Pricing and Hedging in Affine Models with Possibility of Default and Characteristic Functions of Log Stock Prices

This thesis consists of two parts, both of which are applications of characteristic functions to finance.;In the first part, we propose a general class of models for the simultaneous treatment of equity, corporate bonds, government bonds and derivatives. The noise is generated by a general affine Markov process. The framework allows for stochastic volatility, jumps, the possibility of default and correlations between different assets. We extend the notion of a discounted characteristic function of the log stock price to the case where the underlying can default and show how to calculate it in terms of a coupled system of generalized Riccati equations. This yields an efficient method to compute prices of power payoffs and Fourier transforms. European calls and puts as well as binaries and asset-or-nothing options can then be priced with the fast Fourier transform methods of Carr and Madan (1999) and Lee (2005). Other European payoffs can be approximated by a linear combination of power payoffs and vanilla options. We show the results to be superior to using only power payoffs or vanilla options. We also give conditions for our models to be complete if enough financial instruments are liquidly tradable and study dynamic hedging strategies. As an example we discuss a Heston-type stochastic volatility model with possibility of default and stochastic interest rates.;In the second part, we consider pricing models where the stock price is defined as a solution of a stochastic differential equation with jumps. We derive a simple set of sufficient conditions, in order for the characteristic function of the log stock price to be the solution of the corresponding Kolmogorov partial integro-differential equation (PIDE). Moreover, we prove that explosions of the solution of the PIDE correspond to explosions of the moments of the stock price. We also analyze under what sufficient conditions the explosion times do not depend on the initial value and provide a counter-example when these do not hold. The results are extended to the case of stochastic interest rates as well as the case of possibility of default.