AMC-SS: Research on Dependence of Stochastic Processes and on Mathematical Aspects of Credit Derivatives and Convertible Bonds

AMC-SS：随机过程依赖性以及信用衍生品和可转换债券的数学方面的研究

• 批准号: 0604789
• 负责人: Tomasz Bielecki
• 金额: --
• 依托单位: Illinois Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2010-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604789&HistoricalAwards=false
• 关键词: AMC; SS; Research; Dependence; Stochastic

This research project will continue development of new methodologies and models in the areas of stochastic analysis, and stochastic methods in finance and financial engineering, for the purpose of solving complex problems in financial decision making and risk management. Particular emphasis will be put on applications to valuation and hedging of credit derivatives, with focus on credit default swaps and basket credit derivatives, as well on applications to valuation and hedging of convertible bonds. One goal of the project is therefore development of a sound mathematical theory of basket credit derivatives and convertible bonds and related issues of hedging, valuation, and management of credit risk and convertible risk. This in particular will require new results in representation for stochastic processes, as well as new results for modeling of stochastic dependence between random processes. Thus, another goal of the project will be to use stochastic analysis in finite and infinite dimensions for studying of dependence between stochastic processes. In particular, a new theory of semimartingale copulae and Markov copulae will be worked out and applied. Moreover, new applications of stochastic analysis will be developed, in particular with regard to martingale representations and backward stochastic differential equations with reflections (also driven by jump martingales).This research project will be of fundamental importance for several reasons, both from the applications point of view as well as from the purely theoretical perspective. First, the booming credit derivative industry will benefit from it, as development of tractable mathematical tools for the purpose of valuing and managing of basket credit derivatives, such as basket swaps, collateralized debt obligations, and credit indices, will provide the industry with new methodologically sound procedures. Likewise, the convertible bond industry will benefit from this research as our new decomposition results specified for the regime switching market model should provide a better quantitative tools for this industry, which suffered great losses in the Spring of 2005 -- possibly because the nature of such complex hybrid derivatives as convertible bonds was not really well understood. In addition, valuation and hedging of credit default swaps, which is essential for the finance industry, will be specifically emphasized and new analytical tools will be developed for this purpose. On theoretical side, project will also be of fundamental importance for several reasons. First, if true, then an analog of Sklar's theorem for the case of probability measures on canonical spaces of stochastic processes will be an important extension of the classical theorem of A. Sklar (1959), which was proved for real valued random variables. Perhaps, an analog of Sklar's theorem for probability measures on some general vector spaces (such as Polish spaces) will be derived in the process. Second, for those semimartingale processes for which their local characteristics determine their laws it will be important to study the following question: what is the class of multivariate (vector valued) semimartingales with given univariate local characteristics. Given the strategic importance of basket products for financial industry there will be a practical importance of studying of the above problems in view of potential applications, such as valuation and hedging of basket derivatives (basket options, basket credit derivatives, etc.).

该研究项目将继续在金融和金融工程中的随机分析和随机方法领域开发新的方法和模型，以解决金融决策和风险管理中的复杂问题。将特别强调对信用衍生品的估值和套期保值的应用，重点是信用违约互换和一篮子信用衍生品，以及对可转换债券的估值和套期保值的应用。因此，该项目的目标之一是发展一套健全的一篮子信用衍生品和可转换债券的数学理论，以及信用风险和可转换风险的套期保值、估值和管理等相关问题。这尤其需要在随机过程的表示上有新的结果，以及在随机过程之间的随机依赖的建模上有新的结果。因此，该项目的另一个目标将是利用有限维和无限维的随机分析来研究随机过程之间的依赖性。特别地，将提出并应用半鞅和马尔可夫共轭的新理论。此外，将开发随机分析的新应用，特别是关于鞅表示和带反射的倒向随机微分方程（也由跳跃鞅驱动）。无论从应用的角度还是从纯理论的角度来看，这个研究项目都将具有根本性的重要性。首先，蓬勃发展的信用衍生品行业将从中受益，因为用于评估和管理一篮子信用衍生品（如一篮子掉期、债务抵押债券和信用指数）的易于处理的数学工具的发展，将为该行业提供新的方法上健全的程序。同样，可转换债券行业将从这项研究中受益，因为我们为制度转换市场模型指定的新分解结果应该为该行业提供更好的定量工具，该行业在2005年春季遭受了巨大损失-可能是因为可转换债券等复杂混合衍生品的性质没有得到很好的理解。此外，将特别强调对金融业至关重要的信用违约掉期的估值和对冲，并为此开发新的分析工具。在理论方面，由于几个原因，项目也将是至关重要的。首先，如果成立，那么关于随机过程正则空间上概率测度的Sklar定理的类比将是A. Sklar（1959）经典定理的重要扩展，该定理已被证明适用于实值随机变量。也许，在这个过程中，一些一般向量空间（如波兰空间）上的概率测度的类似的Sklar定理将被推导出来。其次，对于那些局部特征决定其规律的半鞅过程，研究以下问题将是重要的：具有给定单变量局部特征的多变量（向量值）半鞅是哪一类？鉴于篮子产品对金融业的战略重要性，考虑到篮子衍生品（一篮子期权、一篮子信用衍生品等）的估值和套期保值等潜在应用，研究上述问题具有现实意义。