A framework for adaptive Monte-Carlo procedures

TL;DR

This paper introduces a flexible mathematical framework for adaptive Monte Carlo methods, relaxing previous assumptions, and demonstrating convergence and normality, with practical applications in financial derivative valuation.

Contribution

It proposes a new, less restrictive theoretical setting for adaptive importance sampling, including a stochastic algorithm for parameter approximation and real-world financial applications.

Findings

Proved convergence and asymptotic normality of the estimator.

Developed a stochastic algorithm for importance sampling parameter approximation.

Applied the method to financial derivatives valuation examples.

Abstract

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Vazquez-Abad and Dufresne, Fu and Su, and Arouna. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives.