Markov-Modulated Affine Processes: Theory and Applications

The present thesis deals with Markov-modulated affine processes, a class of continuous time Markov processes that are created from affine processes by allowing some of their coefficients to be a functon of an exogenous Markov process. Affine processes have been formally studied and characterized by D. Duffie, D. Filipovic, and W. Schachermayer. This class of processes has proven to be an essential tool in modelling financial data as it is capable of capturing many stylized facts, but is at the same time remarkably tractable. However, for many specific applications in mathematical finance adding Markov-modulation to the drift, volatility or jump component of affine processes has facilitated the modelling of certain empirically observed phenomena that cannot be replicated by the standard affine model class.
The first part of this thesis is devoted to the study of theoretical aspects of Markov-modulated affine processes. We formally introduce this class of processes and prove its existence via a martingale problem approach. We show that the formula for the characteristic function of the marginals of Markov-modulated affine processes has a computationally convenient functional form. This in turn preserves much of the tractability of standard affine processes and allows for efficient calibration to data. Our setup is a substantial generalization of earlier work: First, we consider the case where the generator of the exogenous process is an unbounded operator (as is the case for diffusions or jump processes with infinite activity). Second, we allow for discontinuities in those coefficients of the modulated process that are controlled by the exogenous Markov process.
In the second part of this thesis we shift our focus away from the mathematical properties of Markov-modulated affine processes towards their applications in finance and economics. In the first application, we investigate European Safe Bonds (ESBies), a recently proposed class of synthetic securities that are supposed to improve the functioning of the European monetary union. In its core, ESBies form the senior tranche of a CDO backed by a diversified portfolio of sovereign bonds from all members of the euro area. The potential benefits of ESBies and other bond-backed securities hinge on the assertion that these products are really safe. We provide a comprehensive quantitative study of the risks associated with ESBies and related products, using an affine credit risk model with regime switching as vehicle for our analysis. We discuss a recent proposal of Standard and Poor’s for the rating of ESBies, we analyse the impact of model parameters and attachment points on the size and the volatility of the credit spread of ESBies and we consider several approaches to assess the market risk of ESBies. Moreover, we compare ESBies to synthetic securities created by pooling the senior tranche of national bonds. We conclude our analysis with a brief discussion of possible policy implications from our results.
The second application deals with financial bubbles. In particular, we aim to obtain an understanding of the microstructural mechanisms of stock price processes that lead to the formation of a bubble in the long run. We build a microscopic model that captures the most relevant stylized facts of tick-by-tick data and then study an appropriate scaling limit to infer which market conditions enable bubble formation in the long run. Our main findings are that liquidity asymmetry is a decisive factor in the formation of bubbles, that high market activity favors the development of a bubble, and that bubbles can even form in markets that prevent statistical arbitrage opportunities.