Multidimensional Quasi-Monte Carlo Malliavin Greeks

TL;DR

This paper introduces an advanced Malliavin calculus-based method for efficiently computing Greeks of complex multidimensional path-dependent options, improving variance reduction and computational cost in Monte Carlo simulations.

Contribution

It extends Malliavin calculus formulas to the multidimensional case, enhancing variance reduction techniques and computational efficiency for option Greeks estimation.

Findings

Improved variance reduction in Greek calculations.

Reduced computational cost for multidimensional options.

Effective application to complex Asian and exotic options.

Abstract

We investigate the use of Malliavin calculus in order to calculate the Greeks of multidimensional complex path-dependent options by simulation. For this purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin Calculus approach over different techniques that have been previously proposed. Indeed, these techniques may be computationally expensive and do not provide flexibility for variance reduction. In contrast, the Malliavin approach exhibits a higher flexibility by providing a class of functions that return the same expected value (the Greek) with different accuracies. This versatility for variance reduction is not possible without the use of the generalized integral by part formula of Malliavin Calculus. In the multidimensional context, we find convenient formulas that permit to improve the localization technique, introduced in Fourni\'e et al and reduce both the computational cost and the variance. Moreover, we show that the parameters employed for variance reduction can be obtained \textit{on the flight} in the simulation. We illustrate the efficiency of the proposed procedures, coupled with the enhanced version of Quasi-Monte Carlo simulations as discussed in Sabino, for the numerical estimation of the Deltas of call, digital Asian-style and Exotic basket options with a fixed and a floating strike price in a multidimensional Black-Scholes market.