Mathematical and Statistical Methods for Insurance and Credit Risk Management

该研究项目提出了发展数学和统计方法的保险和信贷风险的管理。2008年雷曼兄弟等大型投资银行的破产，引起了人们对信用风险模型分析的兴趣。 我们的重点是当基础过程允许有跳跃时，衍生品合约（如公司债券、贷款、其他分期付款债务或更复杂的产品）的财务损失风险。信用衍生品是路径依赖的金融产品，因此其分析是基于潜在风险过程的历史。他们的市场不完整。这些特征使分析变得非常复杂，这解释了为什么这些产品的经典风险管理技术是在假设基础过程是连续的，衍生工具是路径独立的，或者概率度量是风险中性的情况下进行的。我们的项目扩展了对可违约债权的定价和套期保值问题的研究，而没有强加任何上述简化的假设。 然后，我们建议使用风险措施来衡量违约债券的信用质量，作为替代穆迪或标准普尔的经典排名。后者基于计量经济学模型，在2008年国际金融危机后的近几个月里受到严厉批评。根据这些可违约债券的潜在风险过程，使用风险衡量标准，应能提供更准确、客观和统计上稳定的评级。 该项目的第二部分解决了保险风险的管理问题。我们建议将破产理论工具Gerber-Shiu（G-S）函数的使用扩展到有限时间范围。定义了G-S函数用于分析无限期保险偿付能力问题。这限制了G-S函数在实际精算偿付能力或金融对冲问题中的使用，这些问题通常是在有限时间、短期内设定的。

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.