Search

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 c\`adl\`ag 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., Comment: Doctoral Dissertation (PhD thesis), Graz University of Technology, approved November 13, 2015

Interest rate market models, like the LIBOR market model, have the advantage that the basic model quantities are directly observable in financial markets. Inflation market models extend this approach to inflation markets, where zero-coupon and year-on-year inflation-indexed swaps are the basic observable products. For inflation market models considered so far closed formulas exist for only one type of swap, but not for both. The model in this paper uses affine processes in such a way that prices for both types of swaps can be calculated explicitly. Furthermore call and put options on both types of swap rates can be calculated using one-dimensional Fourier inversion formulas. Using the derived formulas we present an example calibration to market data., Comment: 19 pages

The class of affine LIBOR models is appealing since it satisfies three central requirements of interest rate modeling. It is arbitrage-free, interest rates are nonnegative and caplet and swaption prices can be calculated analytically. In order to guarantee nonnegative interest rates affine LIBOR models are driven by nonnegative affine processes, a restriction, which makes it hard to produce volatility smiles. We modify the affine LIBOR models in such a way that real-valued affine processes can be used without destroying the nonnegativity of interest rates. Numerical examples show that in this class of models pronounced volatility smiles are possible., Comment: 18 pages