Chi-square simulation of the CIR process and the Heston model

TL;DR

This paper introduces new methods for simulating chi-square and generalized Gaussian distributions, improving accuracy and efficiency, and applies these techniques to variance sampling in the Heston model, especially for small degrees of freedom.

Contribution

It provides novel representations and sampling algorithms for chi-square and generalized Gaussian distributions, enhancing simulation accuracy and robustness in financial models.

Findings

Marsaglia's polar method extends to generalized Gaussian distributions.

The inverse CDF approximation achieves tenth decimal place accuracy.

Methods are effective for small degrees of freedom in the Heston model.

Abstract

The transition probability of a Cox-Ingersoll-Ross process can be represented by a non-central chi-square density. First we prove a new representation for the central chi-square density based on sums of powers of generalized Gaussian random variables. Second we prove Marsaglia's polar method extends to this distribution, providing a simple, exact, robust and efficient acceptance-rejection method for generalized Gaussian sampling and thus central chi-square sampling. Third we derive a simple, high-accuracy, robust and efficient direct inversion method for generalized Gaussian sampling based on the Beasley-Springer-Moro method. Indeed the accuracy of the approximation to the inverse cumulative distribution function is to the tenth decimal place. We then apply our methods to non-central chi-square variance sampling in the Heston model. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. Using the additivity property of the chi-square distribution, our methods apply in all parameter regimes.