Applications of Levy processes in finance.

Abstract

The option pricing theory set forth by Black and Scholes assumes that the underlying asset can be modeled by Geometric Brownian motion, with the Brownian motion being the driving force of uncertainty. Recent empirical studies, Dotsis, Psychoyios & Skiadopolous (2007) [17], suggest that the use of Brownian motion alone is insufficient in accurately describing the evolution of the underlying asset. A more realistic description of the underlying asset’s dynamics would be to include random jumps in addition to that of the Brownian motion. The concept of including jumps in the asset price model leads us naturally to the concept of a L'evy process. L'evy processes serve as a building block for stochastic processes that include jumps in addition to Brownian motion. In this dissertation we first examine the structure and nature of an arbitrary L'evy process. We then introduce the stochastic integral for L'evy processes as well as the extended version of Itˆo’s lemma, we then identify exponential L'evy processes that can serve as Radon-Nikod'ym derivatives in defining new probability measures. Equipped with our knowledge of L'evy processes we then implement this process in a financial context with the L'evy process serving as driving source of uncertainty in some stock price model. In particular we look at jump-diffusion models such as Merton’s(1976) [37] jump-diffusion model and the jump-diffusion model proposed by Kou and Wang (2004) [30]. As the L'evy processes we consider have more than one source of randomness we are faced with the difficulty of pricing options in an incomplete market. The options that we shall consider shall be mainly European in nature, where exercise can only occur at maturity. In addition to the vanilla calls and puts we independently derive a closed form solution for an exchange option under Merton’s jump-diffusion model making use of conditioning arguments and stochastic integral representations. We also examine some exotic options under the Kou and Wang model such as barrier options and lookback options where the solution to the option price is derived in terms of Laplace transforms. We then develop the Kou and Wang model to include only positive jumps, under this revised model we compute the value of a perpetual put option along with the optimal exercise point. Keywords Derivative pricing, L'evy processes, exchange options, stochastic integration.