American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods

TL;DR

This paper introduces a novel Monte Carlo approach for American option pricing using Malliavin calculus without localization functions, combined with nonparametric variance reduction techniques and dynamic programming for bias reduction.

Contribution

It develops Malliavin calculus tools for exponential multi-dimensional diffusions with deterministic coefficients and applies nonparametric variance reduction methods, avoiding localization functions.

Findings

Effective variance reduction without localization functions

Numerical efficiency demonstrated on multi-core CPU/GPU systems

Improved accuracy in American option pricing

Abstract

This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f(St)/Ss] using the Malliavin calculus without localization. Then the variance of the estimator of E[f(St)/Ss] is reduced using closed formulas, techniques based on a conditioning and a judicious choice of the number of simulated paths. Finally, we perform the stopping times version of the dynamic programming algorithm to decrease the bias. On the one hand, we will develop the Malliavin calculus tools for exponential multi-dimensional diffusions that have deterministic and no constant coefficients. On the other hand, we will detail various nonparametric technics to reduce the variance. Moreover, we will test the numerical efficiency of our method on a heterogeneous CPU/GPU multi-core machine.