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Some remarks on first passage of Levy processes, the American put and pasting principles

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  • L. Alili
  • A. E. Kyprianou

Abstract

The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Levy process and the solution of Gerber and Shiu [Astin Bull. 24 (1994) 195-220], Boyarchenko and Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663-1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100 (2002) 75-107], Mordecki [Finance Stoch. 6 (2002) 473-493], Asmussen, Avram and Pistorius [Stochastic Process. Appl. 109 (2004) 79-111] and Chesney and Jeanblanc [Appl. Math. Fin. 11 (2004) 207-225] to the American perpetual put optimal stopping problem. Furthermore, we make folklore precise and give necessary and sufficient conditions for smooth pasting to occur in the considered problem.

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  • L. Alili & A. E. Kyprianou, 2005. "Some remarks on first passage of Levy processes, the American put and pasting principles," Papers math/0508487, arXiv.org.
    • Handle: RePEc:arx:papers:math/0508487
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    References listed on IDEAS

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    1. Ernesto Mordecki, 1999. "Optimal stopping for a diffusion with jumps," Finance and Stochastics, Springer, vol. 3(2), pages 227-236.
    2. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    3. Marc Chesney & M. Jeanblanc, 2004. "Pricing American currency options in an exponential Levy model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 11(3), pages 207-225.
    4. Ariel Almendral & Cornelis W. Oosterlee, 2007. "On American Options Under the Variance Gamma Process," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(2), pages 131-152.
    5. Ernesto Mordecki, 2002. "Optimal stopping and perpetual options for Lévy processes," Finance and Stochastics, Springer, vol. 6(4), pages 473-493.
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