Convergence of the Euler–Maruyama Particle Scheme for a Regularised McKean–Vlasov Equation Arising from the Calibration of Local-Stochastic Volatility Models

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.