stochastic volatility model of heston and the smilerweron/assets/pdf/rweron04...uwe wystup...

Stochastic volatility model of Hestonand the smile

Rafa l Weron

Hugo Steinhaus CenterWroc law University of TechnologyPoland

In collaboration with:Piotr Uniejewski (LUKAS Bank)Uwe Wystup (Commerzbank Securities)

Rafa l Weron: The Third Nikkei Econophysics Symposium

http://www.im.pwr.wroc.pl/~rweron/

2

Agenda

1. FX markets and the smile

2. Heston’s model

3. Calibration


FX markets and the smile 3

FX markets

EUR/USD and USD/JPY are two of the most

liquid underlying markets with trading in:

• Spot/forward (ca. 90% of activity, very smallmargins)

• Vanilla options (9%, small margins)

• Exotic options (1%, potentially high margins)



Global markets

USD/JPY market activity



Black-Scholes type formula

• Assumes that asset prices follow GBM:

dSt = St(µdt + σdBt) (1)

• European FX call option price(Garman and Kohlhagen, 1983):

Ct = Ste−rfτΦ(d1)−Ke−rτΦ(d2),

where d1 =log(St/K)+(r−rf+ 12σ

2)τσ√

τ, d2 = d1 − σ

√τ



BS formula is flawed

• Implied volatility σi is the volatility thatequates the BS price:

BS(St, K, r, σi, τ) = Option market price

• Model implied volatilities for different strikesand maturities are not constant

• Volatility smile or smirk/grin is observed



The smile1W, 1M, 3M, 6M, 1Y, 2Y EUR/USD smiles

20 40 60 80

Delta [%]

1010

.511

11.5

Impl

ied

vola

tility

[%

]

STFhes07.xpl

1W (black), 1M (red), 3M (green), 6M (blue), 1Y (cyan), and 2Y(yellow) EUR/USD implied volatility smiles on July 1, 2004


http://www.quantlet.de/codes//STFhes07.html


The smirk/grin

0.6 0.8 1 1.2 1.4 1.60.3

0.4

0.5

0.6

0.7

0.8

0.9

1

moneyness=K/St

σi

ODAX options implied σ’s on Oct. 12, 1998

τ=10 days

τ=38 days

τ=66 days τ=157 days



Correcting the BS formula (1/3)

• Allow the volatility to be a deterministicfunction of time (Merton, 1973): σ = σ(t)

• Explains the different σi levels for differentτ ’s, but cannot explain the smile shape for

different strikes




• Allow not only time, but also statedependence of σ (Dupire, 1994; Derman and

Kani, 1994; Rubinstein, 1994): σ = σ(t, St)

• Lets the local volatility surface to be fitted,but cannot explain the persistent smile shape

which does not vanish as time passes




• Allow the volatility coefficient in the BSdiffusion equation (1) to be random: σ = σt

• Pioneering work of Hull and White (1987),Stein and Stein (1991), and Heston (1993)

led to the development of stochastic volatility

models


Heston’s model 12

Heston’s model

dSt = St

(µ dt +

√vtdW

(1)t

), (2)

dvt = κ(θ − vt) dt + σ√

vtdW(2)t , (3)

dW(1)t dW

(2)t = ρ dt (4)

• Variance process (3) is non-negative andmean-reverting (as observed in the markets)

• It has CIR dynamics


Heston’s model 13

GBM vs. Heston dynamics

0 0.5 1

Time [years]

0.7

0.75

0.8

0.85

0.9

Exc

hang

e ra

te

GBM vs. Heston volatility

0 0.5 1

Time [years]

1520

25

Vol

atili

ty [

%]

STFhes01.xpl

GBM vs. Heston: ρ = −0.05, initial (=GBM) variance v0 = 4%, longterm var. θ = 4%, speed of mean reversion κ = 2, vol of vol σ = 30%



Heston’s model 14

Gaussian vs. Heston densities

-1 -0.5 0 0.5 1

x

00.

51

1.5

2

PDF(

x)

Gaussian vs. Heston log-densities

-1 -0.5 0 0.5 1

x

-10

-50

Log

(PD

F(x)

)STFhes02.xpl

Unlike Gaussian tails, tails of Heston’s marginals are exponential:log-densities resemble hyperbolas (Dragulescu and Yakovenko, 2002)



Heston’s model 15

Option pricing in Heston’s model

• PDE for the option price can be solvedanalytically using the method of characteristic

functions (Heston, 1993)

• Closed-form solution for vanilla options:

h(t) = HestonVanilla(κ, θ, σ, ρ, λ, rd, rf , v0, S0, K, τ)

= e−rfτStP+(φ)−Ke−rdτP−(φ) (5)


Calibration 16

Calibration

1. Look at a time series of historical data:

• Use GMM, SMM, EMM, or BayesianMCMC to fit the price process

• Fit empirical distributions of returns to themarginal distributions

• Cannot estimate the market price of risk λ

2. Calibrate the model to derivative prices or

better to the volatility smile


Calibration 17

Calibration to the smile

• Take the smile of the current vanilla optionsmarket as a given starting point

• Find the optimal set of model parameters fora fixed τ and a given vector of market BS

implied volatilities {σ̂i}ni=1 for a given set ofdelta pillars {∆i}ni=1

• No need to worry about estimating λ as it isalready embedded in the market smile


Calibration 18

3M market and Heston volatilities

20 40 60 80

Delta [%]

10.2

10.4

10.6

10.8

11

Impl

ied

vola

tility

[%

]

6M market and Heston volatilities

20 40 60 80

Delta [%]

10.5

11

Impl

ied

vola

tility

[%

]STFhes06.xpl

EUR/USD volatility surface on July 1, 2004: the fit is very good formaturities between three and eighteen months



Calibration 19

1W market and Heston volatilities

20 40 60 80

Delta [%]

9.5

1010

.511

11.5

Impl

ied

vola

tility

[%

]

2Y market and Heston volatilities

20 40 60 80Delta [%]

10.5

1111

.5Im

plie

d vo

latil

ity [

%]

STFhes06.xpl

Unfortunately, Heston’s model does not perform satisfactorily forshort maturities and extremely long maturities



Calibration 20

Only 3 parameters to fit

Long-run variance and the smile

20 40 60 80

Delta [%]

9.5

1010

.511

11.5

Impl

ied

vola

tility

[%

]

θ(≈ v0) – ATM level of the smileCorrelation and the smile

20 40 60 80

Delta [%]

1010

.511

11.5

Impl

ied

vola

tility

[%

]

ρ – skew (quoted as risk reversals)Vol of vol and the smile

20 40 60 80

Delta [%]

910

1112

Impl

ied

vola

tility

[%

]

σ(≈ κ) – convexity (butterflies)


Calibration 21

Application

1. Calibrate the model to vanilla options

2. Employ it for pricing exotics, like one-touch or

barrier options (finite difference, Monte Carlo)


References 22

References

1. Hakala, J. and Wystup, U. (2002) Heston’s Stochastic

Volatility Model Applied to Foreign Exchange Options,

in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk,

Risk Books.

2. Weron, R. and Wystup, U. (2005)

Heston’s model and the smile, in

P. Cizek, W. Härdle, R. Weron (eds.)

Statistical Tools for Finance

and Insurance, Springer.

http://www.xplore-stat.de/ebooks/ebooks.html

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Statistical Tools for Finance and Insurance

1

P.Čížek W. Härdle R. Weron

1 237 83540 2218909

ISBN 3-540-22189-1

Statistical Tools for Finance and Insurance presents ready-to-use solutions, theoretical developments andmethod construction for many practical problems in quantitative finance and insurance. Written by practi-tioners and leading academics in the field this book offers a unique combination of topics from whichevery market analyst and risk manager will benefit.

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Pavel Čížek is an Assistant Professor of Econometrics at Tilburg University, Tilburg,The Netherlands. He is teaching general econometric courses as well as courses focusedon the analysis of (financial) time series for the Master’s Program in Economics and Econo-metrics. His research interests include, besides theoretical econometrics, nonparametricand computationally intensive methods, primarily with applications to finance.

Wolfgang Härdle is Professor of Statistics at Humboldt-Universität zu Berlin and Directorof CASE – Center for Applied Statistics and Economics. He teaches Quantitative Financeand Semiparametric Statistical methods. His research focuses on dynamic factor models,multivariate statistics in finance and computational statistics. He is an elected ISI memberand advisor to the Guanghua School of Management, Peking University.

Rafał Weron is Assistant Professor of Mathematics at Wrocław University of Technology,Poland. His research focuses on risk management in the power markets and computa-tional statistics as applied to finance and insurance. His professional experience includesdevelopment of insurance strategies for the Polish Power Grid Co. (PSE S.A.) and Hydro-storage Power Plants Co. (ESP S.A.) as well as implementation of option pricing softwarefor LUKAS Bank S.A. (Crédit Agricole Group). He has also been a consultant and executiveteacher to a large number of banks and corporations.

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eron

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Stochastic volatility model of Heston and the smileAgendaFX marketsGlobal marketsBlack-Scholes type formulaBS formula is flawedThe smileThe smirk/grinCorrecting the BS formula (1/3)Correcting the BS formula (2/3)Correcting the BS formula (3/3)Heston's modelOption pricing in Heston's modelCalibrationCalibration to the smileOnly 3 parameters to fitApplicationReferences

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