Option Pricing Under New Classes of Jump-Diffusion Processes

Abstract

In this dissertation, we introduce novel exponential jump-diffusion models for pricing options. Firstly, the normal convolution gamma mixture jump-diffusion model is presented. This model generalizes Merton's jump-diffusion and Kou's double exponential jump-diffusion. We show that the normal convolution gamma mixture jump-diffusion model captures some economically important features of the asset price, and that it exhibits heavier tails than both Merton jump-diffusion and double exponential jump-diffusion models. Secondly, the normal convolution double gamma jump-diffusion model for pricing options is presented. We show that under certain configurations of both the normal convolution gamma mixture and the normal convolution double gamma jump-diffusion models, the latter exhibits a heavier left or right tail than the former. For both models, the maximum likelihood procedure for estimating the model parameters under the physical measure is fairly straightforward; moreover, the likelihood function is given in closed form thereby eliminating the need to embed a probability density function recovery procedure such as the fast Fourier transform or the Fourier-cosine expansion methods in the parameter estimation procedure. In addition, both models can reproduce the implied volatility surface observed in the options data and provide a good fit to the market-quoted European option prices.