Unconstrained Recursive Importance Sampling

TL;DR

This paper introduces an unconstrained stochastic approximation method for optimal measure change in Monte Carlo simulations, leveraging Robbins-Monro procedures without projections, applicable to multidimensional distributions and diffusion processes.

Contribution

It presents a novel unconstrained Robbins-Monro algorithm for variance reduction in Monte Carlo methods, removing the need for projections or truncations.

Findings

Proven convergence for a wide class of multidimensional distributions.

Effective variance reduction demonstrated in pricing Basket options and barrier options.

Algorithm applicable without assuming payoff smoothness.

Abstract

We propose an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponential-tilting). In a multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro procedure to select the parameter minimizing the variance. 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 intensively use the regularity of the density of the law without assume smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions and diffusion processes. We illustrate the effectiveness of our algorithm via pricing a Basket payoff under a multidimensional NIG distribution, and pricing a barrier options in different markets.