Volatility Smile expansions in Lévy models

Abstract

This thesis is about investigating tail expansions for the call price and implied volatility at large strikes in exponential Levy jump-diffusion models. Furthermore, the asymptotics of the density function and the tail probability are studied. To get these expansions, we use the saddle-point method (method of deepest descent) on the Mellin transform of the call price, respectively density function and tail probability. Expansions for the implied volatility skew are derived by using transfer theorems, sharpening previous results of Benaim and Friz. We consider the double exponential Kou and the Merton Jump Diffusion model in this work.