FRG: The Mathematics of Financial Risk Management

该项目的目标是制定合理和可实施的方法来衡量和控制金融行业的风险。这个话题很及时，因为金融业已经出现了一些引人注目的失败，因为正在讨论将定量风险措施制度化的法规，因为主要的风险措施“风险价值”具有不受欢迎的性质。其中最主要的是，在某些情况下，在信贷风险市场中经常出现的情况下，满足监管的风险价值要求会鼓励风险集中，而不是分散风险。该提案建立在一个团队开发的一致风险度量的想法之上，该团队包括该提案的主要研究人员之一。这个团队开发了一套被认为是可取的公理，然后描述了那些满足这些公理的“一致的”风险度量。他们的工作表明，每一个连贯的风险度量都可以用一组对可能的未来情景的概率度量来表征。提出了一个由三部分组成的项目。第一部分是在一致的风险度量的基础上发展分散的风险控制实施。这需要发展对偶理论和企业内部的均衡定价。第二是在真实设置中引出场景并估计模型参数。这就要求对数学模型进行精化，以便在未来情景中呈现出可管理的测量方法，并利用统计理论来做出这些选择。最后，提出了对已有的相干风险测度理论的扩展。该项目得到了数学科学活动重点研究小组的支持。

The goal of this project is to develop rational and implementable methods to measure and control risk in the finance industry. The topic is timely because there have been some spectacular failures in the finance industry, because regulations are under discussion which would institutionalize quantitative risk measures, and because the leading risk measure, "value-at-risk," has undesirable properties. Chief among these is that in certain situations, situations that often arise in credit-risky markets, meeting regulatory value-at-risk requirements encourages concentration of risk rather than diversification. The proposal builds on the idea of coherent risk measures developed by a team, which includes one of the principal investigators on this proposal. This team developed a set of axioms deemed to be desirable, and then characterized those "coherent" risk measures which satisfy these axioms. Their work showed that every coherent risk measure can be characterized by a set of probability measures over possible future scenarios. A three-part project is proposed. The first part is to develop a decentralized implementation of risk control based on coherent risk measurement. This requires development of duality theory and equilibrium pricing within a firm. The second is to elicit scenarios and estimate model parameters in real settings. This mandates refining the mathematical models to the point that a manageable number of measures over future scenarios present themselves, and appealing to statistical theory to make those choices. Finally, it is proposed to extend the existing theory of coherent risk measures. This project is being supported under the Focused Research Groups in the Mathematical Sciences activity.