On Lévy Processes in Mathematical Finance

Abstract: The focus of the first article, On the Modelling of Financial Data with Generalized Hyperbolic Distributions, lies in studying the performance of the generalized hyperbolic distribution (GH), when fitted to historical data. Four different areas were selected. Within these four areas the GH distribution was compared with distributions more traditionally used. A modified version of the GH distribution MGH, with a more versatile behaviour of the tails, was introduced. In the second article, On Semi-parametric Modelling of Stock Prices with Lévy Processes, a Lévy process model for logarithmic asset returns, called Combined Gaussian and Multiple Poisson (Combined), is investigated. This model consists of a Wiener process combined with several rescaled and independent Poisson processes. In order to see how well the model performed, it was compared with two other Lévy processes models, namely the Normal Inverse Gamma process (NIG), and the Variance gamma process (VG), as well as the Wiener process. In order to compare the models, they were fitted to devolatilized logarithmic returns of empirical data from the S&P 500 index and the ABB stock, listed on the New York stock exchange. In order to see how well the models performed when the data were otherwise distributed, eight new data sets were simulated, with the parameters obtained for the four different models. The models were then fitted to these simulated data sets. The performance of the models was investigated by calculating the Kolmogorov-Smirnov distance.