Numerical analysis of a particle system for the calibrated Heston-type local stochastic volatility model

Abstract

We analyse a Monte Carlo particle method for the simulation of the calibrated Heston-type local stochastic volatility (H-LSV) model. The common application of a kernel estimator for a conditional expectation in the calibration condition results in a McKean-Vlasov (MV) stochastic differential equation (SDE) with non-standard coefficients. The primary challenges lie in certain mean-field terms in the drift and diffusion coefficients and the $1/2$-H\"{o}lder regularity of the diffusion coefficient. We establish the well-posedness of this equation for a fixed but arbitrarily small bandwidth of the kernel estimator. Moreover, we prove a strong propagation of chaos result, ensuring convergence of the particle system under a condition on the Feller ratio and up to a critical time. For the numerical simulation, we employ an Euler-Maruyama scheme for the log-spot process and a full truncation Euler scheme for the CIR volatility process. Under certain conditions on the inputs and the Feller ratio, we prove strong convergence of the Euler-Maruyama scheme with rate $1/2$ in time, up to a logarithmic factor. Numerical experiments illustrate the convergence of the discretisation scheme and validate the propagation of chaos in practice.