Essays on Numerical Evaluation of Derivatives

Abstract

In general, this thesis contemplates the potential of series expansion methods in evaluating financial derivatives. Recently, instead of relying on closed-form solutions, the usage of numerical methods became increasingly popular. The contribution of the thesis at hand is three-folded: First, we present and analyze a new method to price European options that are written on a single underlying asset by introducing Gabor series methods into option pricing. The resulting procedure shows to be a very robust pricing tool with a special strength in calculating short-term contracts. Second, we dedicate our attention to multi-asset derivatives. Multi-asset contracts are notoriously hard to deal with in case the user demands both, advanced stochastic processes and fast evaluation. Compared to single underlying derivatives, these multi-asset exotic options are studied to a much lower degree. We focus on European multi-asset options as well as on discrete barrier multi-asset options. To the best of our knowledge, the valuation of multi-asset barrier options in terms of multi-dimensional Fourier series methods has not been addressed in literature before. Third, as recent events in the market for credit risk have shown, there is a need for further research in methods to evaluate credit derivatives. Therefore, we focused on extending the standard Gaussian factor model to price synthetic collateralized debt obligations. The new models are able to cope with a wide range of market conditions. However, given a crisis as severe as the financial crisis of 2008, questions, such as market liquidity, that are outside the scope of pure modeling overlay the approximation quality. Nevertheless, the models presented are flexible instruments to price synthetic collateralized debt obligations while still staying in the intuitive framework of a factor model.