Abstract
Pricing bond options under the Cox, Ingersoll and Ross (CIR) model of the term structure of interest rates requires the computation of the noncentral chi-square distribution function. In this article, we compare the performance in terms of accuracy and computational time of alternative methods for computing such probability distributions against an externally tested benchmark. All methods are generally accurate over a wide range of parameters that are frequently needed for pricing bond options, though they all present relevant differences in terms of running times. The iterative procedure of Benton and Krishnamoorthy (Comput. Stat. Data Anal. 43:249–267, 2003) is the most efficient in terms of accuracy and computational burden for determining bond option prices under the CIR assumption.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As an example, consider a 10-year 6 % bond with a face amount of 100. In this case, N = 20 since the bond makes 19 semiannual coupon payments of 3 as well as a final payment of 103. That is, a i = 3, i = 1, 2, ⋯ , 19, a 20 = 3 + 100 = 103, and \(s_{1} = 0.5,s_{2} = 1,\cdots \,,s_{19} = 9.5,s_{20} = 10\).
- 2.
We obtained these probabilities by computing the values of \(F(x_{1};\nu,b_{1})\) for this set of parameters.
- 3.
This means that care must be taken if one wants to use the CDF built-in-function of Mathematica for computing the noncentral chi-square distribution function.
- 4.
The CPU time for the gamma series method is 3,303.23 s for probabilities, 3,340.48 s for zero-coupon bond options, and 10,714.40 for coupon bond options.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
Benton, D., Krishnamoorthy, K.: Compute discrete mixtures of continuous distributions: noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient. Comput. Stat. Data Anal. 43, 249–267 (2003)
Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rate. Econometrica 53(2), 385–408 (1985)
Ding, C.G.: Algorithm AS 4: computing the non-central χ 2 distribution function. Appl. Stat. 41, 478–482 (1992)
Jamshidian, F.: An exact bond option formula. J. Finance 44(1), 205–209 (1989)
Knüsel, L., Bablock, B.: Computation of the noncentral gamma distribution. SIAM J. Sci. Comput. 17, 1224–1231 (1996)
Schroder, M.: Computing the constant elasticity of variance option pricing formula. J. Finance 44(1), 211–219 (1989)
Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)
Acknowledgments
Dias is member of the BRU-UNIDE and Larguinho and Braumann are members of the Research Center Centro de Investigação em Matemática e Aplicações (CIMA), both centers financed by the Fundação para a Ciência e Tecnologia (FCT). Dias gratefully acknowledges the financial support from the FCTs grant number PTDC/EGE-ECO/099255/2008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Larguinho, M., Dias, J.C., Braumann, C.A. (2014). Valuation of Bond Options Under the CIR Model: Some Computational Remarks. In: Pacheco, A., Santos, R., Oliveira, M., Paulino, C. (eds) New Advances in Statistical Modeling and Applications. Studies in Theoretical and Applied Statistics(). Springer, Cham. https://doi.org/10.1007/978-3-319-05323-3_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-05323-3_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05322-6
Online ISBN: 978-3-319-05323-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)