A martingale control variate method for option pricing with stochastic volatility

Jean-Pierre Fouque; Chuan-Hsiang Han

ESAIM: Probability and Statistics (2007)

  • Volume: 11, page 40-54
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Fouque, Jean-Pierre, and Han, Chuan-Hsiang. "A martingale control variate method for option pricing with stochastic volatility." ESAIM: Probability and Statistics 11 (2007): 40-54. <http://eudml.org/doc/250129>.

@article{Fouque2007,
abstract = { 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. },
author = {Fouque, Jean-Pierre, Han, Chuan-Hsiang},
journal = {ESAIM: Probability and Statistics},
keywords = {Option pricing; Monte Carlo; control variates; stochastic volatility; multiscale asymptotics.; option pricing; multiscale asymptotics},
language = {eng},
month = {3},
pages = {40-54},
publisher = {EDP Sciences},
title = {A martingale control variate method for option pricing with stochastic volatility},
url = {http://eudml.org/doc/250129},
volume = {11},
year = {2007},
}

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
DA - 2007/3//
PB - EDP Sciences
VL - 11
SP - 40
EP - 54
AB - 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.
LA - eng
KW - Option pricing; Monte Carlo; control variates; stochastic volatility; multiscale asymptotics.; option pricing; multiscale asymptotics
UR - http://eudml.org/doc/250129
ER -

References

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  1. G. Barone-Adesi and R.E. Whaley, Efficient Analytic Approximation of American Option Values. J. Finance42 (1987) 301–320.  
  2. R. Bellman, Stability Theory of Differential Equations. McGraw-Hill (1953).  
  3. E. Clement, D. Lamberton, P. Protter, An Analysis of a Least Square Regression Method for American Option Pricing. Finance and Stochastics6 (2002) 449–471.  
  4. J.-P. Fouque and C.-H. Han, A Control Variate Method to Evaluate Option Prices under Multi-Factor Stochastic Volatility Models, submitted, 2004.  
  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 Finance4 (2004) 597–606.  
  6. J.-P. Fouque, G. Papanicolaou and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press (2000).  
  7. J.-P. Fouque, R. Sircar and K. Solna, Stochastic Volatility Effects on Defaultable Bonds. Appl. Math. Finance13 (2006) 215–244.  
  8. J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility Asymptotics. SIAM J. Multiscale Modeling and Simulation2 (2003) 22–42.  
  9. P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer Verlag (2003).  
  10. F. Longstaff and E. Schwartz, Valuing American Options by Simulation: A Simple Least-Squares Approach. Rev. Financial Studies14 (2001) 113–147.  
  11. B. Oksendal, Stochastic Differential Equations: An introduction with Applications. Universitext, 5th ed., Springer (1998).  
  12. P. Wilmott , S. Howison and J. Dewynne, Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press (1995).  

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