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

The conditional distribution of the short rate in the Cox-Ingersoll-Rossprocess can be expressed in terms of the noncentralχ2-distribution.The three standard methods for evaluating this function are by its representation in terms of a series of gamma functions, by analyticapproximation, and by its asymptotic expansion. We perform numerical tests of these methods over parameter ranges typicalfor the Cox-Ingersoll-Ross process. We find that the gamma series representation is accurate over a wide range of parameters but has a runtimethat increases proportional to the square root of the noncentrality parameter.Analytic approximations and the asymptotic expansion run quickly but havean accuracy that varies significantly over parameter space.We develop a fourth method for evaluatingthe upper and lower tails of the noncentral χ2-distributionbased on aBessel function series representation. We find that the Bessel method is accurate over a wide range of parametersand has a runtime that is insensitive to increases in the noncentralityparameter.We show that by using all four methods it is possible to efficiently evaluate the noncentral χ2-distribution to a relative precisionof six significant figures.