Time - inhomogeneous Lévy processes in interest rate and credit risk models

In this thesis, we present interest rate models and a credit risk model, all driven by time-inhomogeneous Lévy processes, i.e. stochastic processes whose increments are independent but in general not stationary. In the interest rate part, we discuss a Heath–Jarrow–Morton forward rate model (the Lévy term structure model), a model for forward bond prices (the Lévy forward price model) and a Libor model (the Lévy Libor model) which generalizes the Libor market model. In all of these models, explicit valuation formulae are established for the most liquid interest rate derivatives, namely caps, floors, and swaptions. The formulae can numerically be evaluated fast and thus allow to calibrate the models to market data. In the Lévy term structure model, we also price floating range notes. Their payoffs are path-dependent. In the credit risk part, the Lévy Libor model (and therewith, as a special case, the Libor market model) is extended to defaultable forward Libor rates. We present a rigorous construction of the model and price some of the most heavily traded credit derivatives, namely credit default swaps, total rate of return swaps, credit spread options and credit default swaptions.