Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T14:06:15.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Asian Options Under One-Sided Lévy Models

Part of: Stochastic processes Mathematical finance

Published online by Cambridge University Press:  30 January 2018

P. Patie
P. Patie*
Affiliation:
Universit Libre de Bruxelles
*
∗ Postal address: Département de Mathématiques, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050, Bruxelles, Belgique. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We generalize, in terms of power series, the celebrated Geman-Yor formula for the pricing of Asian options in the framework of spectrally negative Lévy-driven assets. We illustrate our result by providing some new examples.

Keywords

Asian optionLévy processexponential functionalhypergeometric-type function

MSC classification

Primary: 91G20: Derivative securities 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 359 - 373
Copyright
© Applied Probability Trust 

References

Albrecher, H., Dhaene, J., Goovaerts, M. and Schoutens, W. (2005). Static hedging of {Asian options under L}évy models. J. Derivatives 12, 63–72.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. and Yor, M. (2005). {Exponential functionals of Lévy processes}. Prob. Surveys 2, 191–212.Google Scholar
Boyarchenko, S. I. and Levendorskii, S. Z. (2000). Option pricing for truncated Lévy processes. Internat. J. Theoret. Appl. Finance 3, 549–552.Google Scholar
Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of {Lé}vy processes. In Exponential Functionals and Principal Values Related to Brownian Motion, ed. Yor, M., Rev. Mat. Iberoamericana, Madrid, pp. 73–130.Google Scholar
Carmona, P., Petit, F. and Yor, M. (1998). Beta-{gamma} random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoamericana 14, 311–367.Google Scholar
Carr, P. and Schröder, M. (2003). Bessel processes, the integral of geometric Brownian motion, and Asian options. Teor. Veroyat. Primen. 48, 503–533. English translation: Theory Prob. Appl. 48 (2004), 400-425.Google Scholar
Collin-Dufresne, P., Goldstein, R. S. and Yang, F. (2010). {On the relative pricing of long maturity S&P 500 index options and CDX tranches}. {NBER working paper 15734}.Google Scholar
Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.CrossRefGoogle Scholar
Donati-Martin, C., Ghomrasni, R. and Yor, M. (2001). On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options. {Rev. Mat. Iberoamericana} 17, 179–193.CrossRefGoogle Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39–79.Google Scholar
Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407–428.Google Scholar
Eberlein, E. and Papapantoleon, A. (2005). Equivalence of floating and fixed strike Asian and lookback options. Stoch. Process. Appl. 115, 31–40.Google Scholar
Eberlein, E. and Madan, D. B. (2010). Short positions, rally fears and option markets. {Appl. Math. Finance} 17, 83–98.Google Scholar
Eberlein, E., Jacod, J. and Raible, S. (2005). Lévy term structure models: no-arbitrage and completeness. Finance Stoch. 9, 67–88.Google Scholar
Fu, M. C., Madan, D. B. and Wang, T. W. (1999). Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods. J. Comput. Finance 2, 49–74.Google Scholar
Geman, H. and Yor, M. (1992). Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques. C. R. Acad. Sci. Paris 314, 471–474.Google Scholar
Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3, 349–375.Google Scholar
Gjessing, H. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144.Google Scholar
Gradshteyn, I. S. and Ryshik, I. M. (2000). Table of Integrals, Series, and Products, 6th edn. Academic Press, San Diego, CA.Google Scholar
Henderson, V. and Wojakowski, R. (2002). {On the equivalence of floating- and fixed-strike Asian options}. J. Appl. Prob. 39, 391–394.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Lebedev, N. (1972). Special Functions and Their Applications. Dover Publications, New York.Google Scholar
Linetsky, V. (2004). The spectral decomposition of the option value. Internat. J. Theoret. Appl. Finance 7, 337–384.Google Scholar
Madan, D. and Schoutens, W. (2008). {Break on through to the single side}. Working paper, Katholieke Universiteit Leuven.Google Scholar
Maulik, K. and Zwart, B. (2006). {Tail asymptotics for exponential functionals of Lévy processes}. Stoch. Process. Appl. 116, 156–177.Google Scholar
Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.Google Scholar
Patie, P. (2008). q-invariant functions for some generalizations of the {Ornstein–Uhlenbeck} semigroup. ALEA Lat. Amer. J. Prob. Math. Statist. 4, 31–43.Google Scholar
Patie, P. (2009). Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 45, 667–684.Google Scholar
Patie, P. (2009). Law of the exponential functional of one-sided Lévy processes and Asian options. C. R. Acad. Sci. Paris 347, 407–411.Google Scholar
Patie, P. (2012). Law of the absorption time of some positive self-similar Markov processes. Ann. Prob. 40, 765–787.Google Scholar
Rogers, L. C. G. and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077–1088.CrossRefGoogle Scholar
Schoutens, W. (2003). Lévy Processes in Finance. Pricing Finance Derivatives. John Wiley, New York.Google Scholar
Schröder, M. (2005). Laguerre series in contingent claim valuation, with applications to Asian options. Math. Finance 15, 491–531.Google Scholar
Schröder, M. (2008). On constructive complex analysis in finance: explicit formulas for Asian options. Quart. Appl. Math. 66, 633–658.Google Scholar
Večeř, J. and Xu, M. (2004). Pricing Asian options in a semimartingale model. Quant. Finance 4, 170–175.Google Scholar
Yor, M. (2001). {Exponential Functionals of Brownian Motion and Related Processes}. Springer, Berlin.Google Scholar