### A martingale control variate method for option pricing with stochastic volatility

Jean-Pierre Fouque, Chuan-Hsiang Han
2007 E S A I M: Probability & Statistics
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
more » ... y 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. Introduction Monte Carlo pricing for options is a popular approach in particular since efficient algorithms have been developed for optimal stopping problems, see for example  . The advantage of Monte Carlo simulations is that it is less sensitive to dimensionality of the pricing problems and suitable for parallel computation; the main disadvantage is that the rate of convergence is limited by the central limit theorem so it is slow. To increase the efficiency besides parallel computing, Quasi Monte Carlo and variance reduction techniques are two possible approaches. We refer to  for an extensive review. Quasi Monte Carlo, unlike pseudo-random number generators, forms a class of methods where low-discrepancy numbers are generated in deterministic ways. Its efficiency heavily relates to the regularity of the option payoffs, which in most cases are poorly posted. The pros are that such an approach can be always implemented regardless to the pricing problems and it is easy to combine with other sampling techniques such as those involving the Brownian bridge. On the other hand, variance reduction methods seek probabilistic ways to reformulate the pricing problem considered in order to gain significant variance reduction. For example control variate methods take into account correlation properties of random variables, and importance sampling methods utilize changes of probability measures. The cons are that the efficiency of these techniques is often restricted to certain pricing problems.