Levy option pricing models: Theory and application

The goal of this dissertation is to provide both the underlying theory and applications of Levy option pricing models which have been developed recently. The dissertation is composed of five main parts. Part 1 covers the mathematical theory of Levy processes which constitute a wide class of stochastic processes whose sample paths can be continuous, mostly continuous with occasional discontinuities, and purely discontinuous. Each Levy process is defined and characterized using theorems such as the Levy-Ito decomposition and the Levy-Khinchin representation. They also are characterized in terms of their infinite divisibilities and the Levy measures. Part 2 provides the introduction to Fourier transform and its application to the option pricing. Fourier transform option pricing method has become the mainstream pricing method because of its generality in the sense that it only requires the characteristic function of the log terminal asset price. Part 3 describes the mathematics of the Levy option pricing models. The classic Black-Scholes (BS) model is characterized as the only continuous Levy model. The classic Merton jump-diffusion (MJD) model is characterized as the non-Gaussian Levy model in which asset price dynamics is modeled by the jump diffusion process. The recently developed variance gamma (VG) model, normal inverse Gaussian (NIG) model, and Carr, Geman, Madan, and Yor (CGMY) model are characterized as pure jump Levy models in which the asset price dynamics are modeled using pure jump Levy processes. These pure jump Levy models are calibrated to the S&P 500 futures options and we compare their pricing performance relative to classic models along with the implied dynamics of the log return density and the Levy measure. Part 4 treats the regularization using the relative entropy in the Merton model to overcome the illposed problem of unregularized calibration with pure jump Levy models. Part 5 studies the other approaches to recover the risk-neutral probability density function of terminal asset prices. One is the semiparametric volatility-interpolation method developed by Shimko which makes no assumptions regarding the underlying stochastic process of asset prices. The other is the mixture of two lognormals approach which is a parametric approach by Bahra.