We propose an unconstrained stochastic approximation method for finding the optimal change of measure (in an a priori parametric family) to reduce the variance of a Monte Carlo simulation. We consider different parametric families based on the Girsanov theorem and the Esscher transform (exponential-tilting). In [Monte Carlo Methods Appl. 10 (2004) 1–24], it described a projected Robbins–Monro procedure to select the parameter minimizing the variance in a multidimensional Gaussian framework. In our approach, the parameter (scalar or process) is selected by a classical Robbins–Monro procedure without projection or truncation. To obtain this unconstrained algorithm, we extensively use the regularity of the density of the law without assuming smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions as well as for diffusion processes.
We illustrate the efficiency of our algorithm on several pricing problems: a Basket payoff under a multidimensional NIG distribution and a barrier options in different markets.
Source: Vincent Lemaire and Gilles Pagès, Ann. Appl. Probab. Volume 20, Number 3 (2010), 1029-1067.