Numerical evaluation of complex logarithms in the Cox–Ingersoll–Ross model

Abstract

The Cox–Ingersoll–Ross (CIR) model has been a benchmark in finance for many years because of its analytical and structural tractability. The wide applications and extensions of the CIR model requires to evaluate the cumulative distribution function (CDF) of the integrated CIR process in financial modelling. Usually the characteristic function of the integrated CIR process is known analytically and one can use the option pricing method of Carr and Madan to transform it into the corresponding CDF. This characteristic function is defined via complex logarithms which often leads to numerical instabilities when it is integrated using the inverse Fourier transform. Typically, this instability is expected to be apparent for wide ranges of model parameters. In this work, we adapt the recent approach by Kahl and Jäckel for the Heston model to deal with such instability problems. Our new strategy allows to construct a very robust routine to determine numerically a highly accurate CDF of the integrated CIR process for almost any choices of parameters.

Keywords:

• Cox–Ingersoll–Ross model
• interest-rate dynamics
• Fourier inversion methods
• complex logarithm
• oscillatory integral
• Gauss–Lobatto quadrature

2010 AMS Subject Classifications:

• 91G30
• 60E10
• 91G60
• 42B20
• 62P05