Importance Sampling and Mm-Algorithms with Applications to Options Pricing

Abstract

In this paper we study the problem of finding unbiased variance reducing Monte Carlo estimators. An optimal estimator is characterized by a minimization problem. Our novel approach to solve this problem is based on maximization/minimization algorithms (MM-algorithms). First, we prove a general global convergence result for this type of algorithm. Then we construct a specific MM-algorithm for the minimization problem associated with variance reducing Monte Carlo estimators. In general, it is not possible to evaluate the objective function. Therefore, we construct large sample approximations of the objective function and associate minimization problems with these approximations. The minimization problems determine M-estimators which we use to approximate the minimum point of the original minimization problem. We prove consistency and asymptotic normality of these M-estimators. Furthermore, we modify the MM-algorithm of the original minimization problem to obtain a MM-algorithm for calculating approximations to the M-estimators. These approximations are approximations to optimal unbiased variance reducing Monte Carlo estimators. We perform some numerical experiments in the context of derivatives pricing which show that computing these optimal estimators by a MM-algorithm is efficient.

Keywords: Monte Carlo Methods, Importance Sampling, Majorization/Minimization-Algorithm, Surrogate-Function, M-Estimators, Financial Derivatives

JEL Classification: G13