Two Financial Engineering Problems Related to the Current Financial Crisis

The dissertation consists of the studies on two research problems in Financial Engineering that are closely related to the current financial crisis: (1) what risk measures are good for external financial regulation? (2) how to build a better model to price collateralized debt obligations (CDOs), the mispricing of which contributed significantly to the crisis? In the first part, we investigate whether Value-at-Risk (VaR) in combination with scenario analysis is a good risk measure for external regulation. Although VaR has been adopted by Basel Accord as a preferred risk measure for regulatory capital measurement, it has been criticized for its lack of subadditivity. We argue that while many risk measures may be suitable for internal risk management, risk measures used for external regulation should have robustness with respect to modeling assumptions and data. We propose new data-based risk measures called natural risk statistics that are characterized by a new set of axioms. The new axioms only require subadditivity for comonotonic random variables, which is consistent with the prospect theory. Comparing to other risk measures, natural risk statistics include tail conditional median, which is more robust than tail conditional expectation suggested by coherent risk measures; and, unlike insurance risk measures, natural risk statistics can incorporate scenario analysis. Natural risk statistics include VaR with scenario analysis as a special case and therefore provide a theoretical basis for using VaR along with scenario analysis as a robust risk measure for the purpose of external, regulatory risk measurement. The recent financial crisis has witnessed the powerful impact of the default clustering effect (i.e., one default event tends to trigger more default events in the future and cross-sectionally), especially on the CDO market. In the second part of the dissertation, we propose a new portfolio credit risk model for CDO pricing, which is based on cumulative default intensities. The model can generate a substantially high degree of default clustering and hence capture the possibility of extreme tail events. In particular, the model can generate simultaneous defaults of many names, which has been observed in historical data. Furthermore, the model is tractable enough to provide a direct link between single-name credit securities, such as credit default swaps (CDS), and multi-name credit securities, such as CDOs. The model automatically calibrates to single-name CDS spreads and allows fast CDO pricing and calculation of sensitivities of CDO tranche spreads with respect to the underlying single-name CDS spreads. The result of calibration to the recent market data on March 14, 2008 and September 16, 2008, when Bear Sterns, Lehman Brothers, etc. collapsed and default correlation among firms was substantially high, shows that the model is promising.