Convergence of the Euler--Maruyama particle scheme for a regularised McKean--Vlasov equation arising from the calibration of local-stochastic volatility models

TL;DR

This paper proves the strong convergence of an Euler--Maruyama particle scheme for a regularised McKean--Vlasov equation in local-stochastic volatility models, with practical calibration applications.

Contribution

It establishes well-posedness and convergence rates for a regularised particle scheme in McKean--Vlasov dynamics, addressing open questions in model calibration.

Findings

Convergence rate of 1/2 for the Euler--Maruyama scheme.

Explicit error dependence on regularisation parameters.

Successful implementation for Heston-type models.

Abstract

In this paper, we study the Euler--Maruyama scheme for a particle method to approximate the McKean--Vlasov dynamics of calibrated local-stochastic volatility (LSV) models. Given the open question of well-posedness of the original problem, we work with regularised coefficients and prove that under certain assumptions on the inputs, the regularised model is well-posed. Using this result, we prove the strong convergence of the Euler--Maruyama scheme to the particle system with rate 1/2 in the step-size and obtain an explicit dependence of the error on the regularisation parameters. Finally, we implement the particle method for the calibration of a Heston-type LSV model to illustrate the convergence in practice and to investigate how the choice of regularisation parameters affects the accuracy of the calibration.