Portfolio optimization and option pricing under defaultable Lévy driven models

In this thesis we study some portfolio optimization and option pricing problems in market models where the dynamics of one or more risky assets are driven by Lévy processes, and it is divided in four independent parts. In the first part we study the portfolio optimization problem, for the logarithmic terminal utility and the logarithmic consumption utility, in a multi-defaultable Lévy driven model. In the second part we introduce a novel technique to price European defaultable claims when the pre-defaultable dynamics of the underlying asset follows an exponential Lévy process. In the third part we develop a novel methodology to obtain analytical expansions for the prices of European derivatives, under stochastic and/or local volatility models driven by Lévy processes, by analytically expanding the integro-differential operator associated to the pricing problem. In the fourth part we present an extension of the latter technique which allows for obtaining analytical expansion in option pricing when dealing with path-dependent Asian-style derivatives