A weak approximation with asymptotic expansion and multidimensional Malliavin weights

TL;DR

This paper introduces a novel approximation scheme for SDE expectations using asymptotic expansion and multidimensional Malliavin weights, validated by theory and numerical experiments, notably improving deep OTM option pricing accuracy.

Contribution

It presents a new method combining asymptotic expansion with Malliavin weights for more accurate expectation approximations of SDE solutions.

Findings

Enhanced accuracy in deep OTM option pricing.

Theoretical validation based on Watanabe and Kusuoka theories.

Numerical experiments confirm scheme effectiveness.

Abstract

This paper develops a new efficient scheme for approximations of expectations of the solutions to stochastic differential equations (SDEs). In particular, we present a method for connecting approximate operators based on an asymptotic expansion with multidimensional Malliavin weights to compute a target expectation value precisely. The mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin calculus. Moreover, numerical experiments for option pricing under local and stochastic volatility models confirm the effectiveness of our scheme. Especially, our weak approximation substantially improves the accuracy at deep Out-of-The-Moneys (OTMs).