## Analytical Methods For Levy Processes With Applications To Finance

##### Résumé

This dissertation is divided into two parts: the first part is a literature review and the second describes three new contributions to the literature. The literature review aims to provide a self-contained introduction to some popular Levy models and to two key objects from the theory of Levy processes: the Wiener-Hopf factors and the exponential functional. We pay special attention to techniques and results associated with two “analytically tractable” families of processes known as the meromorphic and hyper-exponential families. We also demonstrate some important numerical techniques for working with these families and for solving numerical integration and rational approximation problems.

In the second part of the dissertation we prove that the exponential functional of a meromorphic Levy process is distributed like an infinite product of independent Beta random variables. We also identify the Mellin transform of the exponential functional, and then, under the assumption that the log-stock price follows a meromorphic process, we use this to develop a fast and accurate algorithm for pricing continuously monitored, fixed strike Asian call options. Next, we answer an open question about the density of the supremum of an alpha-stable process. We find that the density has a conditionally convergent double series representation when alpha is an irrational number. Lastly, we develop an effective and simple algorithm for approximating any process in the class of completely monotone processes –some members of this class include the popular variance gamma, CGMY, and normal inverse Gaussian processes – by a hyper-exponential process. Under the assumption that the log-stock price follows a variance gamma or CGMY process we use this

approximation to price several exotic options such as Asian and barrier options. Our algorithms are easy to implement and produce accurate prices.

In the second part of the dissertation we prove that the exponential functional of a meromorphic Levy process is distributed like an infinite product of independent Beta random variables. We also identify the Mellin transform of the exponential functional, and then, under the assumption that the log-stock price follows a meromorphic process, we use this to develop a fast and accurate algorithm for pricing continuously monitored, fixed strike Asian call options. Next, we answer an open question about the density of the supremum of an alpha-stable process. We find that the density has a conditionally convergent double series representation when alpha is an irrational number. Lastly, we develop an effective and simple algorithm for approximating any process in the class of completely monotone processes –some members of this class include the popular variance gamma, CGMY, and normal inverse Gaussian processes – by a hyper-exponential process. Under the assumption that the log-stock price follows a variance gamma or CGMY process we use this

approximation to price several exotic options such as Asian and barrier options. Our algorithms are easy to implement and produce accurate prices.