Financial models with Lévy processes

This dissertation studies option pricing, portfolio selection, and risk management assuming exponential-Lévy models in financial markets. Option pricing of European, American, and path-dependent derivatives is dealt with the markovian approach. Markovian approach has been introduced by Duan and Simonato [26] to price American options under Wiener and GARCH processes, and then Duan et al. [27] has shown how to price barrier options. This dissertation proposes to extend the markovian approach to Lévy processes and shows numerical results where the price convergence is observed. European, American, and barrier options are priced using the same procedure of Duan et al., while for compound and lookback options we propose a new pricing method. Specifically, we explain how to price compound and lookback options assuming a Markov chain evolutions of the asset price. Portfolio selection is studied assuming financial markets where asset log returns follow subordinated Lévy processes. Firstly, we propose a Mean-Value at Risk analysis under two financial markets, one without transaction costs, and the other one with proportional and constant transaction costs. Secondly, we study a multi-period model with unlimited short sales where investors look only at the mean and variance of the final wealth. Finally, we propose a Mean-Variance-Skewness analysis assuming a financial market with no short sales and without transaction costs. Our numerical results confirm the better performance of the studied subordinated Lévy processes with respect the Normal model. Risk management is studied proposing two conditional heteroscedastic models of portfolio returns. The first one is an extension of the EWMA RiskMetrics model and assumes L´evy distributed returns. The second one is a more sophisticated analysis and consists in a generalization of the GHICA model of Chen et al. [17].