An explicit positivity-preserving scheme for the Heston 3/2-model with order-one strong convergence

Abstract

This article is concerned with numerical approximations of 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, which can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size $h>0$. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is highly non-trivial, by noting that the diffusion coefficient grows super-linearly. The above theoretical results can be then used to justify the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model. Indeed, the unconditional positivity-preserving property is particularly desirable in the MLMC setting, where large discretization time steps are used. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity $O\left({ϵ}^{-2}\right)$ for the MLMC approach, where $ϵ>0$ is the required target accuracy. Numerical experiments are finally reported to confirm the above results.