The thesis is divided into two parts. The first, consisting of Chapters 1, 2, and 3, is devoted to the study of stock price models involving exponential Lévy processes. The second part of the thesis studies term structure models driven by Lévy processes. This part is a continuation of research that started with the author’s diploma thesis [“Modellierung der Zinsstrukturkurve unter Verwendung von Lévy-Prozessen” (Freiburg i. Br., 1996)] and the article of E. Eberlein and the author [Math. Finance 9, No. 1, 31-53 (1999)].
The content of the chapters is as follows. Chapter 1 examines a general stock price model where the price of a single stock follows an exponential Lévy process. Chapter 2 is devoted to the study of the Lévy measure of infinitely divisible distributions, in particular of generalized hyperbolic distributions. This yields information about what changes in the distribution of a generalized hyperbolic Lévy motion can be achieved by a locally equivalent change of the underlying probability measure. Implications for option pricing are discussed. Chapter 3 examines the numerical calculation of option prices. Based on the observation that the pricing formulas for European options can be represented as convolutions, a method to calculate option prices by fast Fourier transforms is derived, making use of bilateral Laplace transformations.
Chapter 4 examines the Lévy term structure model introduced by Eberlein and Raible (loc. cit.). Several new results related to the Markov property of the short-term interest rate are presented. Chapter 5 presents empirical results on the non-normality of the log returns distribution for zero bonds. In Chapter 6 it is shown that in the Lévy term structure model the martingale measure is unique. This is important for option pricing. Chapter 7 presents an extension of the Lévy term structure model to multivariate driving Lévy processes and stochastic volatility structures. In theory, this allows for a more realistic modeling of the term structure by addressing three key features: Non-normality of the returns, term structure movements that can only be explained by multiple stochastic factors, and stochastic volatility.