick product, denoted : · :,is defined by

: W (h)n := Hen(W (h))

with n ∈ N, ||h||L2([0,1]) = 1 and W (h) ≡∫ 1

0h(s)dω(s).

REMARK 11.3 Summing this expression over n, we obtain that : eW (h) :is an exponential martingale

: eW (h) := eW (h)− 12


Example 11.2

: W (h) : = W (h): W (h)2 : = W (h)2 − 1 (11.10): W (h)3 : = W (h)3 − 3W (h)

An equivalent (useful although formal) definition for the Wick product isdirectly given on the white noise ω(t):

DEFINITION 11.8 Wick product 2

: ω(t1) · · · ω(tn) : = ω(t1) · · · ω(tn)− (ω(t3) · · · ω(tn)δ(t1 − t2) + perm.)+ (δ(t1 − t2)δ(t3 − t4)ω(t5) · · · ω(tn) + perm.)− · · ·

(−)k (δ(t1 − t2)δ(t3 − t4) · · · δ(t2k−1 − t2k) + perm.)+ · · ·

For example, we have

: ω(t)ω(s)ω(u) : = ω(t)ω(s)ω(u)− (δ(t− s)ω(u) + perm.)

We prove the equivalence of these two definitions with a simple example(11.10):

: W (h)2 : =∫ 1

0

ds

∫ 1

0

duh(s)h(u) : ω(s)ω(u) :

=∫ 1

0

ds

∫ 1

0

duh(s)h(u)ω(s)ω(u)−∫ 1

0

ds

∫ 1

0

duh(s)h(u)δ(s− u)

=(∫ 1

0

h(s)ω(s)ds)2

−∫ 1

0

h(s)2ds

Here is a list of a few useful properties:

1. Orthogonality :

E[: W (f)n :: W (g)m :] = δnm

∫ 1

0

f(s)g(s)ds


2. Commutativity of the Wick product and Malliavin derivative:

Ds : F (ω) :=: DsF (ω) :

3. Commutativity law: : FG :=: GF :

4. Associativity law: :: FG : H :=: F : GH ::

5. Distributive law: : H(Y + Z) :=: HY : + : HZ :

6. : u(t, ω)ω(t) :=(ω(t)u(t, ω)− δu(t,ω)

δω(t)

)Here are some examples:

Example 11.3

1. : eWt := eWt− t2 . In particular, : eWt : is a martingale.

2. : W (f)W (g) := W (f)W (g)−∫ 1

0f(s)g(s)ds

3. : Wt1Wt0 := Wt1Wt0 −min(t1, t0)

Everything is now in place to give a second definition of the Skorohod integral.

DEFINITION 11.9 Skorohod integral 2

∂(u) =∫ 1

0

: u(s, ω)ω(s) : ds

The integral should be understood in the “Riemann sense.” We prove belowthat this definition satisfies the duality pairing (11.8):

PROOF

EP[< DF, u >] = EP[∫ 1

0

DtFu(t, ω)dt]

=∫Dωe− 1

2

R 10 (ω)2ds

∫ 1

0

δF

δω(t)u(t, ω)dt

=∫Dωe− 1

2

R 10 (ω)2dsF

∫ 1

0

(ω(t)u(t, ω)− δu(t, ω)

δω(t)

)dt using an IPP

=∫Dωe− 1

2

R 10 (ω)2dtF

∫ 1

0

: u(t, ω)ω(t) : dt using the property above (6)

= EP[F∂u]


Here are two examples of computation of a Skorohod integral using the Wickcalculus:

Example 11.4

1.

∂ (ω(t)(ω(T )− ω(t))) =∫ T

0

: ω(t)(ω(T )− ω(t))ω(t) : dt

=16

: ω(T )3 :=16

(ω(T )3 − 3Tω(T ))

2. Let t0 < T .

∂(ω(t0)2) =∫ T

0

: ω(t0)2ω(t) : dt

= : ω(t0)2ω(T ) := :: ω(t0)2 : ω(T ) : +t0ω(T )= : ω(t0)2 : ω(T )− 2t0ω(T ) + t0ω(T )= ω(t0)2ω(T )− 2t0ω(T )

11.5 Fock space and Wiener chaos expansion

DEFINITION 11.10 Fock space A Fock structure is defined by aquadruple

H, a(hi), a†(hi), |0 >

with hi an orthonormal basis of L2([0, 1]). H is a separable Hilbert space,|0 >∈ H a unit vector called the (Fock) vacuum and a(h), a†(h) are operatorsdefined on a domain Dom of vectors dense in H. Furthermore, Dom is a linearset containing |0 > and the vectors a(h) carry vectors in Dom into vectors inDom. If Φ,Ω ∈ Dom, then (Φ, a(h)†,Ω) is a tempered distribution regardedas a functional of h. The Fock space F(H)n is the Hilbert space generated bythe vectors in Dom

a†(h1) · · · a†(hn)|0 >


and we set

F(H) = ⊕∞n=1F(H)n

DEFINITION 11.11 cyclic vector |0 > is called a cyclic vector if

H ' F(H)

Fock space realization on L2(W,P)

In the following, we show how to define a Fock space on the Hilbert spaceH = L2(W,P). We recall that the Malliavin derivative Dh is

Dh =∫ 1

0

h(s)dsδ

δω(s)

and we set

Ah =∫ 1

0

h(s)dω(s)

These operators acting on D1,2 satisfy the Heisenberg Lie algebra

[Dh1 , Dh2 ] = 0[Ah1 , Ah2 ] = 0[Dh1 , Ah2 ] = < h1, h2 > 1

with the commutator [A,B] ≡ AB −BA. Setting

a(h) = Dh

a†(h) = Ah −Dh

the “annihilation-creation” algebra is given by

[a(h1), a(h2)] = 0[a†(h1), a†(h2)] = 0[a(h1), a†(h2)] = < h1, h2 > 1 (11.11)

a(h) and a†(h) are adjoint in the Hilbert space L2(W,P). Indeed from theintegration by parts formula (11.6), we have

(F, a†(h)G) = (a(h)F,G)

The vacuum |0 >∈ F is the constant function equal to 1. In particular,a(h)1 = 0.


(L2(W,P), a(hi), a†(hi), 1) defines a Fock structure. Moreover, one can provethat |0 >= 1 is a cyclic vector: It gives the Ito-Wiener-Segal chaos decompo-sition

THEOREM 11.3 Wiener chaos

L2(W,P) ' F(L2(W,P))

In particular, F(H)n is generated by∏ni=1 a(hi)†|0 > (iterated integrals):

• F(H)0 is generated by 1.

• F(H)1 is generated by∫ 1

0h1(s)dω(s).

• F(H)2 is generated by∫

0≤s≤t≤1h1(s)h2(t)dω(s)dω(t)− < h1, h2 >.

PROOF In this part, only a sketch of the proof is provided. Let J denotethe set of indices I = ni such that ni ∈ N∗ and almost all of them are equalto zero. Denote |I| = n1 + n2 + · · · . For each I ∈ J , define

HI =∏i=1

a†(hi)ni1

where a†(hi)ni means the product of a†(hi) ni times. Following the definitionof a†(hi), each HI belongs to C2 ⊂ L2(W,P). The fact that the Hermitepolynomials form an orthonormal basis of L2([0, 1]) implies that HII∈Jgives a basis of the algebra C2. Using that C2 is dense in L2(P,W), the resultfollows immediately.

11.5.1 Ornstein-Uhlenbeck operator

We introduce the (quantum) number operator N

N =∞∑i=0

a†(hi)a(hi)

As N is a self-adjoint operator on L2(W,P), it can be diagonalized over anorthogonal basis. We next show that HI is an eigenvector of N with eigenvalue|I|. This can be achieved by recurrence. For example,

Na(hi)†|0 > = a†(hi)a(hi)a(hi)†|0 >= a†(hi)(1 + a(hi)†a(hi))|0 > using (11.11)= a†(hi)|0 > using a(hi)|0 >= 0


Note that N can be represented on the Hilbert space L2(W,P) as

N = −∂D

Indeed it is easy to check that the functions HI(ω) are precisely the eigenvec-tors of N .

11.6 Applications

We illustrate the use of Malliavin calculus in finance with three applications:Firstly, we apply the Wiener chaos expansion by computing convexity ad-justment for CMS rate (see section 3.4.4 for definitions). We focus on affineshort-rate models, mainly the HW2 model. This part follows from [57].Secondly, we obtain a probabilistic representation of Greeks and finally wecompute the local volatility associated to SVMs.

11.6.1 Convexity adjustment

The fair value of a CMS in the forward measure PTα is

CMSαβ(t) = PtTαEPTα [sαβ,Tα |Ft]

By definition of the swap rate, the expression above becomes

CMSαβ(t) = PtTαEPTα [PTαTα − PTαTβ∑βi=α+1 τiPTαTi

|Ft] (11.12)

For an n-factor affine model, the forward bond PtTi satisfies in the risk-neutralmeasure

dPtTiPtTi

= rtdt+n∑k=1

σk(t, Ti)dW kt

with σ(t, Ti) function of the time t and the maturity Ti and dW kt dW

pt = ρkpdt.

In the forward measure PTα , we have

dPtTiPtTi

= (rt − σ(t, Tα).σ(t, Ti))dt+n∑k=1

σk(t, Ti)dZkt

where we have set σ(t, Tα).σ(t, Ti) ≡∑nk,p=1 ρkpσk(t, Tα)σp(t, Ti). The solu-

tion is

PtTi = P0TieR t0 rsdse−

R t0 σ(s,Tα).σ(s,Ti)ds : e

R t0

Pnk=1 σk(s,Ti)dZ

ks : (11.13)


where we have used the Wick product to represent the exponential martingaleterm. By plugging the solution (11.13) into (11.12), we obtain

CMSαβ(t) = PtTαEPTα [φα : eΦα :∑β

i=α+1 τiφi : eΦi :− φβ : eΦβ :∑β

i=α+1 τiφi : eΦi :|Ft]

(11.14)

where Φi ≡∫ Tα

0

∑nk=1 σk(s, Ti)dZks and φi ≡ P0Tie

−R Tα0 σ(s,Tα).σ(s,Ti)ds. We

consider the first term; the second term can be treated similarly.The exponential martingale can be decomposed as

: eΦi := 1+ : Φi : + : Φ2i : + · · ·

where · · · indicates higher-order volatility terms. This corresponds to aWiener chaos expansion. The first term in (11.14) can therefore be writtenat this order as

φα1+ : Φα : + : Φ2

α :N

(1− 1

N

β∑i=α+1

τiφi : Φi : − 1N

β∑i=α+1

τiφi : Φ2i :

+1N2

β∑i=α+1

τiτjφiφj : ΦiΦj :

)

where N =∑βi=α+1 τiφi. Using that

EPTα

[: Φni :] = 0 , ∀ n ∈ N

EPTα

[: Φi :: Φj :] = cij ≡∫ Tα

0

σ(s, Ti).σ(s, Tj)ds

we obtain finally the convexity adjustment for the CMS rate

P−1tTα

CMSαβ(t) =(φα − φβ)

N− 1N2

β∑i=α+1

τiφi(φαciα − φβciβ)

+(φα − φβ)

N3

β∑i,j=α+1

τiτjφiφjcij

In the 2-factor affine short-rate model, HW2 model, we have σ(s, Ti) =σ(t)

(e−a(Ti−t)−1

a , e−b(Ti−t)−1

b

). Therefore the correlation matrix is

cij =∫ Tα

0

σ(s)2 ds

(e−a(Ti−t) − 1

a

e−a(Tj−t) − 1a

+e−b(Ti−t) − 1

b

e−b(Tj−t) − 1b

+ρe−a(Ti−t) − 1

a

e−b(Tj−t) − 1b

+ ρe−b(Ti−t) − 1

b

e−a(Tj−t) − 1a

)


11.6.2 Sensitivities

Suppose that our market model depends on n parameters λii=1,··· ,n suchas the volatility, the spot price of each asset, the interest yield curve, thecorrelation matrix between assets, · · · . The fair value C of an option willdepend on these parameters: C = C(λ). For hedging purpose, we need tocompute the sensitivities (the so-called Greeks) of the option with respect tothe model parameters:

∂nC∂λi1 · · ·λin

For example, the Delta, denoted ∆ii=1,··· ,n, is the sensitivity with respectto spot prices Si0i=1,··· ,n:

∆i =∂C∂Si0

and the (Black-Scholes) Vega, denoted Vi, is the sensitivity with respect tothe Black-Scholes volatility σi of asset i:

Vi =∂C∂σi

We restrict ourselves to first-order sensitivities although a full treatmentcan be obtained straightforwardly using the techniques presented below. Weconsider also payoffs φ = φ(St1 , · · · , Stn) which are functions of a single un-derlying St at date t1, · · · , tn. We recall that the fair value C at time t < t1is given by the mean value of the payoff conditional to the filtration Ft:

C(t, λ) = EP[φ(St1 , · · · , Stn)|Ft]

No discount factor has been included as we assume zero interest rate for thesake of simplicity.When the number of underlying assets is large, the option C(t, λ) is estimatedusing a Monte-Carlo simulation (MC). The simplest method to compute sen-sitivities with respect to λi is based on a finite difference scheme:

∂λiC(t, λ) = limε→0

C(t, λi + ε)− C(t, λi − ε)2ε

In practice C(t, λ + ε) and C(t, λ − ε) are computed using the same normalrandom variables. Note that when ε is small, the more accurate the differencescheme is, the more the variance of the MC increases.In the following, we explain how to obtain probabilistic representation ofGreeks as

∂λiC(t, λ) = EP[φω|Ft]

where the universal weight ω is a r.v. The weight is universal as it does notdepend on the payoff φ(·).


Tangent process

We assume that the process St satisfies the one-dimensional SDE

dSt = b(t, St)dt+ σ(t, St)dWt (11.15)

DEFINITION 11.12 Tangent process The tangent process Yt ≡ ∂St∂S0

satisfies the SDE

dYtYt

=∂b(t, St)∂S

dt+∂σ(t, St)∂S

dWt (11.16)

with the initial condition Y0 = 1.

Example 11.5 Black-ScholesFor the Black-Scholes log-normal process, the tangent process is Yt = St

S0.

The next theorem gives the Malliavin derivative of St as a function of thetangent process:

THEOREM 11.4 Malliavin derivative of St

DsSt = σ(s, Ss)YtYs

1(s ∈ [0, t]) (11.17)

PROOF The SDE (11.15) reads as

St = S0 +∫ t

0

b(u, Su)du+∫ t

0

σ(u, Su)dWu

We obtain that (recall that formally Dsω(u) = δ(u− s))

DsSt = σ(s, Ss) +∫ t

s

Dsb(u, Su)du+∫ t

s

Dsσ(u, Su)dWu

Using the chain rule (11.4), we have

DsSt = σ(s, Ss) +∫ t

s

b′(u, Su)DsSudu+∫ t

s

σ′(u, Su)DsSudWu

where the prime means a derivative with respect to S. Then, the processZt = DsSt satisfies the SDE

dZtZt

= b′(t, St)dt+ σ′(t, St)dWt


with the initial condition Zs = σ(s, Ss). Note that this SDE is identical(modulo the initial condition) to (11.16). By scaling Zt = λYt1(s ∈ [0, t]), weobtain (11.17).

Now, everything is in place to compute Greeks: we focus on the Delta andthe (local) Vega.

Delta

Intervening the derivative ∂S0 and the mean value EP[·|Ft] operators, we ob-tain

∆ = ∂S0EP[φ|Ft] = EP[n∑i=1

∂iφ(St1 , · · · , Stn)Yti |Ft] (11.18)

We would like to represent this sensitivity price as

∆ = E[φω|Ft]

As an ansatz, we take ω = ∂(π) with ∂ the Skorohod operator. Using theduality formula (11.8), we obtain

EP[φ∂(π)|Ft] = EP[∫ T

0

Dsφπsds|Ft]

Using the chain rule (11.4) and the formula (11.17), we have

EP[φ∂(π)|Ft] =n∑i=1

EP[∂iφ∫ T

0

DsStiπsds|Ft]

=n∑i=1

EP[∂iφ∫ ti

0

YtiYsσ(s, Ss)πsds|Ft] (11.19)

By identifying the equation (11.18) with the equation (11.19) for any smoothpayoff, we have

n∑i=1

EP[Yti

(1−

∫ ti

0

σ(s, Ss)Ys

πsds

)|St1 , · · · , Stn ] = 0

By setting πs = Ysusσ(s,Ss)

, this reduces to the constraint∫ ti

0

EP[us|St1 , · · · , Stn ]ds = 1 , ∀ i = 1, · · · , n (11.20)

and the Delta is given by

∆ = EP[φ∂(

Ysusσ(s, Ss)

)|Ft]


The simplest solution to (11.20) is

us =n∑k=1

ak1(s ∈ [tk−1, tk)) ,i∑

k=1

ak =1ti

and we obtain

∆ =n∑k=1

akEP[φ∫ tk

tk−1

Ysσ(s, Ss)

dWs|Ft]

where we have used that Ysσ(s,Ss)

is an adapted process and therefore the Sko-rohod integral reduces to an Ito integral (see remark 11.2).

Example 11.6 Black-Scholes

∆ =1S0σ

n∑k=1

akEP[φ(Wtk −Wtk−1

)|Ft]

Note that the weight ω = ∂π which satisfies the condition (11.20) is notunique. Among all the weights such that (11.20) holds, the one which yieldsthe minimum variance is given by the following proposition:

PROPOSITION 11.2The weight with minimal variance denoted by ω0 is the conditional expectationof any weight with respect to the r.v.s (St1 , · · · , Stn)

ω0 = EP[ω|St1 , · · · , Stn ]

PROOF The variance is

σ2 = EP[(φω −∆)2]

We introduce the conditional expectation ω0

σ2 = EP[(φ(ω − ω0) + φω0 −∆)2]= EP[φ2(ω − ω0)2] + EP[(φω0 −∆)2] + EP[φ(ω − ω0)(φω0 −∆)]

We have

EP[φ(ω − ω0)(φω0 −∆)] = EP[EP[φ(ω − ω0)(φω0 −∆)|St1 , · · · , Stn ]]= EP[EP[(ω − ω0)|St1 , · · · , Stn ]φ(φω0 −∆)]= 0


where we have used EP[φ(ω − ω0)|St1 , · · · , Stn ] = 0.

As a further example, we compute the sensibility according to a (local) defor-mation of the Dupire local volatility.

Local Vega

The asset S0t is assumed to follow a local volatility model

dS0t = σ(t, S0

t )dWt

We deform locally the local volatility by

dSεt = σε(t, Sεt )dWt

where σε(t, S) = σ(t, S) + εδ(t, S) and the initial condition Sε0 = S0.Then, the process Zεt ≡ ∂εSεt satisfies the SDE

dZεt = (∂Sσε(t, Sεt )Zεt + δ(t, Sεt ))dWt (11.21)

with the initial condition Zε0 = 0. Similarly, the tangent process Y εt ≡SεtS0

satisfies the SDE

dY εt = ∂Sσε(t, Sεt )Y

εt dWt

with the initial condition Y ε0 = 1. Using Ito’s lemma, the solution of (11.21)is

Zεt =(−∫ t

0

1Y εs

∂Sσε(s, Sεs)δ(s, S

εs)ds+

∫ t

0

1Y εs

δ(s, Sεs)dWs

)Y εt (11.22)

The payoff sensitivity with respect to ε (i.e., local Vega) is

∂εCε|ε=0 =n∑i=1

EP[∂iφZ0ti |Ft]

that we write as

∂εCε|ε=0 =n∑i=1

EP[∂iφYtiZ0ti |Ft] (11.23)

with

Z0t =

Z0t

Y 0t

As in the previous section, we would like to represent the Vega as

E[φ∂(π)|Ft] =n∑i=1

E[∂iφ∫ ti

0

YtiYsσ(s,Xs)πsds|Ft] (11.24)


Identifying equation (11.23) with equation (11.24) for any smooth payoff, wehave

n∑i=1

E[Yti

(∫ ti

0

σ(s, Ss)Ys

πsds− Z0ti

)|St1 , · · · , Stn ] = 0 (11.25)

The simplest solution to (11.25) is

πs =Ys

σ(s, Ss)

n∑j=1

a(s)(Z0tj − Z

0tj−1

)1(s ∈ [tj−1, tj ])

with a(·) such that ∫ tj

tj−1

a(s)ds = 1 , ∀j = 1, · · · , n

Using (11.9), we obtain finally

∂(π) =n∑j=1

((Z0

tj − Z0tj−1

)∫ tj

tj−1

Ysa(s)σ(s, Ss)

dWs

−∫ tj

tj−1

Ysa(s)σ(s, Ss)

(DsZ0tj −DsZ

0tj−1

)ds

)

Note that the Skorohod integral involves an infinite number of processes pa-rameterized by s: DsZ

0t s∈[0,T ]. Other weights for higher-order Greeks can

be found in [58].

11.6.3 Local volatility of stochastic volatility models

The SV models introduced in chapter 6 are defined by the following SDEs

dft = atftΦ(t, ft)(ρdZt +√

1− ρ2dBt) (11.26)dat = b(at)dt+ σ(at)dZt

Here Wt, Bt are two uncorrelated Brownian motions. In order to be able tocalibrate exactly the implied volatility surface, we have decorated the volatilityof the forward by a local volatility function Φ(t, f). By definition, the modelis exactly calibrated to the implied volatility if and only if

σloc(T, f)2 = Φ(T, f)2EP[a2T |fT = f ]

with σloc(T, f) the Dupire local volatility. We use the following result [86, 84]to find a simpler probabilistic representation of the conditional mean value.


PROPOSITION 11.3Let fT , a2

T ∈ D1,2. Assume that Dta2T is non-degenerate P-almost sure for

almost all t ∈ [0, T ] and there exists a process ut ∈ Dom(∂) such that

EP[∫ T

0

(DsfT ).usds|σ(f, a)] = 1 (11.27)

EP[∫ T

0

(Dsa2T ).usds|σ(f, a)] = 0 (11.28)

where σ(f, a) is the σ-algebra generated by the processes ft, at0≤t≤T . Thenthe following formula holds

EP[a2T |fT = f ] =

EP[a2T 1(fT − f)∂(u)]

EP[1(fT − f)∂(u)](11.29)

The proof follows closely [86, 84].

PROOF Denote with δ(·) the Dirac function and 1(x) the Heaviside func-tion. Formally, we have

EP[a2T |fT = f ] ≡ EP[a2

T δ(fT − f)]EP[δ(fT − f)]

Using condition (11.27), the chain rule (11.4) and an integration by parts, weobtain

EP[a2T δ(fT − f)] = EP[a2

T δ(fT − f)E[∫ T

0

(DsfT )usds|σ(f, a)]]

= EP[a2T

∫ T

0

(Ds1(fT − f))usds]

= EP[∫ T

0

Ds

(a2T 1(fT − f)

)us]

− EP[1(fT − f)∫ T

0

(Dsa2T )usds]

= EP[a2T 1(fT − f)∂(u)]− EP[1(fT − f)

∫ T

0

(Dsa2T )usds]

The second term cancels thanks to the condition (11.28) and, finally, we obtain(11.29).

The SDEs satisfied by the Malliavin derivatives of the forward are

dDZs ft =

(ftΦ(t, ft)DZ

s at + at(ftΦ(t, ft))′DZs ft)

(ρdZt +√

1− ρ2dBt)

dDBs ft =

(at(ftΦ(t, ft))′DB

s ft)

(ρdZt +√

1− ρ2dBt)


with the initial conditions

DZs fs = ρasfsΦ(s, fs)

DBs fs =

√1− ρ2asfsΦ(s, fs)

Note that as at is not driven by the Brownian Bt, we have

DBs at = 0

Let the functions uB , uZ be

uBs =1

TDBs fT

1(s ∈ [0, T ])

uZs = 0

They satisfy trivially the conditions (11.27) and (11.28). Note that

DBs fT =


ΞTΞs

1(s ∈ [0, T ])

with the Ito process Ξt given by

dΞt = at(ftΦ(t, ft))′ΞtdNt (11.30)Ξ0 = 1

where we have set dNt ≡ (ρdZt +√

1− ρ2dBt).Using (11.9), we obtain

T√

1− ρ2ΞT∂(uB) =∫ T

0

ΞsasfsΦ(s, fs)

dBs + Ξ−1T

∫ T

0

ΞsasfsΦ(s, fs)

DBs ΞT ds

(11.31)

where we have

dDBs Ξt = at

((ftΦ(t, ft))′′


Ξ2t

Ξs+ (ftΦ(t, ft))′DB

s Ξt

)dNt

(11.32)

DBs Ξs =

√1− ρ2as(fsΦ(s, fs))′Ξs (11.33)

Effective vector field

We prove below that the process DBs Ξt is an effective vector field [31]: it

can be written as a sum of products of a smooth function by an adapted Itoprocess

DBs Ξt = AsΞt +Bsζt (11.34)


PROOF By plugging equation (11.34) into (11.32), we obtain

Bsdζt = at

((ftΦ(t, ft))′′


Ξ2t

Ξs+Bs(ftΦ(t, ft))′ζt

)dNt

Setting Bs =√

1− ρ2asfsΦ(s,fs)

Ξs, we get

dζt = at((ftΦ(t, ft))′′Ξ2

t + (ftΦ(t, ft))′ζt)dNt (11.35)

where we set ζ0 = 1. The initial condition (11.33) is satisfied if

AsΞs +Bsζs =√

1− ρ2as(fsΦ(s, fs))′Ξs

This gives DBs Ξt as an effective vector field

DBs Ξt =

√1− ρ2as

(((fsΦ(s, fs))′ −

fsΦ(s, fs)ζsΞ2s

)Ξt +

fsΦ(s, fs)Ξs

ζt

)(11.36)

As DBs Ξt is an effective vector field, equation (11.31) simplifies and reduces

to

T√

1− ρ2ΞT∂(uB) =∫ T

0

ΞsasfsΦ(s, fs)

dBs

+√

1− ρ2

(∫ T

0

(Ξs ln(fsΦ(s, fs))′ −

ζsΞs

)ds+ TζTΞ−1

T

)Setting

dΘt =Ξt

atftΦ(t, ft)dBt +

√1− ρ2 (∂f ln(ftΦ(t, ft))Ξt) dt , Θ0 = 0

the local volatility can be written as

σloc(T, f)2 = Φ(T, f)2

EP[1(fT − f)VT(

Ξ−1T ΘT +

√1− ρ2Ξ−1

T

∫ T0sd(ζsΞ−1

s ))

]

EP[1(fT − f)(

Ξ−1T ΘT +

√1− ρ2Ξ−1

T

∫ T0sd(ζsΞ−1

s ))

]

The processes Y (1)t ≡ ftΦ(t, ft)Ξ−1

t Θt and Y(2)t ≡ Ξ−1

t ζt satisfy the followingSDEs

dY(1)t =

dBtat

+ ∂t ln(Φ)Y (1)t dt+

12

(fΦ)(fΦ)′′a2tY

(1)t dt+

√1− ρ2(fΦ)′dt

(11.37)

dY(2)t = (ftΦ)′′Ξt

(atdNt − a2

t (ftΦ)′dt)

(11.38)


with the initial conditions Y (1)0 = 0 and Y (2)

0 = 1. The local volatility becomes

σloc(T, f)2 = Φ(T, f)2

EP[(fTΦ(T, fT ))−11(fT − f)a2T

(Y

(1)T +

√1− ρ2Ξ−1

T fTΦ(T, fT )∫ T

0sdY

(2)s

)]

EP[(fTΦ(T, fT ))−11(fT − f)(Y

(1)T +

√1− ρ2Ξ−1

T fTΦ(T, fT )∫ T

0sdY

(2)s

)]

The term∫ T

0sdY

(2)s should be written as TY (2)

T −∫ T

0Y

(2)s ds. Replacing δ(fT−

f) in the conditional expectation by δ(ln fT − ln f), the expression above canbe replaced by a simpler formula and we get

PROPOSITION 11.4 Local VolatilityThe local volatility associated to the SVMS as defined by (11.26) is

σloc(T, f)2 = Φ(T, f)2EP[1

(ln fT

f

)a2TΩ]

EP[1(

ln fTf

)Ω]

with

Ω = Y(1)T − T

√1− ρ2(fΦ(T, fT ))′ +

√1− ρ2Ξ−1

T (fTΦ)∫ T

0

sdY (2)s

which depends on the Ito processes Ξt (11.30), Y (1)t (11.37) and Y (2)

t (11.38).

REMARK 11.4 Log-normal SVM In the particular case Φ(t, ft) = 1,the equation above for the local volatility reduces to [84]

σloc(T, f)2 =EP[1(ln fT − ln f)a2

T

∫ T0

dBsas

]

EP[1(ln fT − ln f)∫ T

0dBsas

]

We can go one step further and reduce the complexity of this equation usinga mixing solution (see problem 6.1). We have

EP[1(ln fT − ln f)a2T

∫ T

0

dBsas

] = EP[a2T

EP[1

(√1− ρ2

∫ T

0

asdBt − lnf

f0− 1

2

∫ T

0

a2sds+ ρ

∫ T

0

asdZs

)∫ T

0

dBsas|FZ ]]

The r.v. X ≡√

1− ρ2∫ T

0asdBs and Y ≡

∫ T0

dBsas

are normally distributed(conditional to FZ) with variance matrix(

(1− ρ2)∫ T

0a2sds

√1− ρ2T√

1− ρ2T∫ T

0dsa2s

)


We obtain that

EP[1(X −K)Y |FZ ] = N e− K2

2(1−ρ2)RT0 a2

sds√∫ T0a2sds

where N is a constant independent of a2ss. We deduce that the local volatil-

ity is given by

PROPOSITION 11.5 Mixing solution local volatility

σloc(T, f)2 =EP[a2

Te− K2

2(1−ρ2)RT0 a2

sds√R T0 a2

sds]

EP[ e− K2

2(1−ρ2)RT0 a2

sds√R T0 a2

sds]

with K = ln ff0

+ 12

∫ T0a2sds − ρ

∫ T0asdZs. Note that this expression only

requires the simulation of the stochastic volatility at.

11.7 Problems

Exercises 11.1 Brownian bridgeLet T ∈ [0, 1].

1. Using the Malliavin calculus, prove that the conditional mean-value

EP[WT |W1 = x]

can be written as

EP[WT |W1 = x] =EP[WT (W1 −WT )1(W1 − x)]

EP[(W1 −Wt)1(W1 − x)]

2. By noting that WT and W1−WT are independent r.v., deduce from theprevious question that we have

EP[WT |W1 = x] =EP[WT e

− (x−WT )2

2(1−T ) ]

EP[e−(x−WT )2

2(1−T ) ]


3. By doing the integration over WT , prove that

EP[WT |W1 = x] = Tx

Chapter 12

Portfolio Optimization andBellman-Hamilton-Jacobi Equation

Abstract Pricing and Hedging derivatives products is essentially a problemof portfolio optimization. Once a measure of risk has been chosen, the pricecan be defined as the mean value of the profit and loss (P&L) and the besthedging strategy is the optimal control which minimizes the risk.In the Black-Scholes model, the only source of risk is the spot process andthe optimal control is the delta-strategy which cancels the risk. However,under the introduction of stochastic volatility, the market model becomesincomplete. The resulting risk is finite and the delta-strategy is not optimal.A portfolio optimization problem appears also naturally if we assume thatthe market is illiquid and the trading strategy affects the price movements.In the following, we will focus on these optimal control problems when themarket is incomplete and the market is illiquid. Our study involves the useof perturbation methods for non-linear PDEs.

12.1 Introduction

Since the famous papers of Black-Scholes on option pricing [65], some progresshas been made in order to extend these results to more realistic arbitrage-freemarket models. As a reminder, the Black-Scholes theory consists in followinga (hedging) strategy to decrease the risk of loss given a fixed amount of return.This theory is based on three important hypotheses which are not satisfiedunder real market conditions:

• The traders can revise their decisions continuously in time. This firsthypothesis is not realistic for obvious reasons. A major improvement wasrecently introduced in [6] in their time-discrete model. They introducean elementary time τ after which a trader is able to revise his decisionsagain. The optimal strategy is fixed by the minimization of the riskdefined by the variance of the portfolio. The resulting risk is no longerzero and in the continuous-time limit where τ goes to zero, one recoversthe classical result of Black-Scholes: the risk vanishes.

339


• The spot dynamics is a log-normal process (with a constant volatility).As a consequence, the market is complete and the risk cancels. Thissecond hypothesis doesn’t truthfully reflect the market as indicated bythe existence of an implied volatility. In chapters 5 and 6, we have seenhow local and stochastic volatility models (SVMs) can account for animplied volatility. The SVMs are incomplete as it is not possible totrade the volatility. The resulting risk is non-zero.

• Markets are assumed to be completely elastic. This third hypothesismeans that small traders don’t modify the prices of the market by sellingor buying large amounts of assets. The limitation of this last assumptionlies with the fact it is only justified when the market is liquid.

While so far we have mainly focused on relaxing the second hypothesis de-scribed above by introducing local and stochastic volatility models, in thischapter we shall explain how to obtain optimal hedging strategy when themarket is incomplete and is illiquid. Our main tool, we use, is the stochasticHamilton-Bellman-Jacobi that we review.In the first part, we analyze the hedging in an incomplete market. The aimof the second part is to analyze the feedback effect of hedging in portfoliooptimization. In this section, we model the market as composed of smalltraders and a large one whose demand is given by a hedging strategy. Wederive from this toy-model the dynamics of the asset price with the influenceof the large trader. Taking into account the influence on the volatility andthe return, we compute the hedging strategy of the large trader in order tooptimize dynamically a portfolio composed of a risky asset and a bond. Thisleads to a well-defined stochastic optimization problem.

12.2 Hedging in an incomplete market

Let us assume that a trader holds at time t a certain number of shares ∆t ofwhich the price at time t is St. The price St satisfies the SVM in the historicalmeasure Phist

dSt = µtStdt+ atC(St)dWt

dat = b(at)dt+ σ(at)dZt

with Wt and Zt two correlated Brownian processes (dWtdZt = ρdt) and withthe initial conditions St=0 = S0, at=0 = α. The trader holds also a number btof bonds the price of which satisfies the following equation

dBt = rtBtdt

Portfolio Optimization and Bellman-Hamilton-Jacobi Equation 341

with rt the interest rate of the bond. For the sake of simplicity, rt is assumedto be a deterministic process. The value of his portfolio at time t is

πt = ∆tSt + btBt

Assuming a self-financing portfolio, the variation of its value between t andt+ dt is

dπt = ∆tdSt + btdBt

and its discounted variation

dD0tπt = ∆tD0t (dSt − rtStdt)

At t = 0, the trader sells a (European) option with payoff f(ST ) at maturityT at a price C. The discounted change in his total portfolio from t = 0 tot = T where the holder can exercise his option is

ΠT = −D0T f(ST ) + C +∫ T

0

D0u∆u(dSu − ruSudu) (12.1)

Using the martingale representation theorem, we have

D0T f(ST ) = EPhist[D0T f(ST )] +

∫ T

0

D0u[aC(S)∂SfdWu + σ(a)∂afdZu]

By plugging this expression into (12.1), we obtain

ΠT = C − EPhist[D0T f(ST )] +

∫ T

0

D0u[−σ(a)∂afdZu

+ aC(Su)(∆u − ∂Sf)dWu] +∫ T

0

D0u∆uSu(µu − ru)du

We impose that the option price should be fixed such that the mean value ofthe portfolio EPhist

[ΠT ] should cancel. We obtain

C = EPhist[D0T f(ST ) +

∫ T

0

D0u∆uSu(µu − ru)du]

and therefore the portfolio Πt follows the SDE

dΠt = −D0tσ(a)∂afdZt +D0taC(S)(∆t − ∂Sf)dWt

Note that the mean-value is under the historical measure. As seen in chapter2, if we impose the non-arbitrage condition, the mean-value will be under arisk-neutral measure.


Once the option price has been fixed, we must focus on the optimal hedgingstrategy. The hedging function ∆t is obtained by requiring that the risk,measuring by the variance of the portfolio, is minimal

∆ such as min∆

[EPhist

t [Π2T ]− EPhist

t [ΠT ]2] = min∆

[EPhist

t [Π2T ]]

where we have used that by construction EPhist

t [ΠT ] = 0. We introduce thefunction

J(t, St,Πt, a) = min∆

EPhist

t [Π2T ]

A small digression: The Hamilton-Jacobi-Bellman equation

The Hamilton-Jacobi-Bellman theorem states that the function J satisfies theparabolic PDE

∂tJ + min∆

(DJ +

12D2

0t ∂2ΠJ (12.2)(

((∆− ∂Sf) aC(S)− ρσ(a)∂af)2 + (1− ρ2)σ(a)2(∂af)2)

+ D0t∂ΠaJ σ(a) ((∆− ∂Sf) aC(S)ρ− σ(a)∂af)+ D0t∂ΠSJ aC(S) ((∆− ∂Sf) aC(S)− ρσ(a)∂af)) = 0 (12.3)

with the boundary condition J(T, S,Π, S) = Π2 and D the Ito generator

D =12(a2C(S)2∂SS + σ(a)2∂aa

)+ ρaC(S)σ(a)∂aS

Taking the functional derivative according to ∆ in the equation above, weobtain the optimal control

∆ = ∂Sf + ρσ(a)aC(S)

∂af −∂ΠaJ

σ(a)aC(S)ρ+ ∂ΠSJD0t∂2

ΠJWe try as an ansatz for the solution of the HJB equation

J(t, S,Π, a) = Π2 +A(t, S, a)

with A(T, S, a) = 0. By plugging this ansatz in (12.3), we obtain easily that

∂tA+DA+12D2

0t(1− ρ)2σ(a)2(∂af)2 = 0

and therefore

∆ = ∂Sf + ρσ(a)aC(S)

∂af

A similar expression was obtained in [61]. When the volatility of the volatilityσ(a) cancels, we reproduce the Black-Scholes Delta-hedging strategy and therisk cancels.In the next section, we focus on the problem of optimal hedging when themarket is illiquid.


12.3 The feedback effect of hedging on price

By definition, a market is liquid when the elasticity parameter is small. Theelasticity parameter ε is given by the ratio of relative change in price St tochange in the net demand D:

dStSt

= εdDt (12.4)

We observe empirically that when the demand increases (resp. decreases),the price rises (decreases). The parameter ε is therefore positive and we willassume that it is also constant. Another interesting (because more realistic)assumption would be to define ε as a stochastic variable.In a crash situation, the small traders who tend to apply the same hedgingstrategy can be considered as a large trader and the hedging feedback effectsbecome very important. This can speed up the crash. In the following, weuse the simple relation (12.4) to analyze the influence of dynamical tradingstrategies on the prices in financial markets. The dynamical trading φ(t, St)which represents the number of shares of a given stock that a large trader holdswill be determined in order to optimize his portfolio. An analytic solution willnot be possible and we will resolve the non-linear equations by expanding thesolution as a formal series in the parameter ε. The first order correction willbe obtained.Let us call D(t,W, St) the demand of all the traders in the market whichdepends on time t, a Brownian process Wt and the price St. St is assumed tosatisfy the local volatility model in Phist

dStSt

= µ(t, St)dt+ σ(t, St)dWt (12.5)

with µ(t, St) the (historical) return and σ(t, St) the volatility. The processWt models both the information the traders have on the demand and thefluctuation of the price St. Applying Ito’s lemma on the function D(t, St,W ),we obtain

dDt = (∂tD +12σ2S2∂2

SD +12∂2WD + σS∂SWD)dt+ ∂SDdSt + ∂WDdWt

(12.6)

As we have also

dDt =1ε

dStSt

=1ε

(µ(t, St)dt+ σ(t, St)dWt) (12.7)

the identification of the coefficients for dt and dWt in (12.6) and (12.7), one


obtains µ and σ as a function of the derivatives of D:

σ(t, St) =ε∂WD

(1− εS∂SD)(12.8)

µ(t, St) =ε[∂tD + 1

2σ2S2∂2

SD + σS∂SWD + 12∂

2WD]

(1− εS∂SD)(12.9)

We will now assume that the market is composed of a group of small traderswho don’t modify the prices of the market by selling or buying large amountsof assets on the one hand and a large trader one on the other hand. One canconsider a large trader as an aggregate of small traders following the samestrategy given by his hedging position φ. We will choose the demand Dsmall

of small traders in order to reproduce the Black-Scholes log-normal processfor St (12.5) with a constant return µ = µ0 and a constant volatility σ = σ0.The solution is given by

Dsmall =1ε

(µ0t+ σ0Wt) (12.10)

Now, we include the effect of a large trader whose demand Dlarge is generatedby his trading strategy

Dlarge = φ(t, St)

It is implicitly assumed that φ only depends on S and t and not explicitly onWt. The hedging position φ is then added to the demand of the small tradersDsmall and the total net demand is given by

Dt = Dsmall + φ(t, St) (12.11)

By inserting (12.11) in (12.8)-(12.9), one finds the volatility and return as afunction of φ:

σ(t, S) =σ0

(1− εS∂Sφ)(12.12)

µ(t, S) =µ0 + ε(∂tφ+ 1

2σ2S2∂2

Sφ)(1− εS∂Sφ)

(12.13)

This relation describes the feedback effect of dynamical hedging on volatilityand return. The hypothesis that the hedging function only depends on Sand t allows to obtain a volatility and a return independent of historic effectsand therefore a Markovian process for St. As we will see in the next section,for derivatives and portfolio hedging strategies, the feedback effect on thevolatility will not be the same.


12.4 Non-linear Black-Scholes PDE

The Black-Scholes analysis can be trivially modified to incorporate the hedg-ing feedback effect. Let Π be the self-financing portfolio and C(t, St) the valueof the option at time t:

Πt = −C(t, St) + φ(t, St)St (12.14)

As a result of Ito’s lemma, dΠ is given by

dΠt = −dC + φdSt (12.15)

= −(∂tC +12σ2S2∂2

SC)dt+ (φ− ∂SC)dSt (12.16)

A free-arbitrage condition gives dΠt = r(t)Πt with r(t) the interest rate. Therisk is therefore zero if φ = ∂SC which is the usual hedging position. The non-linear Black-Scholes PDE (modulo adequate boundary conditions) is then

∂tC +12

σ20

(1− εS∂2SC)2

S2∂2SC + r(−C + S∂SC) = 0 (12.17)

This non-linear PDE can be solved by expanding C (and so φ) as an asymptoticseries in ε.Note that if the option’s gamma ∂2

SC is positive, then ∂Sφ(t, S) > 0 meaningthat the trader buys additional shares when the price rises. The volatility σis then greater than σ0. A destabilizing effect is then obtained.On the other hand, as we will see in the next section, in dynamical portfoliooptimization, we have the expression ∂Sφ(t, S) < 0 according to which atrader should sell stocks when his price increases and buy more stocks whenhis price decreases. As a result σ < σ0 and the price increases. In thissituation, a trader could buy large amounts of shares at a price S and theprice would then move to a higher price S′. By selling his shares, it wouldmake a free-risk profit S′ − S per share. The key feature that allows thismanipulation is that the price reacts with a delay that allows the trader tobuy at low price and sell at a higher price before the price goes down.

12.5 Optimized portfolio of a large trader

Let us assume that a large trader holds at time t a certain number of sharesφ(t, St) of which the price at time t is St. The price St satisfies the SDE (12.5)which depends implicitly on the hedging position φ(t, St) (see the volatility


(12.12) and the return (12.13)). The trader holds also a number bt of bondsthe price of which satisfies the following equation dBt = r(t)Btdt with r(t)the deterministic interest rate of the bond. The change in his portfolio Πt =φ(t, St)St + btBt during an infinitesimal time dt is then

dΠt = φ(t, St)dSt + r(t)btBtdt= r(t)Πtdt+ φ(t, St)St ((µ(t, St)− r(t))dt+ σ(t, St)dWt) (12.18)

One should note that φ can be negative (which is equivalent to a short posi-tion). At a maturity date T , the value of the portfolio ΠT is

ΠT = Π0 +∫ T

0

dΠ

The basic strategy of the trader is to find the optimal strategy φ∗(t, S) so thatthe risk R is minimized for a given fixed value of the profit E[ΠT ] = G. Bydefinition, the risk is given by the variance of the portfolio ΠT :

R ≡ EPhist[Π2T ]− EPhist

[ΠT ]2 = EPhist[Π2T − G2]

It is clear that the result of this section depends on the above mentioneddefinition for the risk, its main advantage being that it gives simple compu-tations. The mean-variance portfolio selection problem is then formulated asthe following optimization problem parameterized by G:

minφ

EPhist[Π2T − G2]

subject to

EPhist[ΠT ] = G , (St,Πt) satisfying (12.5)− (12.18)

This problem is equivalent to the following one in which we have introduceda Lagrange multiplier ζ:

minφ

EPhist[Π2T − G2 + ζ(ΠT − G)]

subject to

(St,Πt) satisfying (12.5)− (12.18)

For the sake of simplicity, we assume in the following that the interest rate isnegligible (i.e., r(t) = 0) although this hypothesis can be easily relaxed.To derive the optimality equations, the problem above can be restated in adynamic programming form so that the Bellman principle of optimality canbe applied [32]. To do this, let us define

J (t, S,Π) = min EPhist

t [Π2T − G2 + ζ(ΠT − G)]


where EPhist

t is the conditional expectation operator at time t with St = S andΠt = Π. Then, J satisfies the HJB equation:

minφ,ζ∂tJ + µS (∂SJ + φ∂ΠJ ) +12σ2S2

(∂2SJ + φ2∂2

ΠJ + 2φ∂SΠJ)

= 0(12.19)

subject to

J (T, S,Π) = Π2 − G2 + ζ(Π− G) (12.20)

= (Π +ζ

2)2 − G2 − ζG − ζ2

4(12.21)

The equation (12.19) is quite complicated due to the explicit dependence ofthe volatility and the return in the control φ. A simple method to solvethe H-J-B equation (12.19) perturbatively in the liquidity parameter ε is touse a mean-field approximation well known in statistical physics: one com-putes the optimal control φ∗(t, S) up to a given order εi, say φ∗i (t, S). Thisgives a mean volatility σi(t, S) = σ(t, S, φ∗i (t, S)) and mean return µi(t, S) =µ(t, S, φ∗i (t, S)).The parameters µ, σ can be written as a formal series in ε and we will see inthe following section that this is also the case for the hedging function

φ =∑i=0

φi(t, S)εi

We give the first order expansion of σ (12.12) and µ (12.13):

σ = σ0(1 + εS∂Sφ) + o(ε2) (12.22)

µ = µ0 + ε(∂tφ+12σ2

0S2∂2Sφ+ µ0S∂Sφ) + o(ε2) (12.23)

The optimal control, at the order εi+1, is then easily found by taking thefunctional derivative of the expression (12.19) according to φ:

φ∗i+1 = − (µi∂ΠJ + σ2i S∂SΠJ )

σ2i S∂

2ΠJ

(12.24)

By inserting this expression in the HJB equation (12.19), one obtains:

∂tJ + µiS∂SJ +12σ2i S

2∂2SJ −

(µi∂ΠJ + σ2i S∂SΠJ )2

2σ2i ∂

2ΠJ

= 0 (12.25)

The resolution of the equation above gives φ(t, S)i+1 and the procedure canbe iterated. We apply this method up to order one. The form of (12.25) andthe boundary conditions (12.21) suggest that J takes the following form:

J (t, S,Π) = A(t, S)(Π +ζ

2)2 − G2 − ζG − ζ2

4(12.26)


with ζ such that ∂ζJ = 0 and with the boundary condition A(T, S) = 1.By inserting this expression in (12.25) and (12.24), one then obtains the equa-tions:

φ∗i+1(t, u) = −S−1(µiσ2i

+ ∂uX)(Π +ζ

2) (12.27)

∂tX − (µi +12σ2i )∂uX +

12σ2i (∂2

uX − (∂uX)2) =µ2i

σ2i

(12.28)

with u = ln(S) and A(t, S) = eX(t,S). The Lagrange condition ∂ζJ = 0 gives

Π +ζ

2=G −ΠeX − 1

(12.29)

Let us define the parameter λ ≡ µ20σ2

0. In the zero-order approximation, the

volatility and return are constant and in this case, A will be a function of tonly given by

A(t, S) = eλ(t−T )

By inserting this expression in (12.27) and using (12.29), one finds the optimalcontrol (at zero-order):

φ∗(t, S,Π) =λ

µ0S−1 Π− G

eλ(t−T ) − 1

This expression allows to derive a SDE for the portfolio Π subject to thecondition Π(0) = Π0:

dΠ(Π− G)

=1

(eλ(t−T ) − 1)(λdt+

√λdW )

The mean of the portfolio E[Π] satisfies the ODE

dE[Π]dt

= λE[Π]− Geλ(t−T ) − 1

The solution is

E[Π]− G = (Π0 − G)(1− e−λ(t−T ))

(1− eλT )

One can then verify that the condition E[ΠT ] = G is well satisfied. Finally,the optimal control φ∗0 at order zero satisfies the following SDE:

d[φ∗0S]φ∗0S

= −λt+

√λ

(eλ(t−T ) − 1)dWt


and the solution is

φ∗0 =λ

µ0S−1 (G −Π0)

(1− e−λT )32

(1− e−λ(t−T ))12

e−λt+ 1

2(eλ(t−T )−1)− 1

2(e−λT−1)+R t0

√λ

(eλ(t′−T )−1)dW ′t

As explained in the first section, the hedging strategy depends explicitly onthe history of the Brownian motion. To obtain a return and a volatility thatonly depend on S and t, we will take φ0

∗(t, S) = φ∗(t, S,EP[Π]) as the demandof the large trader:

φ∗0(t, S) =λ

µ0S−1(G −Π0)

e−λt

(1− e−λT )(12.30)

We should note that the only models considered by Merton are models ofdynamical portfolio optimization with no age effects [32].Let us derive now the one-order correction to this hedging function. First, wegive from (12.22)-(12.23) µ2

σ2 up to the first order in ε (using (12.30)):

µ2

σ2=

1σ2

0

(µ20 + 2ε(∂tφ∗0 +

12σ2

0S2∂2Sφ∗0)) (12.31)

=1σ2

0

(µ20 + 2εφ∗0(−λ+ σ2

0)) (12.32)

Then the equation (12.28) at order one is

∂tX1 − (µ0 +12σ2

0)∂uX1 −12σ2

0∂2uX1 =

2φ∗0σ2

0

(−λ+ σ20) (12.33)

with boundary condition X1(0, u) = 0. The solution is then given by

X1(t, S) = 2S−1 (G −Π0)(eλT − 1)

λ

µ0

(σ20 − λ)

(µ0 − λ)σ20

(eλ(T−t) − eµ0(T−t)) (12.34)

and after a sightly long computation we find the correction to φ∗0 which de-pends on S, t and Π:

φ∗(t, S,Π) =(Π− G)S−1λ

µ0(eλ(t−T ) − 1)

(1 + εφ∗0

(−λ+ σ20 + µ0)

µ0

−εµ0

λX1 − ε

X1

(1− e−λ(t−T ))

)To obtain the next corrections for the implied volatility and return, one shouldtake φ∗(t, S,E[Π]) as the demand of the large trader. By plugging this ex-pression into the volatility (12.22), we found that the large trader strategy


induces a local volatility model (and therefore an implied volatility) which isgiven at the first-order by:

σ(t, S) = σ0(1− εαe−λtS−1) + o(ε2) (12.35)

with α a constant depending on the characteristic of the portfolio (Π0, G, T ,µ0, σ0).

Conclusion

In this chapter, we have explained how to incorporate easily the effect ofthe hedging strategy in the market prices. The derivative hedging gives anegative effect and the portfolio optimization hedging a positive one. Themain difficulty with our optimization scheme was that the hedging functiondepends on age effects, which was simply solved by taking the mean over theportfolio value for S fixed. As explained, the portfolio optimization and theoption pricing lead to relatively simple non-linear PDEs (12.25)-(12.17) thatcan be numerically solved.

Appendix A

Saddle-Point Method

Classical references for the saddle-point methods are [5] and [15].Let Ω be a bounded domain on Rn, S : Ω → R, f : Ω → R and λ > 0be a large positive parameter. The Laplace method consists in studying theasymptotics as λ→∞ of the multi-dimensional Laplace integrals

F (λ) =∫

Ω

f(x)e−λS(x)dx

Let S and f be smooth functions and we assume that the function S has aminimum only at one interior non-degenerate critical point x0 ∈ Ω:

∇xS(x0) = 0 , ∇2xS(x0) > 0 , x0 ∈ Ω

x0 is called the saddle-point. Then in the neighborhood of x0 the function Shas the following Taylor expansion

S(x) = S(x0) +12

(x− x0)†∇2xS(x0)(x− x0) + o((x− x0)3)

As λ→∞, the main contribution of the integral comes from the neighborhoodof x0. Replacing the function f by its value at x0, we obtain a Gaussianintegral where the integration over x can be performed

F (λ) ≈ f(x0)e−λS(x0)

∫Ω

e−λ2 (x−x0)†∇2

xS(x0)(x−x0)dx

≈ f(x0)e−λS(x0)

∫Rne−

λ2 (x−x0)†∇2

xS(x0)(x−x0)dx

One gets the leading asymptotics of the integral as λ→∞

F (λ) ∼ f(x0)e−λS(x0)

(2πλ

)n2

[det(∇2xS(x0))]−

12

More generally, doing a Taylor expansion at the n-th order for S (resp. n− 2order for f) around x = x0, we obtain

F (λ) ∼ e−λS(x0)

(2πλ

)n2

[det(∇2xS(x0))]−

12

∞∑k=0

akλ−k

351


with the coefficients ak expressed in terms of the derivatives of the functionsf and S at the point x0. For example, at the first-order (in one dimension),we find

F (λ) ∼

√2π

λS′′(c)e−λS(x0) (f(x0)

− 1λ

(− f ′′(x0)

2S′′(x0)+f(x0)S(4)(x0)

8S′′(x0)2+f ′(x0)S(3)(x0)

2S′′(x0)2− 5 (S′′′(x0))2

f(x0)24S′′(x0)3

))

Appendix B

Monte-Carlo Methods and HopfAlgebra

B.1 Introduction

Since the introduction of the Black-Scholes paradigm, several alternative mod-els which allow to better capture the risk of exotic options have emerged : localvolatility models, stochastic volatility models, jump-diffusion models, mixedstochastic volatility-jump diffusion models, etc. With the growing complica-tion of Exotic and Hybrid options that can involve many underlyings (equityassets, foreign currencies, interest rates), the Black-Scholes PDE, which suf-fers from the curse of dimensionality (the dimension should be strictly lessthan four in practice), cannot be solved by finite difference methods. Wemust rely on Monte-Carlo methods.This appendix is organized as follows:In the first section, we review basic features in Monte-Carlo simulation froma modern (algebraic) point of view: generation of random numbers and dis-cretization of SDEs. A precise mathematical formulation involves the use ofthe Taylor-Stratonovich expansion (TSE) that we define carefully. Details canbe found in the classical references [20], [28].In the second section, we show that the TSE can be framed in the setting ofHopf algebras.1 In particular, TSEs define group-like elements and solutionsof SDEs can be written as exponentials of primitive elements (i.e., elementsof the universal Lie algebra associated to the Hopf algebra). This section isquite technical and can be skipped by the reader.This Hopf algebra structure allows to prove easily the Yamato theorem thatwe explained in the last section. As an application, we classify local volatilitymodels that can be written as a functional of a Brownian motion and thereforecan be simulated exactly. The use of Yamato’s theorem allows us to reproduceand extend the results found by P. Carr and D. Madan in [69].

1The author would like to thank his colleague Mr. C. Denuelle for fruitful collaborationson this subject and for help in the writing of this appendix.

353


B.1.1 Monte Carlo and Quasi Monte Carlo

Let Xt be an n-dimensional Ito process. According to (2.31), the pricing of aderivative product requires the evaluation of

E[f(XT )] (B.1)

Through a discretization scheme discussed in the next paragraph, we assumethat XT can be approached by a function g of a multidimensional Gaussianvariable G ∼ N(0,Σ). To figure out an approximation of (B.1), one simulatesM independent random variables (Gm ∼ N(0,Σ))1≤m≤M and computes theassociated values (Xm

T )1≤m≤M . Under reasonable hypothesis on the payofff and the function g, the strong version of the law of large numbers [26]entails that the empirical average 1

M

∑Mm=1 f(Xm

T ) provides a good proxy forE[f(XT )]. Better yet the central limit theorem [26] states that the error

εM =1M

M∑m=1

f(XmT )− E[f(XT )]

is asymptotically normally distributed, with mean 0 and covariancetΣΣM . Thus

there exists a constant C such that the L2 Monte Carlo error is

||εM ||L2 ≤ C√M

Even though the O( 1√M

) bound is steady, variance reduction techniques suchas control variates can considerably reduce the constant C. We refer to [20]for further discussions on these methods and the way they speed up the con-vergence of MC estimates.As liquidity on the market expands, option pricing requires increasing pre-cision, and the O( 1√

M) MC bound is simply not good enough. To gain a

factor 10 in accuracy the number of simulations must be increased by a fac-tor 100. To elude this pitfall, Quasi Monte-Carlo (QMC) methods have beendeveloped [20]. They consist in low-discrepancy sequences (Van Der Corput,Faure, Sobol...) filling up (0, 1)d uniformly. Contrary to what is often as-sumed these sequences are not at all random. QMC trajectories are muchtoo uniform to be random. Standard transformations convert the (0, 1)d-valued sequences into d-dimensional Gaussian sequences, and lead to wellchosen solutions (Xm

T )1≤m≤M . The error coming from the QMC estimator1M

∑Mm=1 f(Xm

T ) can be as good as O( (lnM)d

M ), which is nearly O( 1M ).

B.1.2 Discretization schemes

For the sake of simplicity, we assume that we are trying to simulate on thetime interval [0, T ] a stochastic process driven by the one dimensional SDE :

dXt = µ(Xt)dt+ σ(Xt)dWt

Monte-Carlo Methods and Hopf Algebra 355

Let (tn)0≤n≤N be a subdivision of [0, T ]

(∆nt = tn+1 − tn)0≤n≤N−1

and∆W = (∆nW = Wtn+1 −Wtn)0≤n≤N−1

From the basic properties of the Brownian motion it is clear that ∆W ∼N(0,∆t) where ∆t is the diagonal matrix whose entries are the (∆nt). MCsimulation or QMC can therefore be employed to draw M paths for ∆W .As stated in the previous paragraph, discretization schemes will permit thetransformation of these paths into paths of the underlying process. The SDEabove yields

Xtn+1 = Xtn +∫ tn+1

tn

µ(Xt)dt+∫ tn+1

tn

σ(Xt)dWt

Hence a very intuitive discretization is provided by the well-known Eulerscheme ([28],[20]) which generates the following path X conditionally on ∆W :

Xt0 = X0

Xtn+1 = Xtn + µ(Xtn)∆nt+ σ(Xtn)∆nW

However the Euler scheme does have an inconvenient: the two approximationsit consists in are not of the same order∫ tn+1

tn

µ(Xt)dt = µ(Xtn)∆nt+ o(∆nt)

whereas ∫ tn+1

tn

σ(Xt)dWt = σ(Xtn)∆nW + o(√

∆nt)

It would therefore make sense to develop :∫ tn+1

tn

σ(Xt)dWt =∫ tn+1

tn

σ

(Xtn +

∫ t

tn

µ(Xs)ds+∫ t

tn

σ(Xs)dWs

)dWt

= σ(Xtn)∆nW + σ′(Xtn)σ(Xtn)∫ tn+1

tn

(∫ t

tn

dWs)dWt + o(∆nt)

= σ(Xtn)∆nW +σ′(Xtn)σ(Xtn)

2((∆nW )2 −∆nt

)+ o(∆nt)

This refinement is at the origin of the one-dimensional Milstein scheme:

Xt0 = X0

Xtn+1 = Xtn + µ(Xtn)∆nt+ σ(Xtn)∆nW +σ′(Xtn)σ(Xtn)

2((∆nW )2 −∆nt

)


To study the performance of discretization schemes the literature distinguishesbetween strong and weak order of convergence ([28], [20]):

DEFINITION B.1 Strong/Weak weak order of convergence Not-ing h = max

n=0,..,N−1(∆nt), a scheme X is said to be of strong order β > 0 if

there exists a constant c such that

E[‖ X(tN )−X(T ) ‖] ≤ chβ

for some vector norm ‖ · ‖. It is said to be of weak order β > 0 if there existsa constant c such that

| E[f(X(tN )]− E[f(X(T ))] |≤ chβ

for all f polynomially bounded in C2β+2.

Under Lipschitz-type conditions on µ and σ the Euler scheme can be shownto be of strong order 1

2 and of weak order 1, whereas the Milstein scheme isof strong and weak order 1.Writing a multi-dimensional generalization of the Milstein scheme would in-volve adding antisymmetric Wiener iterated integrals, called Levy areas

Aij =∫ 1

0

W it dW

jt −

∫ 1

0

W jt dW

it (B.2)

The standard approximation for this term requires several additional randomnumbers [28]. There are however approaches to avoid the drawing of manyextra random numbers by using the relation of this integral to the Levy areaformula [124]:

E[eiλAij

|W i1 + ıW j

1 = z] =λ

sinhλe−|z|2

2 (λ cothλ−1)

From this characteristic function, Levy areas and Brownian motion W it can

be simulated exactly. However, this can become quite costly in computationalexpense.

B.1.3 Taylor-Stratonovich expansion

Using the tools described in the previous sections, we explain the Taylor-Stratonovich expansion (TSE) equivalent to a Taylor expansion in the de-terministic case. Note that a similar expansion, called Taylor-Ito expansion,that uses the Ito calculus exists. In order to be simulated, the discretizationscheme eventually needs to be set in Ito form. However, we prefer to presentTSE as its algebraic structure is simpler.


Let Xt be an n-dimensional Ito process following the Stratonovich SDE

dXt = V0dt+m∑i=1

VidW it , Xt=0 = X0 ∈ Rn

By introducing the notation dW 0t ≡ dt, we have

dXt =m∑i=0

VidW it (B.3)

Without loss of generality, we assume that we have a time-homogeneous SDE.A time-inhomogeneous SDE can be written in this normal form (B.3) byincluding an additional state Xn+1

t

dXn+1t = dt

The Stratonovich calculus entails that for f in C1(Rm,R)

f(XT ) =m∑i=0

∫ T

0

Vif(Xt)dW it

By iterating this equation, we obtain

f(XT ) = f(X0) +m∑i=0

(Vif(X0)

∫ T

0

dW it

+m∑j=0

∫ T

0

(∫ t

0

ViVjf(Xs)dW js

)dW i

t

With a repeated application of the Stratonovich formula, the iterated integraldefined as ∫

0≤t1<...<tk≤TdW i1

t1 ...dWiktk

appears naturally. From a L2-norm perspective, dW 0t counts as [dt] whereas

dW it (i = 1, ...,m) counts as [

√dt]. As a consequence, following [28], we

introduce a graduation on multi-indexes (i1, ..., ik) by

deg(i1, · · · , ik) = k + ]j such as ij = 0

and takeAr = (i1, ..., ik) such as deg(i1, ..., ik) ≤ r

Finally, continuing the calculation sketched above, we obtain [28]:


THEOREM B.1 Taylor-Stratonovich expansion

f(XT ) =∑

(i1,...,ik)∈Ar

Vi1 ...Vikf(X0)∫

0≤t1<...<tk≤TdW i1

t1 ...dWiktk

+ Rr(T,X0, f)

where supX0∈Rn

||Rr(T,X0, f)||2 ≤ CTr+1

2 sup(i1,...,ik)∈Ar+2\Ar

||Vi1 ...Vikf ||∞.

REMARK B.1 Actually, the TSE remains correct if we replace dWt by

any continuous semi-martingale.

TSE provides a link with the discretization schemes mentioned earlier. Takingfor instance r = 1 and f(Xt) = Xt yields the Euler scheme (if writing in Itoform)

Xtn+1 = Xtn + V0 (Xtn) ∆nt+m∑i=1

Vi (Xtn) ∆nW i

For r = 2, we have

Xtn+1 = Xtn + V0 (Xtn) ∆nt+m∑i=1

Vi (Xtn) ∆nW i

+12

∑1≤i≤j≤m

(12

[Vi, Vj ] (Xtn)Aij + ViVj (Xtn) ∆nW i ∆nW j

)which happens to be strictly equivalent to the multidimensional Milsteinscheme. Here [·, ·] is the Lie bracket (see exercise 4.3) defined by

[Vi, Vj ] = ViVj − VjViNote that when the Lie algebra generated by the Vii=1,··· ,m is Abelian

[Vi, Vj ] = 0 , ∀ i, j = 1, · · · ,m

the Milstein scheme simplifies as it does not require the simulation of Levyareas.

B.2 Algebraic Setting

As seen in the previous section, the MC methods rely on discretization ofSDEs. Convergence issues require in certain cases to select a very small dis-cretization time-step. The simulation can become quite time-consuming for


multi-asset hybrid options. A possible resolution of this problem is to repre-sent asset prices as functionals of simple processes that can be simulated ex-actly (Brownian motion, Ornstein-Uhlenbeck, ...). This is the starting point ofMarkov functional LMMs (see exercise 8.1) and forward variance swap models[62], [68]. In the last section, we state the Yamato theorem giving a necessaryand sufficient condition for representing solutions of SDEs as functionals ofBrownian motions. This theorem originates from the Hopf algebra structureof (Chen) iterated integrals which appear in TSEs.

B.2.1 Hopf algebra

To study algebraic relations between iterated integrals we replace the vectorfields V0, V1, ..., Vm by letters ε0, ε1, ..., εm.

B.2.1.1 Algebra

We endow the real vector space H generated by εii=0,··· ,m with an unitalgraded algebra structure (H,+, ., ε). The noncommutative product corre-sponds to the concatenation product

. : (εi, εj) 7−→ εi.εj = εiεj

The words are obtained by sums of concatenations of generators. The neutralelement . is the empty word ε. It satisfies for any word w:

w.ε = ε.w = w

Consistently with TSEs, we introduce a graduation on words equivalent tothe graduation introduced for multi-indices

deg(εi1 ...εik) = deg(i1, ..., ik)

We have

H =∞⊕r=0

Hr

We draw the reader’s attention to the fact that in Hr all words of graduationgreater than r have a coefficient set to 0. For instance in H3 we get

(2ε1 + ε0εd − ε1ε0ε1).ε1 = 2ε1ε1 + ε0εdε1

(H,+, ., ε) then has the structure of a graded associative algebra with unit ε.The product . is associative and commutes with the identity Id on H, whichwe will summarize by the diagram B.1.

REMARK B.2 Note that the tensor algebra (H ⊗ H,+, .) is a gradedalgebra (its graduation is induced by deg(x ⊗ y) = deg(x) + deg(y)) withoperations + and . deriving from those of H.


TABLE B.1: Associativity diagram..⊗ Id

Hr ⊗Hr ⊗Hr −→ Hr ⊗HrId⊗ . ↓ ↓ .

Hr ⊗Hr −→ Hr.

TABLE B.2: Co-associativity diagram.∆⊗ Id

Hr ⊗Hr ⊗Hr ←− Hr ⊗HrId⊗∆ ↑ ↑ ∆

Hr ⊗Hr ←− Hr∆

B.2.1.2 Co-product

Next we define the co-product ∆ : H⊗H → H on the generators

∆(εi) = εi ⊗ ε+ ε⊗ εi

Then we extend the action of ∆ on words by imposing that ∆ is a mor-phism for the concatenation product, meaning that for two noncommutativepolynomials x and y

∆(x.y) = ∆(x).∆(y)

The action of ∆ can be seen (see diagram B.2) as reversing the arrows in thediagram B.1

B.2.1.3 Co-unit

We also introduce a co-unit η : Hr → R by

η(x) = 1 , x ∈ Hr

B.2.1.4 Antipode

To complete the setting we bring up a final operation a : Hr → Hr called anantipode. a is a linear application characterized by its action on words:

a(εi1εi2 ...εik) = (−1)kεik ...εi2εi1

It follows from this definition that for any two noncommutative polynomialsx, y in Hr we have an anti-homomorphism

a(xy) = a(y)a(x)


Finally (H,+, .,∆, a, ε) endowed with these five operations is a Hopf algebra(see [17] for details).2

DEFINITION B.2 Hopf algebra A Hopf algebra H is a vector spaceendowed with five operations

· : H⊗H → H multiplicationε : R→ H unit map

∆ : H → H⊗H coproductη : H → R counita : H → H antipode

which possess the following properties

m (Id×m) = m (m× Id) (associativity)m (Id× ε) = id = m (ε× Id) (existence of unit)(Id×∆) ∆ = (∆× Id) ∆ (coassociativity)(η × Id) ∆ = Id = (Id× η) ∆ (existence of counit)m (Id× a) ∆ = ε η = m (a× Id) ∆∆ m = (m×m) (∆×∆)

There happens to be very particular elements of H called primitive elements

DEFINITION B.3 Primitive element An element L satisfying

∆(L) = L ⊗ ε+ ε⊗ L

is called a primitive element.

By equipping the set G of primitive elements with the bracket defined by

[L,L′] ≡ L.L′ − L′.L

it follows that (G, [·, ·]) forms a Lie algebra.

PROOF Indeed if L,L′ ∈ G,

∆(L.L′) = ∆(L).∆(L′)= (L ⊗ 1 + 1⊗ L).(L′ ⊗ 1 + 1⊗ L′)= LL′ ⊗ 1 + 1⊗ LL′ + L ⊗ L′ + L′ ⊗ L

2These operations are so rich and beautiful that some physicists conjecture that God shouldsurely be a Hopf algebra.


Hence

∆([L,L′]) = ∆(L.L′)−∆(L′.L)= LL′ ⊗ 1 + 1⊗ LL′ − L′L ⊗ 1− 1⊗ L′L

∆([L,L′]) = [L,L′]⊗ 1 + 1⊗ [L,L′]

DEFINITION B.4 Grouplike elements Elements g of H satisfying

∆(g) = g ⊗ g

are called grouplike elements.

The set G of grouplike elements forms a group for the concatenation product.

PROOF Indeed if g, h ∈ G

∆(g.h) = ∆(g).∆(h)= (g ⊗ g).(h⊗ h)

∆(g.h) = gh⊗ gh

DEFINITION B.5 Exponential map We define an exponential mapon Hr by the usual series expansion

exp(L) =∞∑k=0

1k!Lk

The expression above is a finite sum because deg(Lk) = k deg(L) and Lk isequal to 0 as soon as k deg(L) > r.

This definition takes its sense from the fact that the exponential of a primitiveelement in Gr is a group-like element in Gr:

PROPOSITION B.1

Gr = exp(Gr)


PROOF We do the proof in one way: If L ∈ Gr then

∆(exp(L)) = ∆(∞∑k=0

Lk

k!)

=∞∑k=0

1k!

∆(L)k

=∞∑k=0

1k!

(L ⊗ ε+ ε⊗ L)k

=∞∑k=0

1k!

k∑j=0

k!j!(k − j)!

(Lj ⊗ Lk−j)

= exp(L)⊗ exp(L)

B.2.2 Chen series

Now that this setting has been introduced we can almost reach our goal andsee how it appears in TSE. The idea is to use a morphism

Γr : εi1 ...εik .ε ∈ Hr 7−→ Vi1 ...Vik(X0)

For a semi-martingale ω we define its Chen series in Hr as the pre-image ofthe Taylor-Stratonovich expansion:

X0,1(ω) =∑

(i1,...,ik)∈Ar

εi1 ...εik

∫0≤t1<...<tk≤1

dωi1(t1)...dωik(tk)

The introduction of the Hopf algebra setting allows us to provide a simpleproof for the major result that follows

THEOREM B.2 Chen [74]

X0,1(ω) is a group-like element of Hr.

PROOF (Sketch) Assuming m = 2 for simplicity, we prove the result forr = 2 . In that case

X0,1(ω) = ε + ε1

∫0≤t≤1

dω1(t) + ε2

∫0≤t≤1

dω2(t) + ε0

+ ε1ε2

∫0≤t≤1

ω1(t)dω2(t) + ε2ε1

∫0≤t≤1

ω2(t)dω1(t)


Remembering the graduation introduced on Hr ⊗Hr we compute

X0,1(ω)⊗X0,1(ω) = ε⊗ ε + (ε0 ⊗ ε+ ε⊗ ε0)

+ (ε1 ⊗ ε+ ε⊗ ε1)∫

0≤t≤1

dω1(t)

+ (ε2 ⊗ ε+ ε⊗ ε2)∫

0≤t≤1

dω2(t)

+ (ε1ε2 ⊗ ε+ ε⊗ ε1ε2)∫

0≤t≤1

ω1(t)dω2(t)

+ (ε2ε1 ⊗ ε+ ε⊗ ε2ε1)∫

0≤t≤1

ω2(t)dω1(t)

+ (ε1 ⊗ ε2 + ε2 ⊗ ε1)∫

0≤t≤1

dω1(t)∫

0≤t≤1

dω2(t)

On the other hand we have

∆(X0,1(ω)) = ε⊗ ε+ (ε1 ⊗ ε+ ε⊗ ε1)∫

0≤t≤1

dω1(t)

+ (ε2 ⊗ ε+ ε⊗ ε2)∫

0≤t≤1

dω2(t)

+ (ε0 ⊗ ε+ ε⊗ ε0) + ∆(ε1ε2)∫

0≤t≤1

ω1(t)dω2(t)

+ ∆(ε2ε1)∫

0≤t≤1

ω2(t)dω1(t)

Noting that

∆(ε1ε2) = ∆(ε1)∆(ε2) = (ε1ε2 ⊗ ε+ ε⊗ ε1ε2 + ε1 ⊗ ε2 + ε2 ⊗ ε1)

we see that∆(X0,1(ω)) = X0,1(ω)⊗X0,1(ω)

is equivalent to∫0≤t≤1

dω1(t)∫

0≤t≤1

dω2(t) =∫

0≤t≤1

ω1(t)dω2(t) +∫

0≤t≤1

ω2(t)dω1(t)

=∫

0<s≤t<1

dω1(s)dω2(t) +∫

0<s≤t<1

dω2(s)dω1(t)

This last is a consequence of Fubini’s theorem. The same proof holds for ageneral r, using the fact that ∆ is a morphism for words.

As a consequence, if ω is a semi-martingale, we can write

X0,1(ω) = exp(L)

for a primitive element (Lie polynomial) L in G.


B.3 Yamato’s theorem

By applying Γr on the Chen series X0,1(W ), we obtain that the TSE up todegree r on the interval [0, 1] can be written as

X0,1(W ) = exp(L)(X0)

X0,1(W ) is the solution of a flow at time t = 1 driven by the (random) vectorfield Γr(L):

dX0,1

dt= Γr(L)

For completeness, we recall the definition of a flow:

DEFINITION B.6 Flow A flow on a manifold M generated by a vectorfield Z at time t, denoted by etZ , is a one parameter group of transformationsφt(x0) ≡ etZx0 which satisfies:

• For each t ∈ [0,∞), φt is a diffeomorphism of M .

• φt φs = φt+s for any t, s ∈ [0,∞).

• limt→0 φt(x0) = x0.

• Solution to the ODE dφt(x0)dt = Z(φt(x0)).

REMARK B.3 Beware, φt(x0) ≡ etZx0 is only a notation, and thesolution at time t to the ODE dφt(x0) = Z(φt(x0))dt is not necessarilyeR t0 Z(φu(x0))dux0. This will only be the case if the vector fields Z(φu(x0))0≤u≤t

commute.

Before stating the Yamato theorem, we introduce a last object, the free Liealgebra. (Note that this free Lie algebra is slightly different (resp. identical)from the one which appears in the Hormander (resp. Frobenius) theorem.)

DEFINITION B.7 Free Lie algebra The free Lie algebra spanned bythe generators (V0, V1, ..., Vm) is the Lie algebra consisting of linear combina-tions of elements of the form

[Vi1 , ..., [Vik−1 , Vik ]]

with (i1, ..., ik) ∈ 0, ...,m.

Finally, from the Chen theorem B.2, we have


THEOREM B.3 Yamato [143]Let Xt be a solution of

dXt = V0dt+m∑j=1

Vj dW jt (B.4)

with Xt=0 = X0. Then it is represented as

Xt = exp (Lt)X0

where

Lt = tV0 + · · ·+Wmt Vm +

∑r=2

∑J∈Ar

cJWJt V

J

with the iterated Brownian integrals

W Jt =

∫0≤tj1<···<tjm≤t

dW j1u1· · · dW jm

um

and

V J = [· · · [Vj1 , Vj2 ] · · ·Vjm ] , J = (j1, · · · , jm)

cJ are some constants.

COROLLARY B.1Let Xt be a solution of (B.4). If V0, V1, · · · , Vm forms an Abelian Lie algebra,i.e., [Vi, Vj ] = 0 for each i and j, then the solution of the SDE above startingat X0 is represented as

Xt = exp(tV0 +W 1

t V1 + · · ·Wmt Vm

)X0 (B.5)

This means that Xt can be represented as a functional of the Brownian mo-tions W i

t i=1,··· ,m. The functional is obtained by solving the flow (B.5). Asa consequence, the SDE can be simulated exactly. In the following example,we classify the local volatility function for which asset prices can be writtenas a functional of a Brownian motion.

Example B.1 Exact discretization of (Abelian) LV modelsLet us consider the local volatility model

dft = σ(t, ft)dWt (B.6)


Using the Stratonovich calculus, this gives the time-homogeneous SDE

dft = −12σ(u, ft)′σ(u, ft)dt+ σ(u, ft) dWt

du = dt

for which we read the vector fields

V0 = −12σ(u, f)′σ(u, f)∂f + ∂u

V1 = σ(u, f)∂f

The prime ′ indicates a derivative with respect to f . From corollary B.1, ftis a functional ft ≡ Φ(t,Wt) of the time and the Brownian Wt if and only

[V0, V1] =(∂uσ(u, f) +

12σ(u, f)2σ(u, f)′′

)∂f (B.7)

= 0

Setting F (u, f) ≡ σ(u, f)2, the constraint above gives

∂uF +FF ′′

2− F ′2

4= 0

A solution is the hyperbolic model

σ(t, f) =√a(t)f2 + b(t)f + c(t)

with

a(t) = a0

b(t) = b0

c(t) = c0e−a0t +

b204a0

(1− e−a0t

)

Example B.2 Nilpotent step 1 LV modelsAccording to the Yamato theorem, the forward ft driven by the SDE (B.6) is

a functional ft = Φ(Wt,W

[01]t

)of the Brownian motion Wt and the stochastic

iterated integral

W[01]t ≡ 1

2

∫0<s<u<t

(dsdWu − dWsdu)

=∫ t

0

udWu −t

2Wt


if and only if the vectors V0, V1 generate a step 1 nilpotent Lie algebra:

[V1, [V0, V1]] = 0 , [V0, [V0, V1]] = 0

From (B.7), we get

[V1, [V0, V1]] =(σ∂uσ

′ − σ′∂uσ +12σ3σ′′′ +

12σ2σ′σ′′

)∂f = 0

[V1, [V0, V1]] =(−1

2σσ′∂uσ

′ + ∂2uσ +

12∂u(σ2σ′′

)+

12∂uσ (σσ′)′

)∂f

+(−1

4σσ′

(σ2σ′′

)′+

14σ2σ′′ (σσ′)′

)∂f = 0

Note that the processes Wt and∫ t

0udWu are normal r.v. with zero mean and

variance (using Ito’s isometry) (t t2

2t2

2t3

3

)

The forward can therefore be simulated exactly and efficiently in a nilpotentstep 1 LV model.When σ = σ(f) is time-homogeneous, the constraints above reduce to

σσ′′′ + σ′σ′′ = 0 (B.8)

− (σ′)2σ′′ − σσ′σ′′′ + σ (σ′′)2 = 0 (B.9)

The first equation (B.8) gives

σσ′′ = A (B.10)

with A an integration constant. The equation (B.9) reduces

A2

σ= 0

for which we deduce that A = 0 and σ(f) = αf + β with α and β twoconstants. Therefore a time-homogeneous nilpotent step 1 LV model is adisplaced diffusion model which is already an Abelian LV model.Finally, we exhibit an example which is a nilpotent step 1 (but not Abelian)LV model:

σ(t, f) = (αt+ β)(f + λ)

with α, β and λ three constants.

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Index

Lk(Ω,F ,P) space, 12H2-model, 173PSL(2,R), 167σ-algebra, 9(Co)-Tangent vector bundle, 902-Sphere, 81

Abelian connection, 92Abelian Lie algebra, 358Abelian LV model, 366Absolutely continuous, 253Adapted process, 14Adjoint operator, 252Almost Markov Libor Market Model,

244Almost surely (a.s), 10American option, 8Analytical call option for the CEV

model, 141Annihilation-creation algebra, 323Antipode, 360Arbitrage, 27Asian option, 8Asset, 29Attainable payoff, 42

Bachelier model, 52Backward Kolmogorov PDE, 78Barrier option, 9Basket implied volatility, 190Basket option, 189Benaim-Friz theorem, 292Bermudan option, 8Black-Scholes formula, 31Black-Scholes PDE, 33Bond, 39, 64Borel σ-algebra, 10Bounded Linear operator, 252

Break-even volatility, 53Brownian filtration, 14Brownian motion, 14Brownian sheet, 218

Cameron-Martin space, 310Caplet, 68Caratheodory theorem, 10Cartan-Hadamard manifold, 156Chart, 80Chen series, 363Cholesky decomposition, 76Christoffel symbol, 93Cliquet option, 63Closable operator, 316Co-cycle condition, 88Co-product, 360Co-terminal swaption, 68Co-unit, 360Collaterized Commodity Obligation,

196Collaterized Debt Obligation, 197Complete market, 42Complete probability space, 10Conditional expectation, 12Confluent hypergeometric potential,

276Connection, 90Constant Elasticity of Variance (CEV),

125Control distance, 298Convexity adjustment, 68Correlation matrix, 76Correlation smile, 195Corridor variance swap, 60Cotangent space, 84Covariant derivative, 92Cumulative normal distribution, 31

379

380 Index

Curvature, 101Cut-locus, 99Cut-off function, 111Cyclic vector, 323Cylindrical function, 310

DeWitt-Gilkey-Pleijel-Minakshisun-daram, 111

Deficiency indices, 258Delta, 43, 327Delta hedging, 43Density bundle, 103Derivation, 83Differential form, 87Diffusion, 15Discount factor, 24Domain of a linear operator, 252Down-and-out call option, 163Drift, 15Dupire local volatility, 134

Effective vector field, 335Eigenvalue, 255Einstein summation convention, 77Elasticity parameter, 343Equity hybrid model, 72Equivalent local martingale, 29Equivalent measure, 13Euclidean Schrodinger equation, 263Euler scheme, 355European call option, 7European put option, 8Exact conditional probability for the

CEV model, 133Expectation, 11Exponential map, 99, 362Exterior derivative d, 87

Feller boundary classification, 266Feller non-explosion test, 129Feynman path integral, 311Feynman-Kac, 33Fiber, 88Filtration, 13First-order asymptotics of implied

volatility for SVMs, 159

Flow, 365Fock space, 322Fokker-Planck, 78Forward, 39Forward implied volatility, 63Forward Kolmogorov, 78Forward measure, 40Forward-start option, 63Free Lie algebra, 365Freezing argument, 212Frobenius theorem, 220Functional derivative, 315Functional integration, 312Functional space D1,2, 315Fundamental theorem of Asset pric-

ing, 29

Gauge transformation, 108Gauss hypergeometric potential, 274Gaussian bounds, 300Gaussian estimates, 296Generalized Variance swap, 60Geodesic curve, 95Geodesic distance on Hn, 228Geodesic equation, 95Geodesics, 94Geometric Brownian, 21Girsanov, 36Grouplike element, 362Gyongy theorem, 158

Hormander form, 118Hormander’s theorem, 118Hamilton-Jacobi-Bellman equation,

342Hasminskii non-explosion test, 153Heat kernel, 105Heat kernel coefficients, 112Heat kernel on H2, 177Heat kernel on H3, 180Heat kernel on Heisenberg group,

118Heat kernel semigroup, 259Hedging strategy, 42

Index 381

Heisenberg Lie algebra, 119, 121,323

Hermite polynomials, 319Heston model, 181Heston solution, 181Hilbert space, 251HJM model, 47Ho-Lee model, 72Hopf algebra, 361Hull-White 2-factor model, 209Hull-White decomposition, 185Hybrid option, 9Hyperbolic manifold Hn, 228Hyperbolic Poincare plane, 167Hyperbolic surface, 95hypo-elliptic, 117

Implied volatility, 55Incomplete market, 42Injectivity radius, 100Isothermal coordinates, 98Ito isometry, 51Ito lemma, 21Ito process, 17Ito-Tanaka, 135

Jensen inequality, 196

Kato class, 296Killing vector, 98Kunita theorem, 124

Levy area formula, 356Laplace method, 351Laplace-Beltrami, 104Laplacian heat kernel, 105Large traders, 344LCEV model, 126Leading symbol of a differential op-

erator, 104Lebesgue-Stieltjes integral, 257Lee moment formula, 291Length curve, 84Levi-Cevita connection, 92Libor market model, 207

Libor market model (LMM), 49Libor volatility triangle, 214Lie algebra, 120Line bundle, 88Linear operator, 252Local martingale, 28Local skew, 141Local Vega, 331Localized Feynman-Kac, 34Log-normal SABR model, 151

Malliavin derivative, 314Malliavin Integration by parts, 316Manifold, 80Manifold H3, 179Market model, 24Markov Libor Market Model, 239Markovian realization, 220Martingale, 28Maturity, 7Measurable function, 10Measurable space, 10Mehler formula, 121Merton model, 57, 136Metric, 84Milstein scheme, 355Minkowski pseudo-sphere, 168Mixed local-stochastic volatility model,

332Mixing solution, 185Moebius transformation, 168Money market account, 24

Napoleon option, 64Natanzon potential, 274Negligible sets, 10Nilpotent step 1 LV model, 367Non-autonomous Kato class, 298Non-explosion, 126Non-linear Black-Scholes PDE, 345Norm, 252Normal SABR model, 176Novikov condition, 36Numeraire, 34Number operator, 324

382 Index

One-form, 84Operator densely defined, 252Ornstein-Uhlenbeck, 22Ornstein-Uhlenbeck operator, 324

P&L Theta-Gamma, 53Parallel gauge transport, 94Partition of unity, 102Path space, 310Payoff, 8Poincare disk, 168Predictor-corrector, 220Primitive element, 361Probability measure, 10Pullback bundle, 93Pullback connection, 93Put-call duality, 163Put-call parity, 126Put-call symmetry, 163

Quadratic variation, 44Quasi-random number, 354

Radon-Nikodym, 13Random variables, 11Rebonato parametrization, 215Reduction method, 249Regular value, 255Regularly varying function, 292Resolution of the identity, 258Resolvent, 255Ricci tensor, 102Riemann surface, 153Riemann tensor, 102Riemann Uniformization theorem,

154Riemannian manifold, 84Risk-neutral measure, 30

SABR formula, 171SABR-LMM, 225Saddle-point, 351Scalar curvature, 102Scholes-Black equation, 270Second moment matching, 195

Second theorem of asset pricing, 43Second-order elliptic operator, 104Section, 88Self-adjoint extension, 254Self-adjoint operator, 253Self-financing portfolio, 26Separable Hilbert space, 251Separable local volatility model, 124Short-rate model, 46Singular value, 255Skew, 56Skew at-the-money, 56Skew at-the-money forward, 56Skew averaging, 147Skorohod integral, 317Small traders, 344Smile, 55Sobolev Hm, 253Spectral theorem, 258Spectrum, 255Spot, 7Spot Libor measure, 216Static replication, 58Sticky rules, 62Stiejles function, 257Stochastic differential equation (SDE),

17Stochastic integral, 15Stochastic process, 13Stochastic volatility Libor market

model, 207Stochastic volatility Model, 150Stochastically complete, 106Stratonovich, 107Stratonovich integral, 16Strike, 7Strong convergence, 257Strong order of convergence, 356Strong solution, 23Supercharge operators, 269Superpotential, 269SVM, 150Swap, 65Swaption, 66Swaption implied volatility, 67

Index 383

Symbol of a differential operator,104

Symmetric operator, 253

Tangent process, 328Tangent space, 83Taylor-Stratonovich expansion, 358Tensor of type (r, p), 86Tensor vector bundle, 90Time-dependent heat kernel expan-

sion, 116Torsion, 93Trivial vector bundle, 89

Unbounded Linear operator, 252Uniformization theorem, 153Unique in law, 24Upper half-plane, 168

Variance, 12Variance swap, 59Vector bundle, 88Vector field, 83Vega, 327Volatility, 15Volatility of volatility (vol of vol),

150Volatility swap, 70

Weak derivative, 253Weak order of convergence, 356Weak solution, 23White noise, 312Wick identity, 50, 313Wick product, 319Wiener chaos, 324Wiener measure, 311

Yamato theorem, 366

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