Statistical Methods Applied to Credit Risk and Reacting Systems

Statistical physics uses probability theory and statistics to provide a macroscopic description of real world systems composed of a large number of units. Its main purpose is to study the properties of complex systems, showing complicated interactions and a high degree of freedom, from the statistical behavior of their components.
The central topic of the thesis is the application of methods from statistical physics on diverse complex systems. Particularly we are interested in the statistics of extreme events. These often tend to have significant consequences and hence need to be understood in detail.

The work is structured in two parts. In the first part we focus on the dynamics of financial markets and credit risk. A portfolio consisting of several credit contracts faces a high risk of large losses, especially when the underlying asset values are correlated. In order to provide a realistic model of the correlated asset values we have to take the non-stationarity of financial markets into account. This was demonstrated in a rather drastic way during the financial crisis.
We introduce a random matrix approach for correlation matrices to model the non-stationarity of financial markets. Based on this approach we review recent progress in modeling credit risk for correlated assets. We discuss the effects of diversification, i.e., reducing the risk by distributing it, and investigate common risk measures for one credit portfolio. We present results of numerical simulations in which mutual dependencies between two non-overlapping credit portfolios are studied.

To obtain a comprehensive understanding of systemic credit risk we present new, analytical results for the multivariate joint loss distribution of several credit portfolios on a non-stationary market.
This distribution gives us the opportunity to calculate the portfolio loss correlation of two credit portfolios. We investigate various portfolio structures, such as two non-overlapping portfolios in one market or one portfolio that operates in two on average uncorrelated markets or a subordinated debt of two creditors. This gives a quantitative understanding of the limitations of diversification.

In the second part of this work we focus on rare event statistics in reacting systems. We calculate the probabilities to find systems of reacting particles in states which largely deviate from typical behavior. We consider various systems where the interactions of particles are described by chemical reactions.


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