Simulation of square-root processes made simple: applications to the Heston model

TL;DR

This paper presents a new numerical scheme for simulating square-root processes, especially useful for the Heston model, offering high accuracy, efficiency, and the ability to reproduce exact distributions with minimal steps.

Contribution

The paper introduces a novel approach that simulates the integrated process first, enabling exact distribution recovery and improved simulation efficiency for square-root processes.

Findings

High precision with few time steps in numerical experiments

Exact inverse Gaussian distribution achieved with one step in certain regimes

Effective for realistic parameters in the Heston model

Abstract

We introduce a simple, efficient and accurate nonnegative preserving numerical scheme for simulating the square-root process. The novel idea is to simulate the integrated square-root process first instead of the square-root process itself. Numerical experiments on realistic parameter sets, applied for the integrated process and the Heston model, display high precision with a very low number of time steps. As a bonus, our scheme yields the exact limiting Inverse Gaussian distributions of the integrated square-root process with only one single time-step in two scenarios: (i) for high mean-reversion and volatility-of-volatility regimes, regardless of maturity; and (ii) for long maturities, independent of the other parameters.