Janek, Agnieszka and Kluge, Tino and Weron, Rafal and Wystup, Uwe (2010): FX Smile in the Heston Model.
There is a more recent version of this item available. 

PDF
MPRA_paper_25491.pdf Download (485Kb)  Preview 
Abstract
The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is nonnegative and meanreverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semianalytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semianalytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.
Item Type:  MPRA Paper 

Original Title:  FX Smile in the Heston Model 
Language:  English 
Keywords:  Heston model; vanilla option; stochastic volatility; Monte Carlo simulation; Feller condition; option pricing with FFT 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C63  Computational Techniques; Simulation Modeling C  Mathematical and Quantitative Methods > C5  Econometric Modeling G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing; Futures Pricing 
Item ID:  25491 
Depositing User:  Rafal Weron 
Date Deposited:  28. Sep 2010 20:17 
Last Modified:  21. Feb 2013 22:15 
References:  Albrecher, H., Mayer, P., Schoutens, W. and Tistaert, J. (2006). The little Heston trap, Wilmott Magazine, January: 83–92. 231–262. Andersen, L. and Andreasen, J. (2000). JumpDiffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing, Review of Derivatives Research 4: 231–262. Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model, The Journal of Computational Finance 11(3), 1–42. Apel, T., Winkler, G., and Wystup, U. (2002). Valuation of options in Heston’s stochastic volatility model using finite element methods, in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk, Risk Books, London. Bakshi, G., Cao, C. and Chen, Z. (1997). Empirical Performance of Alternative Option Pricing Models, Journal of Finance 52: 2003–2049. Bates, D. (1996). Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options, Review of Financial Studies 9: 69–107. Borak, S., Detlefsen, K., and Hardle, W. (2005). FFTbased option pricing, in P. Cizek, W. Hardle, R. Weron (eds.) Statistical Tools for Finance and Insurance, Springer, Berlin. Broto, C., and Ruiz, E. (2004). Estimation methods for stochastic volatility models: A survey, Journal of Economic Surveys 18(5): 613–649. Carr, P. and Madan, D. (1999). Option valuation using the fast Fourier transform, Journal of Computational Finance 2: 61–73. Cont, R., and Tankov, P. (2003). Financial Modelling with Jump Processes, Chapman & Hall/CRC. Derman, E. and Kani, I. (1994). Riding on a Smile, RISK 7(2): 32–39. Dragulescu, A. A. and Yakovenko, V. M. (2002). Probability distribution of returns in the Heston model with stochastic volatility, Quantitative Finance 2: 443–453. Dupire, B. (1994). Pricing with a Smile, RISK 7(1): 18–20. Eberlein, E., Kallsen, J., and Kristen, J. (2003). Risk Management Based on Stochastic Volatility, Journal of Risk 5(2): 19–44. Fengler, M. (2005). Semiparametric Modelling of Implied Volatility, Springer. Fouque, J.P., Papanicolaou, G., and Sircar, K.R. (2000). Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press. Garman, M. B. and Kohlhagen, S. W. (1983). Foreign currency option values, Journal of International Monet & Finance 2: 231–237. Gatheral, J. (2006). The Volatility Surface: A ractitioner’s Guide, Wiley. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering, SpringerVerlag, NewYork. 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, London. Heston, S. (1993). A ClosedForm Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies 6: 327–343. Hull, J. and White, A. (1987). The Pricing of Options with Stochastic Volatilities, Journal of Finance 42: 281–300. Jackel, P. and Kahl, C. (2005). NotSoComplex Logarithms in the Heston Model, Wilmott Magazine 19: 94–103. Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets, Springer. Kluge, T. (2002). Pricing derivatives in stochastic volatility models using the finite difference method, Diploma thesis, Chemnitz Technical University. Lee, R., (2004). Option pricing by transform methods: extensions, unification and error control, Journal of Computational Finance 7(3): 51–86. Martinez, M. and Senge, T. (2002). A JumpDiffusionModel Applied to Foreign Exchange Markets, in J. Hakala, U.Wystup (eds.) Foreign Exchange Risk, Risk Books, London. Merton, R. (1973). The Theory of Rational Option Pricing, Bell Journal of Economics and Management Science 4: 141–183. Merton, R. (1976). Option Pricing when Underlying Stock Returns are Discontinuous, Journal of Financial Economics 3: 125–144. Reiss, O. and Wystup, U. (2001). Computing Option Price Sensitivities Using Homogeneity, Journal of Derivatives 9(2): 41–53. Rogers, L.C.G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales – Volume Two: Ito Calculus, McGrawHill. Rubinstein, M. (1994). Implied Binomial Trees, Journal of Finance 49: 771–818. Rudin, W., (1991). Functional Analysis, McGrawHill. Schoutens, W., Simons, E. and Tistaert, J. (2004). A perfect calibration! Now what? Wilmott March: 66–78. Schmelzle, M., (2010). Option pricing formulae using Fourier transform: Theory and application, Working paper. Stein, E. and Stein, J. (1991). Stock Price Distributions with Stochastic Volatility: An Analytic Approach, Review of Financial Studies 4(4): 727–752. Tompkins, R. G. and D’Ecclesia, R. L. (2006). Unconditional Return Disturbances: A NonParametric Simulation Approach, Journal of Banking and Finance 30(1): 287–314. Weron, R. (2004). Computationally intensive Value at Risk calculations, in J.E. Gentle, W. Hardle, Y. Mori (eds.) Handbook of Computational Statistics, Springer. Weron, R., and Wystup. U. (2005). Heston’s model and the smile, in P. Cizek, W. Hardle, R. Weron (eds.) Statistical Tools for Finance and Insurance, Springer. Wystup, U. (2003). The market price of onetouch options in foreign exchange markets, Derivatives Week, 12(13). Wystup, U. (2006). FX Options and Structured Products, Wiley. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/25491 
Available Versions of this Item
 FX Smile in the Heston Model. (deposited 28. Sep 2010 20:17) [Currently Displayed]