Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance

TL;DR

This paper develops a Milstein finite difference scheme for SPDEs in finance, demonstrating convergence, stability, and efficiency improvements via multilevel Monte Carlo methods, with applications to credit derivatives.

Contribution

It introduces a new Milstein finite difference scheme for SPDEs, analyzes its convergence and stability, and applies multilevel Monte Carlo to improve computational efficiency in financial modeling.

Findings

First-order convergence in timestep

Second-order convergence in spatial grid size

Multilevel Monte Carlo reduces computational complexity

Abstract

In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.