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Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Stochastic Volatility Models
Heston 1.5
Jean-Pierre Fouque
Department of Statistics & Applied ProbabilityUniversity of California Santa Barbara
Modeling and Managing Financial Risks
Paris, January 10-13, 2011
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
References:
Multiscale Stochastic Volatility for Equity, Interest-Rate
and Credit Derivatives
J.-P. Fouque, G. Papanicolaou, R. Sircar and K. SølnaCambridge University Press (in press).
A Fast Mean-Reverting Correction to Heston Stochastic
Volatility Model
J.-P. Fouque and M. LorigSIAM Journal on Financial Mathematics (to appear).
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Volatility Not Constant
1950 1960 1970 1980 1990 2000−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04Daily Log Returns
S&P500Simulated GBM
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Skews of Implied Volatilities (SP500 on June 1, 2007)
0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Moneyness=K/X
Impl
ied
Vol
atili
tyImplied Volatilities on 1 June 2007
1 mo2 mo3 mo6 mo9 mo12 mo18 mo
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
What’s Wrong with Heston?
Single factor of volatility running on single time scale notsufficient to describe dynamics of the volatilty process.
Not just Heston . . . Any one-factor stochastic volatility modelhas trouble fitting implied volatility levels accross all strikesand maturities.
Emperical evidence suggests existence of several stochasticvolatility factors running on different time scales
−→
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Evidence
Melino-Turnbull 1990, Dacorogna et al. 1997,Andersen-Bollerslev 1997,
Fouque-Papanicolaou-Sircar 2000,
Alizdeh-Brandt-Diebold 2001, Engle-Patton 2001,Lebaron 2001,
Chernov-Gallant-Ghysels-Tauchen 2003,
Fouque-Papanicolaou-Sircar-Solna 2003, Hillebrand 2003,Gatheral 2006,
Christoffersen-Jacobs-Ornthanalai-Wang 2008.
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Multiscale SV Under Risk-Neutral
dXt = rXt dt + f (Yt ,Zt)Xt dW(0)⋆t ,
dYt =
(1
εα(Yt) −
1√εβ(Yt)Λ1(Yt ,Zt)
)dt +
1√εβ(Yt) dW
(1)⋆t ,
dZt =(δc(Zt) −
√δ g(Zt)Λ2(Yt ,Zt)
)dt +
√δ g(Zt) dW
(2)⋆t .
Time scales: ε≪ 1 ≪ 1/δFast factor: YSlow factor: ZOption pricing:
Pε,δ(t,Xt ,Yt ,Zt) = IE ⋆{
e−r(T−t)h(XT ) | Xt ,Yt ,Zt
},
∂Pε,δ
∂t+ L(X ,Y ,Z)P
ε,δ − rPε,δ = 0.
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Expand
Pε,δ =∑
i≥0
∑
j≥0
(√ε)i (
√δ)j Pi ,j ,
P0 = PBS computed at volatility σ(z):
σ2(z) = 〈f 2(·, z)〉 =
∫f 2(y , z)ΦY (dy).
√εP1,0 = V ε
2 x2 ∂2PBS
∂x2+ V ε
3 x∂
∂x
(x2 ∂
2PBS
∂x2
),
√δ P0,1 = V δ
0
∂PBS
∂σ+ V δ
1
∂
∂x
(∂PBS
∂σ
).
In terms of implied volatility:
I ≈(b⋆ + τbδ
)+
(aε + τaδ
) log(K/x)
τ, τ = T − t.
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
Time Scales
The previous asymptotics is a perturbation around Black-Scholesleading to Black-Scholes 2.0 where 2.0 = 1+ 2x(0.5)
Why not keeping one volatility factor on a time scale of order one?This would be a perturbation around a one-factor stochasticvolatility model, but which one?
Heston of course!
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Heston Under Risk-Neutral Measure
dXt = rXtdt +√
Zt XtdW xt
dZt = κ(θ − Zt)dt + σ√
Zt dW zt
d 〈W x ,W z〉t = ρdt
One-factor stochastic volatility model
Square of volatility, Zt , follows CIR process
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Formulas!
Explicit formulas for european options:
PH(t, x , z) = e−rτ 1
2π
∫e−ikqG (τ, k, z)h(k)dk
q(t, x) = r(T − t) + log x ,
h(k) =
∫e ikqh(eq)dq,
G(τ, k, z) = eC(τ,k)+zD(τ,k)
C (τ, k) and D(τ, k) solve ODEs in τ = T − t.
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
ODEs
dD
dτ(τ, k) =
1
2σ2D2(τ, k) − (κ+ ρσik)D(τ, k) +
1
2
(−k2 + ik
),
D(0, k) = 0.
dC
dτ(τ, k) = κθD(τ, k),
C (0, k) = 0,
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Pretty Pictures! Heston captures well-documented features ofimplied volatility surface: smile and skew
100
150
K
0.5
1.0
1.5
2.0T
0.10
0.15
0.20
0.25
Σ
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Captures Some . . . Not All Features of Smile
Mispricesfar ITMand OTMEuropeanoptions[Fiorentini-Leon-Rubio2002,Zhang-Shu2003 ...]
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19Days to Maturity = 583
log(K/x)
Impl
ied
Vol
atili
ty
Market DataHeston Fit
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
DynamicsWhy We like HestonProblems with Heston
Simultaneous Fit Across Expirations Is Poor
Particulardifficultyfitting shortexpirations[Gatheral2006]
−0.1 −0.05 0 0.05 0.1 0.150.1
0.11
0.12
0.13
0.14
0.15
0.16Days to Maturity = 65
log(K/x)
Impl
ied
Vol
atili
ty
Market DataHeston Fit
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
Multiscale Under Risk-Neutral Measure
dXt = rXtdt +√
Zt f (Yt)XtdW xt
dYt =Zt
ε(m − Yt)dt + ν
√2
√Zt
εdW y
t
dZt = κ(θ − Zt)dt + σ√
Zt dW zt
d⟨W i ,W j
⟩t= ρijdt i , j ∈ {x , y , z}
Volatility controled by product√
Zt f (Yt)
Yt modeled as OU process running on time-scale ε/Zt
Note: f (y) = 1 ⇒ model reduces to Heston
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
Option Pricing PDE
Price of European Option Expressed as
Pt = E
[e−r(T−t)h(XT )
∣∣∣Xt ,Yt ,Zt
]=: Pε(t,Xt ,Yt ,Zt)
Using Feynman-Kac, derive following PDE for Pε
LεPε(t, x , y , z) = 0,
Lε =∂
∂t+ L(X ,Y ,Z) − r ,
Pε(T , x , y , z) = h(x)
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
Some Book-Keeping of Lε
Lε has convenient form (z factorizes)
Lε =z
εL0 +
z√εL1 + L2,
L0 = ν2 ∂2
∂y2+ (m − y)
∂
∂y
L1 = ρyzσν√
2∂2
∂y∂z+ ρxyν
√2 f (y)x
∂2
∂x∂y
L2 =∂
∂t+
1
2f 2(y)zx2 ∂
2
∂x2+ r
(x∂
∂x− ·
)
+1
2σ2z
∂2
∂z2+ κ(θ − z)
∂
∂z+ ρxzσf (y)zx
∂2
∂x∂z.
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
Expand
Perform singular perturbation with respect to ε
Pε = P0 +√εP1 + εP2 + . . .
Look for P0 and P1 functions of t, x , and z only
Find
P0(t, x , z) = PH(t, x , z)
with effective square volatility 〈f 2〉Zt = Zt ,and effective correlation ρ→ ρxz 〈f 〉
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
More Formulas!
P1(t, x , z) =e−rτ
2π
∫e−ikq
(κθf0(τ, k) + z f1(τ, k)
)
× G(τ, k, z)h(k)dk,
f0(τ, k) =
∫ τ
0f1(t, k)dt,
f1(τ, k) =
∫ τ
0b(s, k)eA(τ,k,s)ds.
b(τ, k) = −(V1D(τ, k)
(−k2 + ik
)+ V2D
2(τ, k) (−ik)
+V3
(ik3 + k2
)+ V4D(τ, k)
(−k2
)).
A(τ, k, s) solves ODE in τ
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Heston + Fast FactorOption Pricing
Goup Parameters: the V’s
P1 is linear function of four constants
V1 = ρyzσν√
2⟨φ′
⟩,
V2 = ρxzρyzσ2ν
√2
⟨ψ′
⟩,
V3 = ρxyν√
2⟨f φ′
⟩,
V4 = ρxyρxzσν√
2⟨f ψ′
⟩.
ψ(y) and φ(y) solve Poisson equations
L0φ =1
2
(f 2 −
⟨f 2
⟩),
L0ψ = f − 〈f 〉 .
Each Vi has unique effect on implied volatility
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Effect of V1 and V2 on Implied Volatility
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2
0.12
0.14
0.16
0.18
0.2
0.22
0.24
log(K/x)
σ imp
V1 Effect
−0.02−0.01 00.005
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2
0.12
0.14
0.16
0.18
0.2
0.22
log(K/x)σ im
p
V2 Effect
−0.02−0.01 00.005
Vi = 0 corresponds to Heston
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Effect of V3 and V4 on Implied Volatility
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2
0.12
0.14
0.16
0.18
0.2
0.22
0.24
log(K/x)
σ imp
V3 Effect
−0.02−0.01 00.005
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2
0.1
0.15
0.2
0.25
log(K/x)σ im
p
V4 Effect
−0.02−0.01 00.005
Vi = 0 corresponds to Heston
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Outline1 Multiscale Models
Motivation for Multiple Time ScalesMultiscale DynamicsSingular-Regular Perturbations: Black-Scholes 2.0
2 Heston ModelDynamicsWhy We like HestonProblems with Heston
3 Perturbation around HestonHeston + Fast FactorOption Pricing
4 Numerical WorkMultiscale Implied Volatility SurfaceMultiscale Fit to Data
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Captures More Features of Smile
Better fitfor far ITMand OTMEuropeanoptions(SPX,T=121days, onMay 17,2006)
−0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20.1
0.11
0.12
0.13
0.14
0.15
0.16
log(K/x)
Impl
ied
Vol
atili
ty
Market DataHeston FitMultiscale Fit
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Simultaneous Fit Across Expirations Is Improved
Vast im-provementfor shortexpirations
65 days tomaturity
−0.1 −0.05 0 0.05 0.1 0.150.1
0.11
0.12
0.13
0.14
0.15
0.16
log(K/x)
Impl
ied
Vol
atili
ty
Market DataHeston FitMultiscale Fit
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Accuracy of Approximation: Smooth Payoffs Usual technique forsmooth payoffs (Schwartz space of rapidly decaying functions):
Write a PDE for the residual (model price minus next-orderapproximated price)
Use probabilistic representation for parabolic PDEs withsource (small of order ε) and terminal condition (small oforder ε)
Deduce residual is small of order ε
Well, not exactly because moments of Yt need to be controlleduniformly in ε
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Moments of Y
dYt =Zt
ε(m − Yt)dt + ν
√2
√Zt
εdW y
t
Yt = m + (y − m)e−1ε
R t0 Zsds +
ν√
2√ε
e−1ε
R t0 Zudu
∫ t
0e
1ε
R s0 Zuduν
√Zs dW y
s
We show that for any given α ∈ (1/2, 1) there is a constant Csuch that.
E|Yt | ≤ C εα−1 ,
and therefore an accuracy of ε1−
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Numerical Illustration for Call Options Monte Carlo simulation fora European call option: standard Euler scheme, time step of 10−5
years, 106 sample paths with ε = 10−3 so that√εV3 = 0.0303 is of
the same order as V ε3 , the largest of the V ε
i ’s obtained in the datacalibration example. The parameters used in the simulation are:
x = 100, z = 0.24, r = 0.05, κ = 1, θ = 1, σ = 0.39, ρxz = −0.35,
y = 0.06,m = 0.06, ν = 1, ρxy = −0.35, ρyz = 0.35,
τ = 1,K = 100,
and f (y) = ey−m−ν2so that
⟨f 2
⟩= 1.
ε√εP1 P0 +
√εP1 PMC σMC |P0 +
√εP1 − PMC
0 0 21.0831 - - -
10−3 −0.2229 20.8602 20.8595 0.0364 0.0007
Jean-Pierre Fouque Heston 1.5
Multiscale ModelsHeston Model
Perturbation around HestonNumerical Work
Multiscale Implied Volatility SurfaceMultiscale Fit to Data
Summary
Heston model provides easy way to calculate option prices instochastic volatility setting, but fails to capture some featuresof implied volatility surface
Multiscale model offers improved fit to implied volatilitysurface while maintaining convenience of option pricingforumulas
Present work with Matt Lorig to appear in the SIAM Journal onFinancial MathematicsWork in preparation for Risk: formulas for fast and slow time
scales around Heston (Heston 2.0).
Jean-Pierre Fouque Heston 1.5