Fouque, Jean-Pierre; Han, Chuan-Hsiang
A martingale control variate method for option pricing with stochastic volatility
ESAIM: Probability and Statistics, Tome 11 (2007) , p. 40-54
Zbl pre05216869 | MR 2299646
doi : 10.1051/ps:2007005
URL stable :

Classification:  65C05,  62P05
A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singular and regular perturbation analysis to characterize the variance reduced by martingale control variates. This variance analysis is done in the regime where time scales of associated driving volatility processes are well separated. Numerical results for European, Barrier, and American options are presented to illustrate the effectiveness and robustness of this martingale control variate method in regimes where these time scales are not so well separated.


[1] G. Barone-Adesi and R.E. Whaley, Efficient Analytic Approximation of American Option Values. J. Finance 42 (1987) 301-320.

[2] R. Bellman, Stability Theory of Differential Equations. McGraw-Hill (1953). MR 61235 | Zbl 0053.24705

[3] E. Clement, D. Lamberton, P. Protter, An Analysis of a Least Square Regression Method for American Option Pricing. Finance and Stochastics 6 (2002) 449-471. Zbl 1039.91020

[4] J.-P. Fouque and C.-H. Han, A Control Variate Method to Evaluate Option Prices under Multi-Factor Stochastic Volatility Models, submitted, 2004. MR 2241299

[5] J.-P. Fouque and C.-H. Han, Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models. Quantitative Finance 4 (2004) 597-606.

[6] J.-P. Fouque, G. Papanicolaou and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press (2000). MR 1768877 | Zbl 0954.91025

[7] J.-P. Fouque, R. Sircar and K. Solna, Stochastic Volatility Effects on Defaultable Bonds. Appl. Math. Finance 13 (2006) 215-244. Zbl pre05127626

[8] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility Asymptotics. SIAM J. Multiscale Modeling and Simulation 2 (2003) 22-42. Zbl 1074.91015

[9] P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer Verlag (2003). MR 1999614 | Zbl 1038.91045

[10] F. Longstaff and E. Schwartz, Valuing American Options by Simulation: A Simple Least-Squares Approach. Rev. Financial Studies 14 (2001) 113-147.

[11] B. Oksendal, Stochastic Differential Equations: An introduction with Applications. Universitext, 5th ed., Springer (1998). MR 1619188 | Zbl 0897.60056

[12] P. Wilmott , S. Howison and J. Dewynne, Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press (1995). MR 1357666 | Zbl 0842.90008