### Title: Asymptotic analysis of options in a jump-diffusion model with binomial jump size distribution

Authors: Lamia Benothman; Faouzi Trabelsi

Addresses: Department of Mathematics, Faculty of Sciences of Bizerte, Carthage University, Jarzouna 7021, Bizerte, Tunisia; Faculty of Sciences of Monastir, University of Monastir, Research Unit 'Ondelettes Et Multifractales' (99UR1504), Tunisia ' Department of Mathematics, Laboratory of Mathematical and Numerical Modelling in Engineering Science, 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, University of Monastir, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia

Abstract: In this paper, we provide an asymptotic analysis of European and American call options in a jump-diffusion model for a single-asset market, where the jump size follows a binomial distribution B(n, p) for n ≥ 1 and p ∈]0, 1[, and the volatility is small compared to the drift terms. An asymptotic formula for the perpetual call option for small volatility is also developed. It is showed that at leading order, the American call option, behaves in the same manner as a perpetual call, except in a boundary layer about the option's expiry date. Next, we apply the obtained asymptotic results to approximate the same options in the Merton's model. Precisely, we approximate the jump size normal distribution by a discrete binomial one for large number n, on the basis of the central limit theorem. Then, we use for small volatility, the binomial asymptotic expansion formulas to approximate European and American call prices, in the Merton's model. Finally, the found expansion formulas for call prices are illustrated graphically. They represent a powerful tool for approximating option prices with a good accuracy. We think that these formulas contribute to the theory of option pricing.

Keywords: jump diffusion Levy market model; Merton model; binomial distribution; asymptotic analysis; central limit theorem; European call; American call; perpetual call; volatility; option prices; option pricing theory.

International Journal of Mathematical Modelling and Numerical Optimisation, 2013 Vol.4 No.1, pp.14 - 55