Applications of stochastic processes to financial risk computation

Ever since the financial crisis the focus on having efficient analytic and numerical methods in the field of financial risk computations has increased significantly. New regulations have been invoked that force banks and other financial entities to much more carefully monitor the various risks involved in their daily practices. One area in which the regulations have increased significantly has been that of the so-called valuation adjustments in derivative pricing: banks are now required to price all components of a trade. These additional factors are collectively called valuation adjustments. The analysis and valuation of these adjustments is crucial to banks, but in turn is also a complex task involving both accounting methodologies as well as the need for efficient mathematical methods. In order to price derivatives with or without valuation adjustments, the arising (non-linear) partial differential equation can then be solved by means of a combination of the COS method and the approximated characteristic function, resulting in an efficient and easy-to-implement valuation method. Another risk metric related to counterparty credit risk, whose importance has increased since the crisis is that of systemic risk. Systemic risk was an important contributor to the financial crisis, where the collapse of individual financial entities triggered a chain of defaults throughout the system. Hawkes processes are able to incorporate an important feature of risk in interconnected systems, in particular that of a cross- and self-excitement in the monetary reserves of the banks. While modeling the full multivariate system in case of a large number of banks is time-consuming, using a weak convergence analysis in which the number of entities in the system tends to infinity allows us to obtain an expression for the behavior of the monetary reserve process in a large system, and quantify the systemic risk present in the system.