Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T06:53:38.623Z Has data issue: false hasContentIssue false
  • English
  • Français

A study of the Hartman–Watson distribution motivated by numerical problems related to the pricing of Asian options

Part of: Distribution theory Distribution theory - Probability Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

P. Barrieu ,
A. Rouault  and
M. Yor
P. Barrieu*
Affiliation:
London School of Economics
A. Rouault*
Affiliation:
University of Versailles
M. Yor*
Affiliation:
University of Paris VI
*
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, University of Versailles, 45 avenue des Etats-Unis, 78035 Versailles cedex, France
∗∗∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, University of Paris VI, 175 rue du Chevaleret, 75013 Paris, France
Get access Rights & Permissions [Opens in a new window]

Abstract

One approach to the computation of the price of an Asian option involves the Hartman–Watson distribution. However, numerical problems for its density occur for small values. This motivates the asymptotic study of its distribution function.

Keywords

Asian optionHartman–Watson distributionlarge deviation

MSC classification

Primary: 60J65: Brownian motion 60E99: None of the above, but in this section
Secondary: 62E20: Asymptotic distribution theory 65C50: Other computational problems in probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1049 - 1058
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramovitz, M., and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Bhattacharya, R., Thomann, E., and Waymire, E. (2001). A note on the distribution of integrals of geometric Brownian motion. Statist. Prob. Lett. 55, 187–192.Google Scholar
Bouaziz, L., Briys, E., and Crouhy, M. (1994). The pricing of forward-starting Asian options. J. Banking Finance 18, 823–839.Google Scholar
Carverhill, A. P., and Clewlow, L. J. (1990). Valuing average rate (Asian) options. Risk 3, 25–29.Google Scholar
Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. Appl. Prob. 33, 223–241.Google Scholar
Dufresne, D. (2004). The log-normal approximation in financial and other computations. Adv. Appl. Prob. 36, 747–773.Google Scholar
Elworthy, K. D., Li, X. M., and Yor, M. (1997). On the tails of the supremum and the quadratic variation of strictly local martingales. In Séminaire de Probabilités XXXI (Lecture Notes Math. 1655), Springer, Berlin, pp. 113–125.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Geman, H., and Yor, M. (1993). Bessel processes, Asian options and perpetuities. Math. Finance 3, 349–375.Google Scholar
Gould, R. (2000). The distribution of the integral of exponential Brownian motion. , Oregon State University.Google Scholar
Hartman, P., and Watson, G. S. (1974). ‘Normal’ distribution on spheres and the modified Bessel functions. Ann. Prob. 2, 593–607.Google Scholar
Ishiyama, K. (2003). The probability densities of integrals of geometric Brownian motion. Submitted.Google Scholar
Kemna, A. G. Z., and Vorst, A. C. F. (1990). A pricing method for options based on average asset values. J. Banking Finance 14, 113–129.Google Scholar
Kendall, D. G. (1991). The Mardia-Dryden shape distribution for triangles: a stochastic approach. J. Appl. Prob. 28, 225–230.Google Scholar
Kent, J. (1997). Distributions, processes and statistics on spheres. , University of Cambridge.Google Scholar
Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitsth. 22, 205–225.Google Scholar
Lévy, E. (1992). Pricing European average rate currency options. J. Internat. Money Finance 11, 474–491.Google Scholar
Linetsky, V. (2004). Spectral expansion for Asian (average price) options. To appear in Operat. Res.Google Scholar
Liu, Q., and Rouault, A. (2000). Limit theorem for Mandelbrot's multiplicative cascades. Ann. Appl. Prob. 10, 218–239.CrossRefGoogle Scholar
Matsumoto, H., and Yor, M. (2003). Dufresne's expressions for the probability densities of integrals of geometric Brownian motions. Adv. Appl. Prob. 35, 184–206.Google Scholar
Rogers, L. C. G., and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077–1088.Google Scholar
Schröder, M. (2003). On the integral of geometric Brownian motion. Adv. Appl. Prob. 35, 159–183.Google Scholar
Schürger, K. (2002). Laplace transforms and suprema of stochastic processes. In Advances in Finance and Stochastics, eds Sandmann, K. and Schönbucher, P. J., Springer, Berlin, pp. 285–294.CrossRefGoogle Scholar
Yor, M. (1980). Loi de l'indice du lacet Brownien et distribution de Hartman–Watson. Z. Wahrscheinlichkeitsth. 53, 71–95.Google Scholar
Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531.Google Scholar
Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.Google Scholar