Lévy Processes, Representations and Models with Application in Finance

Lévy processes are becoming increasingly important in Mathematical Finance. This thesis aims to contribute to the development of theoretical representations of Lévy processes and their financial applications. The first part of the thesis presents a computational explicit formula of the chaotic representation property (CRP) for the powers of increments of a Lévy process. The formula can be used to obtain the integrands of the CRP in terms of the orthogonalised compensated power jump processes and the CRP in terms of Poisson random measures. The second part of the thesis presents hedging strategies for European and exotic options in a Lévy market. By applying Taylor's theorem, dynamic hedging portfolios are constructed and in the case of European options, static hedging is also implemented. It is shown that perfect hedging can be achieved by investing in power jump assets, moment swaps or some traded financial derivatives depending on the same underlying asset. Note that variance swaps are special cases of moment swaps and are traded in OTC (Over-The- Counter) markets. We can also hedge by constructing the minimal variance portfolios that invest in the risk-free bank account, the underlying stock and variance swaps. The numerical algorithms and performance of the hedging strategies are presented. The third part of the thesis contributes to the design of an option trading strategy, where the stock price is driven by a Lévy process. The trading strategy is based on comparing the deviations between the density implied by historical time series and that implied by current market prices of the options. The performance of the trading strategy under di¤erent market conditions is reported and optimal parameters are obtained using efficient frontier analysis. The analysis compares the expected returns with the Conditional Value at Risks (CVaRs). Simulation results show that the trading strategy has a high earning potential