Time-inhomogeneous affine processes and affine market models

Abstract

This thesis is devoted to the study of affine processes and their applications in financial mathematics. In the first part we consider the theory of time-inhomogeneous affine processes on general state spaces. We present a concise setup for time-inhomogeneous Markov processes. For stochastically continuous affine processes we show that there always exists a càdlàg modification. Afterwards we consider the regularity and the semimartingale property of affine processes. Contrary to the time-homogeneous case, time-inhomogeneous affine processes are in general neither regular nor semimartingales and the time-inhomogeneous case raises many new and interesting questions. Assuming that an affine process is a semimartingale, we show that even without regularity the parameter functions satisfy generalized Riccati integral equations. This generalizes an important result for time-homogeneous affine processes. We also show that stochastically continuous affine semimartingales are essentially generated by deterministic time-changes of what we call absolutely continuously affine semimartingales. These processes generalize time-homogeneous regular affine processes. In the second part we consider the class of affine LIBOR market models. We contribute to this class of models in two ways. First, we modify the original setup of the affine LIBOR market models in such a way that next to nonnegative affine processes real-valued affine processes can also be used. Numerical examples show that this allows for more flexible implied volatility surfaces. Second, we introduce the class of affine inflation market models, an extension of the affine LIBOR market models. A calibration example shows that these models perform very well in fitting market-observed prices of inflation derivatives.