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.
Mots clés : option pricing, Monte Carlo, control variates, stochastic volatility, multiscale asymptotics
@article{PS_2007__11__40_0,
author = {Fouque, Jean-Pierre and Han, Chuan-Hsiang},
title = {A martingale control variate method for option pricing with stochastic volatility},
journal = {ESAIM: Probability and Statistics},
pages = {40--54},
publisher = {EDP-Sciences},
volume = {11},
year = {2007},
doi = {10.1051/ps:2007005},
mrnumber = {2299646},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2007005/}
}
TY - JOUR AU - Fouque, Jean-Pierre AU - Han, Chuan-Hsiang TI - A martingale control variate method for option pricing with stochastic volatility JO - ESAIM: Probability and Statistics PY - 2007 SP - 40 EP - 54 VL - 11 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007005/ DO - 10.1051/ps:2007005 LA - en ID - PS_2007__11__40_0 ER -
%0 Journal Article %A Fouque, Jean-Pierre %A Han, Chuan-Hsiang %T A martingale control variate method for option pricing with stochastic volatility %J ESAIM: Probability and Statistics %D 2007 %P 40-54 %V 11 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007005/ %R 10.1051/ps:2007005 %G en %F PS_2007__11__40_0
Fouque, Jean-Pierre; Han, Chuan-Hsiang. A martingale control variate method for option pricing with stochastic volatility. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 40-54. doi : 10.1051/ps:2007005. http://www.numdam.org/articles/10.1051/ps:2007005/
[1] and, Efficient Analytic Approximation of American Option Values. J. Finance 42 (1987) 301-320.
[2] , Stability Theory of Differential Equations. McGraw-Hill (1953). | MR | Zbl
[3] ,,, An Analysis of a Least Square Regression Method for American Option Pricing. Finance and Stochastics 6 (2002) 449-471. | Zbl
[4] and, A Control Variate Method to Evaluate Option Prices under Multi-Factor Stochastic Volatility Models, submitted, 2004. | MR
[5] and, Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models. Quantitative Finance 4 (2004) 597-606.
[6] , and, Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press (2000). | MR | Zbl
[7] , and, Stochastic Volatility Effects on Defaultable Bonds. Appl. Math. Finance 13 (2006) 215-244. | Zbl
[8] ,, and, Multiscale Stochastic Volatility Asymptotics. SIAM J. Multiscale Modeling and Simulation 2 (2003) 22-42. | Zbl
[9] , Monte Carlo Methods in Financial Engineering. Springer Verlag (2003). | MR | Zbl
[10] and, Valuing American Options by Simulation: A Simple Least-Squares Approach. Rev. Financial Studies 14 (2001) 113-147.
[11] , Stochastic Differential Equations: An introduction with Applications. Universitext, 5th ed., Springer (1998). | MR | Zbl
[12] , and, Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press (1995). | MR | Zbl
Cité par Sources :
