  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.
