| This dissertation is concerned with the pricing of path-dependent options where the underlying asset is modeled as a continuous-time exponential Levy process and is monitored at discrete dates. These options enable their users to tailor random payoff outcomes to their particular risk profiles and are widely used by hedgers such as large multinational corporations and speculators alike. The use of continuous--time models since the breakthrough paper of Black and Scholes has been greatly facilitated by advances in stochastic calculus and the mathematical elegance it provides. The recent financial crisis started in 2008 has highlighted the importance of models that incorporate the possibility of sudden, large jumps as well as the higher likelihood of adverse outcomes as compared with the classical Black-Scholes model. Increasingly, exponential Levy processes have become preferred alternatives, thanks in particular to the explicit Levy--Khinchin representation of their characteristic functions. On the other hand, the restriction of monitoring dates to a discrete set increases the mathematical and computational complexity for the pricing of path--dependent options even in the classical Black-Scholes model. This dissertation develops new techniques based on recent advances in the fast evaluation and inversion of Fourier and Hilbert transforms as well as classical results in fluctuation theory, particularly those involving random walk duality and ladder epochs. |
