Barrier Option Under Lévy Model: A PIDE and Mellin Transform Approach

Abstract

We propose a stochastic model to develop a partial integro-differential equation (PIDE) for pricing and pricing expression for fixed type single Barrier options based on the Ito-Levy calculus with the help of Mellin transform. The stock price is driven by a class of infinite activity Levy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for fixed type Barrier options, and apply the Mellin transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Levy processes numerically. Finally, the algorithm for computing numerically is presented with results for a set of Levy processes.