High order approximations and simulation schemes for the log-Heston process

TL;DR

This paper develops high-order weak approximation schemes for the log-Heston process using a method that combines schemes on different random grids, validated through numerical experiments on various financial models.

Contribution

It introduces a novel approach to achieve arbitrary order convergence for the log-Heston process by combining existing schemes on random grids.

Findings

Achieves any order of convergence with the proposed method.

Numerical validation confirms theoretical convergence rates.

Demonstrates applicability to multifactor and rough Heston models.

Abstract

We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this method with either the Ninomiya-Victoir scheme (2008) or a second-order scheme that samples exactly the volatility component, and we show rigorously that we can achieve then any order of convergence. We give numerical illustrations on financial examples that validate the theoretical order of convergence. We also present promising numerical results for the multifactor/rough Heston model and hint at applications to other models, including the Bates model and the double Heston model.