Journal of Applied Probability

Asian options under one-sided Lévy models

P. Patie

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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.

Article information

J. Appl. Probab., Volume 50, Number 2 (2013), 359-373.

First available in Project Euclid: 19 June 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91G20: Derivative securities 60G51: Processes with independent increments; Lévy processes

Asian option Lévy process exponential functional hypergeometric-type function


Patie, P. Asian options under one-sided Lévy models. J. Appl. Probab. 50 (2013), no. 2, 359--373. doi:10.1239/jap/1371648946.

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  • Albrecher, H., Dhaene, J., Goovaerts, M. and Schoutens, W. (2005). Static hedging of Asian options under Lévy models. J. Derivatives 12, 63–72.
  • Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
  • Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Prob. Surveys 2, 191–212.
  • Boyarchenko, S. I. and Levendorskii, S. Z. (2000). Option pricing for truncated Lévy processes. Internat. J. Theoret. Appl. Finance 3, 549–552.
  • Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion, ed. M. Yor, Rev. Mat. Iberoamericana, Madrid, pp. 73–130.
  • 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.
  • 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.
  • 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.
  • Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.
  • 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.
  • Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39–79.
  • Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407–428.
  • Eberlein, E. and Papapantoleon, A. (2005). Equivalence of floating and fixed strike Asian and lookback options. Stoch. Process. Appl. 115, 31–40.
  • Eberlein, E. and Madan, D. B. (2010). Short positions, rally fears and option markets. Appl. Math. Finance 17, 83–98.
  • Eberlein, E., Jacod, J. and Raible, S. (2005). Lévy term structure models: no-arbitrage and completeness. Finance Stoch. 9, 67–88.
  • 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.
  • 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.
  • Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3, 349–375.
  • Gjessing, H. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144.
  • Gradshteyn, I. S. and Ryshik, I. M. (2000). Table of Integrals, Series, and Products, $6$th edn. Academic Press, San Diego, CA.
  • Henderson, V. and Wojakowski, R. (2002). On the equivalence of floating- and fixed-strike Asian options. J. Appl. Prob. 39, 391–394.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • Lebedev, N. (1972). Special Functions and Their Applications. Dover Publications, New York.
  • Linetsky, V. (2004). The spectral decomposition of the option value. Internat. J. Theoret. Appl. Finance 7, 337–384.
  • Madan, D. and Schoutens, W. (2008). Break on through to the single side. Working paper, Katholieke Universiteit Leuven.
  • Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177.
  • Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.
  • Patie, P. (2008). $q$-invariant functions for some generalizations of the Ornstein–Uhlenbeck semigroup. ALEA Lat. Amer. J. Prob. Math. Statist. 4, 31–43.
  • 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.
  • Patie, P. (2009). Law of the exponential functional of one-sided Lévy processes and Asian options. C. R. Acad. Sci. Paris 347, 407–411.
  • Patie, P. (2012). Law of the absorption time of some positive self-similar Markov processes. Ann. Prob. 40, 765–787.
  • Rogers, L. C. G. and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077–1088.
  • Schoutens, W. (2003). Lévy Processes in Finance. Pricing Finance Derivatives. John Wiley, New York.
  • Schröder, M. (2005). Laguerre series in contingent claim valuation, with applications to Asian options. Math. Finance 15, 491–531.
  • Schröder, M. (2008). On constructive complex analysis in finance: explicit formulas for Asian options. Quart. Appl. Math. 66, 633–658.
  • Večeř, J. and Xu, M. (2004). Pricing Asian options in a semimartingale model. Quant. Finance 4, 170–175.
  • Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.