Evaluating the Noncentral Chi-Square Distribution for the Cox-Ingersoll-Ross Process

Abstract

The conditional distribution of the short rate in the Cox-Ingersoll-Ross process can be expressed in terms of the noncentral χ2-distribution. The three standard methods for evaluating this function are by its representation in terms of a series of gamma functions, by analytic approximation, and by its asymptotic expansion. We perform numerical tests of these methods over parameter ranges typical for the Cox-Ingersoll-Ross process. We find that the gamma series representation is accurate over a wide range of parameters but has a runtime that increases proportional to the square root of the noncentrality parameter. Analytic approximations and the asymptotic expansion run quickly but have an accuracy that varies significantly over parameter space. We develop a fourth method for evaluating the upper and lower tails of the noncentral χ2-distribution based on a Bessel function series representation. We find that the Bessel method is accurate over a wide range of parameters and has a runtime that is insensitive to increases in the noncentrality parameter. We show that by using all four methods it is possible to efficiently evaluate the noncentral χ2-distribution to a relative precision of six significant figures.