Lévy Processes in Finance : Theory , Numerics , and Empirical Facts

Preface Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have a simple structure in comparison with general semimartingales. Thus stochastic models based on Lévy processes often allow for analytically or numerically tractable formulas. This is a key factor for practical applications. This 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. In the second part, we study term structure models driven by Lévy processes. This part is a continuation of the research that started with the author's diploma thesis Raible (1996) and the article Eberlein and Raible (1999). The content of the chapters is as follows. In Chapter 1, we study 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, we derive a method to calculate option prices by fast Fourier transforms, making use of bilateral Laplace transformations. Chapter 4 examines the Lévy term structure model introduced in Eberlein and Raible (1999). 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, we show 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 modelling 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 …