Modelling Stock Returns with Lévy Processes

Several approaches to model stock returns with Lévy Processes have been developed in the past years. Firstly, this article will review existing approaches and compare the latest ones through an analysis of the Lévy density. Secondly, this article will provide a simple but general parameterization for the Lévy density which yields a class of Lévy processes that can be used in a financial context. These processes – titled α-β Lévy motions – will allow for excessive arrival rates of average size jumps, in correspondence to humped return distributions at short time scales.