Stochastic volatility model of Heston and the smile

Transcription

1 Stochastic volatility model of Heston and the smile Rafa l Weron Hugo Steinhaus Center Wroc law University of Technology Poland In collaboration with: Piotr Uniejewski (LUKAS Bank) Uwe Wystup (Commerzbank Securities)

2 2 Agenda 1. FX markets and the smile 2. Heston s model 3. Calibration

3 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 small margins) Vanilla options (9%, small margins) Exotic options (1%, potentially high margins)

4 FX markets and the smile 4 Global markets USD/JPY market activity

5 FX markets and the smile 5 Black-Scholes type formula Assumes that asset prices follow GBM: ds t = S t (µdt + σdb t ) (1) European FX call option price (Garman and Kohlhagen, 1983): C t = S t e r f τ Φ(d 1 ) Ke rτ Φ(d 2 ), where d 1 = log(s t/k)+(r r f σ2 )τ σ τ, d 2 = d 1 σ τ

6 FX markets and the smile 6 BS formula is flawed Implied volatility σ i is the volatility that equates the BS price: BS(S t, K, r, σ i, τ) = Option market price Model implied volatilities for different strikes and maturities are not constant Volatility smile or smirk/grin is observed

7 FX markets and the smile 7 The smile 1W, 1M, 3M, 6M, 1Y, 2Y EUR/USD smiles Implied volatility [%] Delta [%] STFhes07.xpl 1W (black), 1M (red), 3M (green), 6M (blue), 1Y (cyan), and 2Y (yellow) EUR/USD implied volatility smiles on July 1, 2004

8 FX markets and the smile 8 The smirk/grin ODAX options implied σ s on Oct. 12, 1998 τ=10 days σ i τ=157 days τ=66 days τ=38 days moneyness=k/s t

9 FX markets and the smile 9 Correcting the BS formula (1/3) Allow the volatility to be a deterministic function of time (Merton, 1973): σ = σ(t) Explains the different σ i levels for different τ s, but cannot explain the smile shape for different strikes

10 FX markets and the smile 10 Correcting the BS formula (2/3) Allow not only time, but also state dependence of σ (Dupire, 1994; Derman and Kani, 1994; Rubinstein, 1994): σ = σ(t, S t ) Lets the local volatility surface to be fitted, but cannot explain the persistent smile shape which does not vanish as time passes

11 FX markets and the smile 11 Correcting the BS formula (3/3) Allow the volatility coefficient in the BS diffusion equation (1) to be random: σ = σ t Pioneering work of Hull and White (1987), Stein and Stein (1991), and Heston (1993) led to the development of stochastic volatility models

12 Heston s model 12 Heston s model ( ds t = S t µ dt + ) v t dw (1) t, (2) dv t = κ(θ v t ) dt + σ v t dw (2) t, (3) dw (1) t dw (2) t = ρ dt (4) Variance process (3) is non-negative and mean-reverting (as observed in the markets) It has CIR dynamics

13 Heston s model 13 GBM vs. Heston dynamics GBM vs. Heston volatility Exchange rate Volatility [%] Time [years] Time [years] STFhes01.xpl GBM vs. Heston: ρ = 0.05, initial (=GBM) variance v 0 = 4%, long term var. θ = 4%, speed of mean reversion κ = 2, vol of vol σ = 30%

14 Heston s model 14 Gaussian vs. Heston densities Gaussian vs. Heston log-densities PDF(x) Log(PDF(x)) x x STFhes02.xpl Unlike Gaussian tails, tails of Heston s marginals are exponential: log-densities resemble hyperbolas (Dragulescu and Yakovenko, 2002)

15 Heston s model 15 Option pricing in Heston s model PDE for the option price can be solved analytically using the method of characteristic functions (Heston, 1993) Closed-form solution for vanilla options: h(t) = HestonVanilla(κ, θ, σ, ρ, λ, r d, r f, v 0, S 0, K, τ) = e r f τ S t P + (φ) Ke rdτ P (φ) (5)

16 Calibration 16 Calibration 1. Look at a time series of historical data: Use GMM, SMM, EMM, or Bayesian MCMC to fit the price process Fit empirical distributions of returns to the marginal distributions Cannot estimate the market price of risk λ 2. Calibrate the model to derivative prices or better to the volatility smile

17 Calibration 17 Calibration to the smile Take the smile of the current vanilla options market as a given starting point Find the optimal set of model parameters for a fixed τ and a given vector of market BS implied volatilities {ˆσ i } n i=1 for a given set of delta pillars { i } n i=1 No need to worry about estimating λ as it is already embedded in the market smile

18 Calibration 18 3M market and Heston volatilities 6M market and Heston volatilities Implied volatility [%] Implied volatility [%] Delta [%] Delta [%] STFhes06.xpl EUR/USD volatility surface on July 1, 2004: the fit is very good for maturities between three and eighteen months

19 Calibration 19 1W market and Heston volatilities 2Y market and Heston volatilities Implied volatility [%] Delta [%] Implied volatility [%] Delta [%] STFhes06.xpl Unfortunately, Heston s model does not perform satisfactorily for short maturities and extremely long maturities

20 Calibration 20 Only 3 parameters to fit Long-run variance and the smile Implied volatility [%] Delta [%] θ( v 0 ) ATM level of the smile Correlation and the smile Implied volatility [%] Delta [%] ρ skew (quoted as risk reversals) Vol of vol and the smile Implied volatility [%] Delta [%] σ( κ) convexity (butterflies)

21 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)

22 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 Cizek U :28 Uhr Seite 1 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. Statistical Tools for Finance and Insurance presents ready-to-use solutions, theoretical developments and method construction for many practical problems in quantitative finance and insurance. Written by practitioners and leading academics in the field this book offers a unique combination of topics from which every market analyst and risk manager will benefit. Features of the book: Offers insight into new methods and the applicability of the stochastic technology Provides the tools,instruments and (online) algorithms for recent techniques in quantitative finance and modern treatments in insurance calculations Covers topics such as heavy tailed distributions,implied trinomial trees,support vector machines,valuation of mortgage-backed securities, pricing of CAT bonds, simulation of risk processes, and ruin probability approximation Presents extensive examples The downloadable electronic edition of the book offers interactive tools This book presents modern tools for quantitative analysis in finance and insurance. It provides a smooth introduction into advanced techniques applicable to a wide range of practical problems. The fact that all examples can be reproduced by the XploRe Quantlet Server technique makes it a sure buy for both practioners and theoretical analysts. Prof. Dr. Helmut Gründl, Dr. Wolfgang Schieren Chair for Insurance and Risk Management, sponsored by Allianz AG and Stifterverband für die Deutsche Wissenschaft springeronline.com 9 ISBN Pavel Čížek is an Assistant Professor of Econometrics at Tilburg University, Tilburg, The Netherlands. He is teaching general econometric courses as well as courses focused on the analysis of (financial) time series for the Master s Program in Economics and Econometrics. His research interests include, besides theoretical econometrics, nonparametric and computationally intensive methods, primarily with applications to finance. Wolfgang Härdle is Professor of Statistics at Humboldt-Universität zu Berlin and Director of CASE Center for Applied Statistics and Economics. He teaches Quantitative Finance and Semiparametric Statistical methods. His research focuses on dynamic factor models, multivariate statistics in finance and computational statistics. He is an elected ISI member and 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 computational statistics as applied to finance and insurance. His professional experience includes development of insurance strategies for the Polish Power Grid Co. (PSE S.A.) and Hydrostorage Power Plants Co. (ESP S.A.) as well as implementation of option pricing software for LUKAS Bank S.A. (Crédit Agricole Group). He has also been a consultant and executive teacher to a large number of banks and corporations. Čížek Härdle Weron 1 Statistical Tools for Finance and Insurance P. Čížek W. Härdle R.Weron Statistical Tools for Finance and Insurance Mail: mdtech@mdtech.de Home: C M Y K designandproduction GmbH Bender Dieser Farbausdruck/pdf-file kann nur annähernd das endgültige Druckergebnis wiedergeben!