Quasi-Monte Carlo for Portfolio Credit Derivates

It is the aim of this project on the one hand to develop refined models for the valuation of portfolio credit derivatives with the help of quasi-Monte Carlo methods, and on the other hand the further development of quasi- Monte Carlo methods (especially with respect to the analysis of low-discrepancy point sequences and of weighted quasi-Monte Carlo techniques), and finally the application and the testing of the new models and techniques in concrete examples. In detail we are planning the following work: a) Concerning Modelling, Simulation and Valuation of Portfolio Credit Derivatives We will investigate credit risk models where the firm values in a portfolio are modeled using Levy processes, which admits much better fit to market data than models ordinarily used by practitioners. For inhomogeneous portfolios with relatively high risk one has to drop some assumptions which otherwise make exact computation possible. One therefore has to resort to simulation methods, which are very time-consuming. We want to speed up the computation by using quasi-Monte Carlo methods. b) Concerning low-discrepancy Point Sets and Weighted Quasi-Monte Carlo Methods A new class of low-discrepancy point sets, which should be especially useful for quasi-Monte Carlo simulations (generalized Halton-Niederreiter sequences) will be analysed in detail with respect to estimation of discrepancy, weighted discrepancy, and with respect to existence properties. Further we will study if weighted quasi-Monte Carlo methods (in the sense of Sloan and Wozniakowski) are applicable for the modelling provided in part a) and how we optimally have to choose the sets of weights for these problems. In exhaustive numerical tests we finally will test the efficiency of the point sequences and the techniques for the valuation of portfolio credit derivatives.

One of the central topics in mathematical finance is the determination of fair prices for complex financial products. On example of such products is the class of credit derivatives. With the help of credit derivatives for example it is possible to transfer the risks of large credit-portfolios to other investors. Credit portfolios were heavily and controversial discussed during the financial crisis of 2008. The determination of fair prices in theory in most cases is possible, but also in most cases there do not exist explicit formulas for these fair prices. That means, that in most cases these values only can be determined approximately by numerical methods or with the help of Monte Carlo simulation. In this project we worked into two different directions. On the one hand we studied different types of credit-derivative products and other complex finance products like for example rainfall-derivatives or cat-bonds and we prepared the theoretical basics to be able to treat them with the help of (quasi-) Monte Carlo methods. On the other hand - on the basis of these application problems - we refined and improved the Monte Carlo methods (and the quasi-Monte Carlo methods (here we use carefully chosen deterministic scenarios for simulation and not random-scenarios)) used in this context. We published 13 papers in international scientific journals on these topics.