Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

TL;DR

This paper analyzes the efficiency of multilevel Monte Carlo path simulation using the Milstein discretisation, demonstrating improved variance convergence and computational benefits over Euler-Maruyama, supported by numerical results for basket options.

Contribution

It provides a theoretical analysis of multilevel Monte Carlo with Milstein discretisation, showing improved variance convergence and computational efficiency.

Findings

Milstein discretisation improves variance convergence in multilevel Monte Carlo.

Theoretical analysis confirms enhanced efficiency over Euler-Maruyama.

Numerical results validate the theoretical improvements for basket options.

Abstract

The multilevel Monte Carlo path simulation method introduced by Giles ({\it Operations Research}, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.