Mathematical Modeling of Systemic Risk in Financial Networks: Managing Default Contagion and Fire Sales

As impressively shown by the financial crisis in 2007/08, contagion effects in financial networks harbor a great threat for the stability of the entire system. Without sufficient capital requirements for banks and other financial institutions, shocks that are locally confined at first can spread through the entire system and be significantly amplified by various contagion channels. The aim of this thesis is thus to investigate in detail two selected contagion channels of this so-called systemic risk, provide mathematical models and derive consequences for the systemic risk management of financial institutions. The first contagion channel we consider is default contagion. The underlying effect is here that insolvent institutions cannot service their debt or other financial obligations anymore - at least partially. Debtors and other directly impacted parties in the system are thus forced to write off their losses and can possibly be driven into insolvency themselves due to their incurred financial losses. This on the other hand starts a new round in the default contagion process. In our model we simplistically describe each institution by all the financial positions it is exposed to as well as its initial capital. In doing so, our starting point is the work of Detering et al. (2017) - a model for contagion in unweighted networks - which particularly considers the exact network configuration to be random and derives asymptotic results for large networks. We extend this model such that weighted networks can be considered and an application to financial networks becomes possible. More precisely, for any given initial shock we deduce an explicit asymptotic expression for the total damage caused in the system by contagion and provide a necessary and sufficient criterion for an unshocked financial system to be stable against small shocks. Moreover, ...