Lewis Model Revisited: Option Pricing with Levy Processes

Abstract

This paper aims to discuss the mathematical details in Lewis' model by considering the analyticity and integrability conditions of characteristic functions and payoff functions of contingent claims. In his seminal paper, Lewis shows that it is much easier to compute the option value in the Fourier space than computing in terminal security price space. He computes the option value as an integral in the Fourier space, the integrand being some elementary functions and the characteristic functions of a wide range of Levy processes. The model also illustrates how the residue calculus leads to several variations of option formulas through the contour integrals. In this paper, we provide with, to a reasonable extent, some rigor into the mathematical background of Lewis' model and validate his results for particular Levy processes. We also simply give the analyticity conditions for the characteristic function of the Carr-Geman-Madan-Yor model and a simple derivation of the characteristic function of Kou's double exponential model.