Abstract
It is the aim of Chap. 13 to introduce computational tools, which can be used to implement the functionals presented in this book. The first part of the chapter focuses on the non-central chi-squared distribution, which had arisen in the context of pricing financial derivatives in the Minimal Market Model introduced in Chap. 3. We provide both theoretical results and also a stable algorithm which can be used to compute the distribution function. In the second part of the chapter we focus on the non-central beta distribution, which had arisen in the context of pricing exchange options in the Minimal Market Model. Again, we provide both theoretical results but also a stable algorithm which can be used to compute the distribution function. The chapter concludes by discussing the inversion of Laplace transforms, which can be used to recover transition densities from the Laplace transforms presented throughout this book. We illustrate this approach in the context of the Minimal Market Model presented in Chap. 3.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abate, J., Whitt, W.: Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7(1), 36–43 (1995)
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)
Chattamvelli, R.: A note on the noncentral beta distribution function. Am. Stat. 49(2), 231–234 (1995)
Craddock, M., Heath, D., Platen, E.: Numerical inversion of Laplace transforms: a survey with applications to derivative pricing. J. Comput. Finance 4(1), 57–81 (2000)
Ding, C.G.: Algorithm AS 275: computing the non-central χ 2 distribution function. Appl. Stat. 41(2), 478–482 (1992)
Dyrting, S.: Evaluating the noncentral chi-square distribution for the Cox-Ingersoll-Ross process. Comput. Econ. 24(1), 35–50 (2004)
Hulley, H.: Strict local martingales in continuous financial market models. PhD thesis, UTS, Sydney (2009)
Hulley, H., Platen, E.: Laplace transform identities for diffusions, with applications to rebates and barrier options. In: Stettner, L. (ed.) Advances in Mathematical Finance. Banach Center Publications, vol. 83, pp. 139–157 (2008)
Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, 2nd edn. Wiley Series in Probability and Mathematical Statistics, vol. 1. Wiley, New York (1994)
Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, 2nd edn. Wiley Series in Probability and Mathematical Statistics, vol. 2. Wiley, New York (1995)
Kuo, F.Y., Dunsmuir, W.T.M., Sloan, I.H., Wand, M.P., Womersley, R.: Quasi-Monte Carlo for highly structured generalised response models. Methodol. Comput. Appl. Probab. 10(2), 239–275 (2008)
Patnaik, P.B.: The non-central χ 2- and F-distributions and their applications. Biometrika 36(1/2), 202–232 (1949)
Posten, H.O.: An effective algorithm for the noncentral chi-squared distribution function. Am. Stat. 43(4), 261–263 (1989)
Posten, H.O.: An effective algorithm for the noncentral beta distribution function. Am. Stat. 47(2), 129–131 (1993)
Sankaran, M.: Approximations to the non-central chi-square distribution. Biometrika 50(1/2), 199–204 (1963)
Schroder, M.: Computing the constant elasticity of variance option pricing formula. J. Finance 44(1), 211–219 (1989)
Seber, G.A.F.: The non-central chi-squared and beta distributions. Biometrika 50(3/4), 542–544 (1963)
Siegel, A.F.: The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity. Biometrika 66(2), 381–386 (1979)
Tang, P.C.: The power function of the analysis of variance tests with tables and illustrations of their use. Stat. Res. Mem. 2, 126–150 (1938)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Baldeaux, J., Platen, E. (2013). Computational Tools. In: Functionals of Multidimensional Diffusions with Applications to Finance. Bocconi & Springer Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-00747-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-00747-2_13
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00746-5
Online ISBN: 978-3-319-00747-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)