Title: Asymptotic analysis of European and American options with jumps in the underlying

Authors: Lamia Benothman; Faouzi Trabelsi

Addresses: Higher Institute of Computer Sciences and Mathematics of Monastir, Department of Mathematics, University of Monastir, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia; Research Unit 'Ondelettes Et Multifractales' (99UR1504) in Faculty of Sciences of Monastir. ' Laboratory of Mathematical and Numerical Modelling in Engineering Science, Department of Mathematics, National Engineering School of Tunis, Tunis El Manar University, B.P. 37, 1002 Tunis-Belvédère, Tunisia; Higher Institute of Computer Sciences and Mathematics of Monastir, Monastir University, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia

Abstract: In a jump-diffusion model for a single-asset market, we present an asymptotic analysis of European and American call options where the volatility is small compared with the drift terms. We precisely derive in the limit where volatility is negligible, relative to drifts, asymptotic expansion formulas for European, American and perpetual American call prices. As in the Black-Scholes model, we find that at leading order, the American call behaves in the same manner as a perpetual call option, except in a boundary layer about the option's expiry date. The derived expansion formulas contribute to the option pricing theory and provide a powerful tool to approximate call prices with a good accuracy, which allows to avoid the use of any numerical method.

Keywords: jump diffusion modelling; Lévy market model; asymptotic analysis; European call options; American call options; perpetual call options; call prices; volatility; Black-Scholes model; partial integro-differential equations; characteristics method; option pricing theory.

DOI: 10.1504/IJMOR.2012.048931

International Journal of Mathematics in Operational Research, 2012 Vol.4 No.5, pp.548 - 585

Published online: 23 Dec 2014 *

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