Mathematics of Quantitative Risk Management under Uncertainty

最近的经济危机导致人们重新审视数学在金融中的作用，特别是风险管理。从这一反省过程中得到的一个极其重要的教训是，认识到模型的最终用户和开发模型的数学家都需要更多地关注模型不确定性在风险度量和管理中的作用。该项目的目的是开发必要的基础数学和实用工具，以促进模型不确定性下的风险管理。将特别关注的两个应用领域是交易对手信用风险（CCR）和担保保险产品。交易对手信贷风险指因场外衍生工具合约交易对手的信誉出现不利变动而导致亏损的风险。雷曼兄弟（Lehman Brothers）违约和美国国际集团（AIG）几近违约对金融体系造成的破坏，戏剧性地说明了它的重要性。CCR损失取决于不利信贷事件发生的时间（由信贷风险因素驱动）以及事件发生时合同组合的价值（由市场风险因素驱动）。金融机构通常分别建立市场风险和信用风险模型;然而，这两种风险类型的联合建模提出了重大挑战，特别是依赖结构是模型不确定性的来源。研究将集中在推导出最坏的情况下的情况和保守的范围为CCR给定的不确定性之间的依赖市场和信用风险。现代人寿保险产品已变得越来越复杂，提供多种保障，并依赖于许多不同的风险来源。人寿保险公司在其担保产品组合中将面临的风险包括市场（例如利率、股本）风险、死亡率风险以及与保单持有人行为相关的风险（例如失效）。对这一主题的研究将集中在两个方向。首先是投保人的最佳行为。其次，研究保险人的决策过程对投保人的影响。

The recent economic crisis has led to a re-examination of the role of mathematics in finance in general, and risk management in particular. A lesson of paramount importance from this process of introspection has been the realization that both end users of models and the mathematicians who develop models need to pay more attention to the role of model uncertainty in the measurement and management of risk. The purpose of this project is to develop the underlying mathematics and practical tools necessary to facilitate the management of risk under model uncertainty. Two areas of application that will receive particular focus are counterparty credit risk (CCR) and insurance products with guarantees. Counterparty credit risk (CCR) refers to the risk of a loss due to an adverse change in the creditworthiness of a counterparty to an over-the-counter derivative contract. Its importance was dramatically illustrated by the devastation caused to the financial system by the default of Lehman brothers and the near default of AIG. CCR losses depend on both the timing of the adverse credit event (driven by credit risk factors), as well as the value of the portfolio of contracts when the event occurs (driven by market risk factors). Financial institutions often have in place models for market risk and credit risk separately; however, the joint modelling of these two risk types presents significant challenges, and the dependence structure in particular is a source of model uncertainty. Research will focus on deriving worst-case scenarios and conservative bounds for CCR given the uncertainty regarding the dependence between market and credit risk. Modern life insurance products have become increasing complex, offering numerous guarantees, and depending on many disparate sources of risk. Included among the risks a life insurer will face on its portfolio of guaranteed products are market (e.g. interest rate, equity) risk, mortality risk, and the risk associated with policyholder behaviour (e.g. lapse). Research on this topic will focus on two directions. The first will be the optimal behaviour for policyholders. Secondly, the impact of the insurer's decision making process on the policyholder will be studied.