Option Pricing Models built from Lévy Processes

This article seeks to studying two different methods of option pricing - one introduced in Carr and Madan (1999), and the other one in F. Fang and Oosterlee (2008) - suitable for stock prices following stochastic processes whose characteristic function is known. The advantage of these methods is that they do not require an explicit formula for the density function. For each method, we determine good computation parameters before comparing them in terms of efficiency and accuracy. As an intermediary step, and because the Carr-Madan method is not compatible with a customised strike grid, we study two interpolation methods : the linear and the natural cubic spline interpolations. We also discuss the calibration problem, explain why it is not as straightforward as it may seem, and compare the results obtained for both models.