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Law of the exponential functional of one-sided Lévy processes and Asian options. (English. Abridged French version) Zbl 1162.60015
Summary: The purpose of this note is to describe, in terms of a power series, the distribution function of the exponential functional, taken at some independent exponential time, of a spectrally negative Lévy process \(\xi =(\xi_t,\,t\geq 0\)) with unbounded variation. We also derive a Geman-Yor type formula for Asian option prices in a financial market driven by \(e^\xi \).

60G51 Processes with independent increments; Lévy processes
91G20 Derivative securities (option pricing, hedging, etc.)
60E05 Probability distributions: general theory
91G80 Financial applications of other theories
Full Text: DOI arXiv
[1] Bernyk, V.; Dalang, R.C.; Peskir, G., The law of the supremum of a stable Lévy process with no negative jumps, Ann. probab., 36, 1777-1789, (2008) · Zbl 1185.60051
[2] Bertoin, J., Lévy processes, (1996), Cambridge University Press Cambridge · Zbl 0861.60003
[3] Bertoin, J.; Yor, M., Exponential functionals of Lévy processes, Probab. surv., 2, 191-212, (2005) · Zbl 1189.60096
[4] Carmona, Ph.; Petit, F.; Yor, M., On the distribution and asymptotic results for exponential functionals of Lévy processes, (), 73-121 · Zbl 0905.60056
[5] Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. ann., 300, 463-520, (1994) · Zbl 0865.90014
[6] Geman, H.; Yor, M., Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques, C. R. acad. sci. Paris, ser. I, 314, 6, 471-474, (1992) · Zbl 0759.60084
[7] Gjessing, H.K.; Paulsen, J., Present value distributions with applications to ruin theory and stochastic equations, Stochastic process. appl., 71, 1, 123-144, (1997) · Zbl 0943.60098
[8] Lamperti, J., Semi-stable Markov processes. I, Z. wahrsch. verw. geb., 22, 205-225, (1972) · Zbl 0274.60052
[9] Lebedev, N.N., Special functions and their applications, (1972), Dover Publications New York · Zbl 0271.33001
[10] P. Patie, Exponential functional of one-sided Lévy processes and self-similar continuous state branching processes with immigration, Bull. Sci. Math. (2008), in press
[11] P. Patie, A Geman-Yor formula for one-sided Lévy processes, Preprint, 2008
[12] P. Patie, Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes, Ann. Inst. H. Poincaré Probab. Statist. (2008), in press
[13] P. Patie, Law of the absorption time of positive self-similar Markov processes, Preprint, 2008
[14] Patie, P., q-invariant functions associated to some generalizations of the ornstein – uhlenbeck semigroup, ALEA lat. am. J. probab. math. stat., 4, 31-43, (2008) · Zbl 1168.60011
[15] Sato, K., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press Cambridge · Zbl 0973.60001
[16] Yor, M., Exponential functionals of Brownian motion and related processes, (2001), Springer Finance Berlin · Zbl 0999.60004
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