Title: Mathematical modeling of systemic risk in financial networks
Abstract: 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, we develop an explicit formula for necessary and sufficient risk capital at the level of single institutions that ensures stability of the financial network. We demonstrate by simulations that our asymptotic results give a good description for financial networks of the size of a few thousand institutions already.
In the next step, we develop a multi-dimensional extension of our model for default contagion such that we can describe the complex structures observed in financial networks - particularly core-periphery-structures but also multi-layered structures, regional concentrations and mixtures thereof. To this end, we assign to each institution in the network an additional parameter describing its type. The network is thereby divided in different subsystems (blocks). In particular, this new model enables us to quantify the impact of a local shock in one of the subsystems (e. g. a certain country) to the global system. Our results show that the additional complexity can significantly affect the stability of the financial system and we develop measures for the individual subsystems to secure themselves against contagion from other subsystems. Furthermore, we accomplish a more realistic modeling of financial obligations whose size may depend on both contracting parties. So far, meaningful analytical results could only be derived under the assumption that the amount of contagion only depends on the exposed party. We demonstrate that this simplifying assumption can lead to a grave underestimation of the risk potential of a system and the additional complexity in our model is thus essential for a realistic assessment of a system's stability.
Next, we develop a model for the contagion channel of fire sales at which institutions react to an initial shock by selling asset shares - forced by regulations for instance. As a result the share prices come under pressure and investors suffer further losses. This in turn leads again to asset sales and the process proceeds. For the modeling of this contagion channel, we describe each institution by the number and kind of its held asset shares as well as its initial capital and the losses suffered due to some initial shock. Additionally we assume that institutions make their decision to sell shares according to some given function and also the price impact of sales is described by a given function. In our modeling we resort to ideas from the default contagion literature and we thus achieve a rigorous description of the process. In particular, we asymptotically determine the total damage to the system caused by the initial shock and the subsequent fire sales, and we provide a classification of stable systems as well as sufficient risk capital to ensure stability of a financial system. Again we verify the applicability of our asymptotic results by suitable simulations.
Finally, we combine the models for default contagion and fire sales to get a more complete picture of contagion effects in periods of crisis. Our results show that the two contagion channels can tremendously amplify each other and thus stress the importance of combined models for the understanding of systemic risk. Also for the combined case we achieve to derive capital requirements sufficient to ensure stability of the system that are hence of great interest to regulatory institutions.
Publication Year: 2019
Publication Date: 2019-05-13
Language: en
Type: article
Access and Citation
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot