On Dufresne’s Translated Perpetuity and Some Black-Scholes Annuities

Christophe Profeta ()
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Christophe Profeta: Laboratoire d’Analyse et Probabilités, Université d’Évry - Val d’Essonne, Bâtiment I.B.G.B.I., Évry, France

Analítika, 2014, vol. 7, issue 1, 7-19

Abstract: Let $(mathcal{E}_t, tgeq0)$ be a geometric Brownian motion. In this paper, we compute the law of a generalization of Dufresne’s translated perpetuity (following the terminology of Salminen-Yor) : $$ int_0^{+infty} frac{mathcal{E}_s^2}{(mathcal{E}_s^2+2 amathcal{E}_s +b)^2} ds,$$ and show that, in some cases, this perpetuity is identical in law with the first hitting time of a three-dimensional Bessel process with drift.We also study the law of the following pair of annuities $$left(int_0^{t}left(mathcal{E}_s-1 right)^+ ds , quad int_0^{t}left(mathcal{E}_s-1 right)^-ds right)$$ via a Feynman-Kac approach, and discuss some particular cases for which we are able to recover the associated perpetuities.

Keywords: Geometric Brownian motion; Bessel processes; Feynman-Kac formula (search for similar items in EconPapers)
JEL-codes: J60 J65 (search for similar items in EconPapers)
Date: 2014
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