Mathematical and Statistical Methods for Insurance and Credit Risk Management

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This research project proposes to develop mathematical and statistical methods for the management of insurance and credit risks. The bankruptcy of major investment banks in 2008, such as Lehman Brothers, has raised interest for the analysis of credit risk models. Our focus is on the risk of the financial losses on derivative contracts such as corporate bonds, loans, other instalment debts or more complicated products, when the underlying process is allowed to have jumps. Credit derivatives are path dependent financial products; hence their analysis is based on the history of the underlying risk process. Their markets are incomplete. These characteristics complicate substantially the analysis, which explains why classical risk management techniques of these products are carried under the assumptions that the underlying process is continuous, the derivative is path independent, or the probability measure is risk neutral. Our project extends the study of the pricing and hedging problem of defaultable claims without imposing any of the above simplifying assumptions. Then we propose to use risk measures to gauge the credit quality of defaultable bonds, as an alternative to the classical rankings of Moody's or Standard and Poor's. The latter, based on econometric models, have been under severe criticism in recent months, following the international financial crisis of 2008. Using risk measures, based on the underlying risk process of these defaultable bonds, should provide more accurate, objective and statistically stable rankings. A 2nd part of the project addresses the management problem of insurance risks. We propose to extend the use of a ruin theory tool, the Gerber-Shiu (G-S) function, to a finite-time horizon. The G-S function was defined for the analysis of an infinite horizon insurance solvency problem. This limits the use of G-S functions in practical actuarial solvency or financial hedging problems, which are usually set over finite-time, short horizons.