Option pricing via Laplace transforms - Petrella, Giovanni (ProQuest Dissertations & Theses, 2003)
This thesis is a collection of essays focused on the pricing of plain-vanilla and path-dependent options using Laplace transforms. In the first essay we derive Laplace transforms to value discretely monitored barrier and lookback options under a wide variety of stochastic, models. We show that these transforms can be obtained via a recursion, involving only the prices of European calls and puts. The numerical inversion of the transforms ensures that the methodology is accurate and efficient when compared to lattice methods and Monte Carlo simulation. Our second essay studies the pricing of plain-vanilla and path-dependent options under a double exponential jump diffusion process. We derive Laplace transforms for plain-vanilla derivatives and, using the memoryless property of the exponential distribution, for continuously monitored barrier and lookback options. We also suggest a transform based approximation for the valuation of American options, representing the early exercise boundary as a piecewise exponential function. A numerical implementation of our theoretical results confirms the accuracy of the method for European and path-dependent contracts and the validity of the approximation for American options. In our applications, we often need to invert transforms of functions defined on the real line. To this end, we use the Euler inversion algorithm, first developed by Abate and Whitt. While the original Euler algorithm is designed for functions defined only on the positive line, in the third essay we present a two-sided extension of the algorithm for functions defined in the entire real domain. We develop bounds on the discretization error, specify implementation procedures and provide numerical examples from option pricing, which confirm the robustness and accuracy of the proposed methodology.