Approximating explicitly the mean reverting CEV process

TL;DR

This paper develops explicit numerical schemes for mean reverting CEV processes in financial models, ensuring positivity and convergence, and extends the approach to two-dimensional stochastic volatility models.

Contribution

It introduces a positivity-preserving semi-discrete scheme for mean reverting CEV processes with proven convergence rates, extending to 2D stochastic volatility models.

Findings

The proposed scheme converges with a rate depending on q.

Numerical experiments confirm the scheme's effectiveness.

Comparison shows advantages over existing positivity-preserving methods.

Abstract

In this paper we want to exploit further the semi-discrete method appeared in Halidias and Stamatiou (2015). We are interested in the numerical solution of mean reverting CEV processes that appear in financial mathematics models and are described as non negative solutions of certain stochastic differential equations with sub-linear diffusion coefficients of the form $(x_t)^q,$ where $\frac{1}{2}<q<1.$ Our goal is to construct explicit numerical schemes that preserve positivity. We prove convergence of the proposed SD scheme with rate depending on the parameter $q.$ Furthermore, we verify our findings through numerical experiments and compare with other positivity preserving schemes. Finally, we show how to treat the whole two-dimensional stochastic volatility model, with instantaneous variance process given by the above mean reverting CEV process.