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

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  • S. Dyrting

Abstract

The conditional distribution of the short rate in the Cox-Ingersoll-Ross process can be expressed in terms of the noncentral χ-super-2-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 χ-super-2-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 χ-super-2-distribution to a relative precision of six significant figures.

Suggested Citation

  • S. Dyrting, 2004. "Evaluating the Noncentral Chi-Square Distribution for the Cox-Ingersoll-Ross Process," Computational Economics, Springer;Society for Computational Economics, vol. 24(1), pages 35-50, August.
    • Handle: RePEc:kap:compec:v:24:y:2004:i:1:p:35-50
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    Cited by:

    1. Yang, Nian & Chen, Nan & Liu, Yanchu & Wan, Xiangwei, 2017. "Approximate arbitrage-free option pricing under the SABR model," Journal of Economic Dynamics and Control, Elsevier, vol. 83(C), pages 198-214.
    2. Padmakumari, Lakshmi & S., Maheswaran, 2017. "A new statistic to capture the level dependence in stock price volatility," The Quarterly Review of Economics and Finance, Elsevier, vol. 65(C), pages 355-362.
    3. Shane Miller, 2007. "Pricing of Contingent Claims Under the Real-World Measure," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 25, July-Dece.
    4. Alan Genaro & Adilson Simonis, 2015. "Estimating doubly stochastic Poisson process with affine intensities by Kalman filter," Statistical Papers, Springer, vol. 56(3), pages 723-748, August.
    5. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 19, July-Dece.
    6. Peter Carr & Vadim Linetsky, 2006. "A jump to default extended CEV model: an application of Bessel processes," Finance and Stochastics, Springer, vol. 10(3), pages 303-330, September.

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