Title: Asymptotic behaviour of random maturity barrier options

Authors: Nawel Khodja; Faouzi Trabelsi; Mohamed Riad Remita

Addresses: Department of Mathematics, LaPS Laboratory, Badji-Mokhtar University, Annaba, Algeria ' Department of Mathematics, Institut Supérieur d'Informatique et de Mathématiques de Monastir, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia; Unité de Recherche 'Multifractales et Ondelettes' (UR11ES53), Faculté des Sciences de Monastir, Université de Monastir, Avenue de l'Environnement, 5000 Monastir, Tunisia ' Department of Mathematics, LaPS Laboratory, Badji-Mokhtar University, Annaba, Algeria

Abstract: In this paper, we study and analyse European barrier options in the Black-Scholes framework with deterministic or random maturity. When the maturity is deterministic, we provide closed form formulas for (up and out) European barrier call and put options. When the maturity is random, we suppose that it follows a Poisson distribution. In this case, we provide an explicit expression for the European barrier call option, presented in the form of a numerical series. In order to make our results more explicit, we give a Taylor expansion's formula for the random maturity European barrier call option, as the volatility decreases to zero. Since there are no, or at least, it is hard to obtain closed-form formulas for random maturity barrier contingent claims, the last formula should be very useful both in practice and theory of option pricing.

Keywords: option pricing; hitting time; random maturity; deterministic maturity; European barrier call; European barrier put; up and out barrier option; small volatility; Taylor expansion; asymptotic behaviour; European barrier options; Black-Scholes.

DOI: 10.1504/IJOR.2016.076302

International Journal of Operational Research, 2016 Vol.26 No.2, pp.221 - 235

Received: 10 Mar 2014
Accepted: 25 Apr 2014
Published online: 01 May 2016 *

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