Nonnegative Simulation of Square-Root Processes for Heston Models

Summary

This paper presents a numerical method for simulating square-root processes while preserving nonnegativity. Its central step is to simulate the integrated process first, using that quantity to construct a scheme for the process itself. The method is applied to the integrated square-root process and to the Heston model, which uses a square-root process for stochastic variance.

Numerical experiments on realistic parameter sets are reported to show high precision with few time steps. The paper also identifies two settings in which a single time step gives the limiting Inverse Gaussian distribution of the integrated process: high mean reversion and volatility of volatility at any maturity, and long maturities irrespective of other parameters. These are stated as results for the specified limiting scenarios, not a general guarantee of accuracy for every parameter choice or simulation task. The document provides no specific error measures or comparisons with competing schemes, so the strength of the reported numerical advantage cannot be assessed from this summary alone.

Key ideas

• The proposed scheme is designed to preserve nonnegativity when simulating square-root processes.
• It simulates the integrated process first as the basis for constructing the simulation method.
• The method is tested on the integrated process and on the Heston model.
• Numerical experiments report high precision with few time steps.
• The limiting Inverse Gaussian distribution is obtained in specified regimes using one time step.

Tags

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


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









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.