FRG: Collaborative Research on Mathematical Methods for Defaultable Instruments

奖项摘要DMS-0456195 / DMS-0455982 / DMS-0456118Rene a . Carmona和K. Ronnie Sircar，普林斯顿大学，jean - pierre Fouque，北卡罗莱纳州立大学，thaleia Zariphopoulou，德克萨斯大学奥斯汀分校：可违约工具数学方法的合作研究本项目研究信贷市场驱动的金融数学问题，其中主要风险来源是债务人对其支付义务的潜在违约。具体而言，考虑的问题是：(1)违约风险的效用无差异估值；Ii)设计工具，以最佳方式提高信用价值；Iii)随机强度模型的渐近分析，研究公司收益率息差的时间尺度内容；Iv)与跨公司违约相关性分析相关的计算问题，建模为交互中的大型系统。第一部分涉及随机控制问题，涉及随机强度模型和无限维利率模型。第二种方法也有重叠，包括对部分观测到的系统进行过滤。第三种方法使用奇异和规则摄动技术来处理在这种情况下产生的相互作用势偏微分方程，第四种方法使用相互作用粒子系统来计算对默认关联敏感的概率，以及为分析罕见事件而设计的蒙特卡罗计算。这个项目的智力价值在于开发适用的科学工具来解决特定类别的优化、设计、校准和计算问题，这些问题对管理违约风险至关重要。可违约工具或信用关联衍生品是一种金融证券，它向持有人支付的金额取决于违约事件的发生（或不发生），如公司（或国家或市政当局）破产、不偿还贷款或未支付抵押贷款。近年来，信贷相关衍生产品市场增长了七倍多，从1997年的1700亿美元未偿名义债券增长到2001年的近1400亿美元。这些工具在建模，分析，计算和估计方面提出了新的挑战，其中一些我们建议在这里通过汇集一个具有应用数学，随机过程和计算统计学专业知识的重点研究小组来研究。该项目更广泛的影响是加深对信贷风险的理解，这些风险影响着从大型商业机构到拥有养老基金和抵押贷款的个人，以及设计和正确评估工具以控制它。该项目还着重于培训五名研究生和一名博士后，他们将通过许多会议，特别是在三年结束时就该研究领域举行一次大型国际会议，从与基础广泛的小组各部分的互动中受益匪浅。PI所获得的经验将反映在他们的专业研究生和本科课程的教学中，并为该领域的高级论文项目提供建议。除了闭幕会议外，研究成果还将通过学术和行业会议、课程和为同行评审期刊撰写的文章进行传播。

Award Abstract DMS-0456195 / DMS-0455982 / DMS-0456118Rene A. Carmona and K. Ronnie Sircar, Princeton UniversityJean-Pierre Fouque, North Carolina State UniversityThaleia Zariphopoulou, University of Texas at AustinFRG: Collaborative Research on Mathematical Methods for Defaultable Instruments This project investigates problems in financial mathematics motivated by credit markets in which a major source of risk is the potential default of debtors on their payment obligations. Specifically, the problems under consideration are i) utility-indifference valuation of default risk; ii) design of instruments to optimally enhance credit worthiness; iii) asymptotic analysis of stochastic intensity models to study the time-scale content of corporate yield spreads; iv) computational issues related to the analysis of correlation between defaults across firms, modeled as large systems in interaction. The first part involves stochastic control problems related to random intensity models and infinite dimensional interest rate models. The second also overlaps and involves filtering of partially observed systems. The third uses singular and regular perturbation techniques for the class of interacting potential partial differential equations arising in this context, and the fourth uses interacting particle systems to compute probabilities which are sensitive to correlation of defaults, as well as Monte Carlo computations designed for the analysis of rare events. The intellectual merit of this project is in developing applicable scientific tools to address the particular class of optimization, design, calibration and computation issues which are essential for managing default risk.Defaultable instruments, or credit-linked derivatives, are financial securities that pay their holders amounts that are contingent on the occurrence (or not) of a default event such as the bankruptcy of a firm (or a country or municipality), non-repayment of a loan or missing a mortgage payment. The market in credit-linked derivative products has grown more than seven-fold in recent years, from $170 billion outstanding notional in 1997, to almost $1400 billion through 2001. These instruments raise new challenges in modeling, analysis, computation and estimation, some of which we propose to study here by bringing together a Focused Research Group with expertise in applied mathematics, stochastic processes and computational statistics. The broader impact of the project is in deeper understanding of credit risks, which affect people from large commercial institutions to individuals with pension funds and mortgages, and designing and correctly valuing instruments to control for it. The project is also strongly geared towards training of five graduate students and one postdoctoral associate, who will benefit enormously from interaction with all parts of the broad-based group through many meetings, and in particular a large international conference on the research area at the end of the three years. The experience gained by the PI's will be reflected in their teaching of specialist graduate and undergraduate classes, and advising Senior Thesis projects in this field. As well as the closing conference, the results of the work will be disseminated through academic and industry meetings, classes and articles written for peer-reviewed journals.