Quasi-Monte Carlo-Based Conditional Malliavin Method for Continuous-Time Asian Option Greeks

TL;DR

This paper introduces a novel method combining Malliavin calculus and quasi-Monte Carlo techniques to efficiently compute Greeks of continuous-time Asian options, including those with discontinuous payoffs.

Contribution

It develops a new integration by parts formula in multi-dimensional Malliavin calculus and applies conditional quasi-Monte Carlo to improve Greek estimation for continuous-time Asian options.

Findings

The proposed method achieves significant efficiency improvements in Greek calculations.

The Greeks estimates are infinitely differentiable for various Asian options.

The method effectively handles discontinuous payoffs with smooth estimates.

Abstract

Although many methods for computing the Greeks of discrete-time Asian options are proposed, few methods to calculate the Greeks of continuous-time Asian options are known. In this paper, we develop an integration by parts formula in the multi-dimensional Malliavin calculus, and apply it to obtain the Greeks formulae for continuous-time Asian options in the multi-asset situation. We combine the Malliavin method with the quasi-Monte Carlo method to calculate the Greeks in simulation. We discuss the asymptotic convergence of simulation estimates for the continuous-time Asian option Greeks obtained by Malliavin derivatives. We propose to use the conditional quasi-Monte Carlo method to smooth Malliavin Greeks, and show that the calculation of conditional expectations analytically is viable for many types of Asian options. We prove that the new estimates for Greeks have good smoothness. For binary Asian options, Asian call options and up-and-out Asian call options, for instance, our estimates are infinitely times differentiable. We take the gradient principal component analysis method as a dimension reduction technique in simulation. Numerical experiments demonstrate the large efficiency improvement of the proposed method, especially for Asian options with discontinuous payoff functions.