Multilevel Monte Carlo methods for positivity-preserving approximations of the Heston 3/2-model

Abstract

This article is concerned with the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model from mathematical finance, which takes values in $(0, \infty)$ and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed to produce independent sample paths. The proposed scheme can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size $h>0$. This positivity-preserving property for large discretization time steps is particularly desirable in the MLMC setting. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is not trivial, as the diffusion coefficient grows super-linearly. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity $\mathcal{O}(\epsilon^{-2})$ for the MLMC approach, where $\epsilon > 0$ is the required target accuracy. Numerical experiments are finally reported to confirm the theoretical findings.