Lévy Processes in Finance: Inverse Problems and Dependence Modelling.

Summary This thesis deals with the modelling of stock prices by the exponentials of Lévy processes. In the first part we develop a non-parametric method allowing to calibrate exponential Lévy models, that is, to reconstruct such models from the prices of market-quoted options. We study the stability and convergence properties of this calibration method, describe its numerical implementation and give examples of its use. Our approach is first to reformulate the calibration problem as that of finding a risk-neutral exponential Lévy model that reproduces the option prices with the best possible precision and has the smallest relative entropy with respect to a given prior process, and then to solve this problem via the regularization methodology, used in the theory of ill-posed inverse problems. Applying this calibration method to the empirical data sets of index options allows us to study some properties of Lévy measures, implied by market prices. The second part of this thesis proposes a method allowing to characterize the dependence structures among the components of a multidimensional Lévy process and to construct multidimensional exponential Lévy models. This is done by introducing the notion of Lévy copula, which can be seen as an analog for Lévy processes of the notion of copula, used in statistics to model dependence between real-valued random variables. We give examples of parametric families of Lévy copulas and develop a method for simulating multidimensional Lévy processes with dependence given by a Lévy copula.