AMC-SS: Mathematical foundations of responsible risk management in credit markets

研究的目的是在随机分析和金融工程领域开发新的数学方法和模型，以解决与金融风险管理和金融决策相关的复杂而重要的问题。由于当前全球市场动荡，并考虑到引发信贷紧缩的已知原因，将特别强调信用衍生品的动态估值和对冲，重点是信用违约掉期和一篮子信用衍生品，以及其他复杂的金融工具，如通胀指数掉期。本研究将构建以下两个主要研究领域：以金融风险管理为目的的数学建模，及其在套期保值、估值和投资组合信用风险管理方面的应用，以及应用随机分析研究随机过程之间的依赖性。虽然即将进行的研究将在随机分析和金融数学领域产生根本性的进展，但预计它将导致新的实用工具最终广泛应用于金融行业和其他应用学科，从而减轻和适当控制所涉及的风险。由于几个原因，这项研究将具有根本性的重要性。首先，陷入困境的信用衍生品行业将直接从中受益，因为基于坚实的数学和金融进步的新方法和技术将被开发出来，用于评估和管理一篮子信用衍生品，如一篮子掉期、现金抵押债务凭证、资产支持证券等。自2007年春季以来，这个行业遭受了巨大损失，主要是因为人们对结构信用衍生品等复杂混合衍生品的性质并没有很好地理解，尤其是在风险管理层面（风险敞口的对冲）。所做的研究将导致为这些证券开发新的健壮的动态模型，并提供与这些模型相关的可靠的动态对冲策略，从而使人们更好、更广泛地理解现代金融市场的一个重要组成部分。第二，将开发随机分析的新应用，特别是在动态可接受性指数方面。动态可接受度指标是对给定现金流绩效的无单位度量，将从抽象概率的角度对其进行研究。所获得的结果将有利于所有市场参与者，包括监管机构和政府机构，并将导致更好地理解市场效率。第三，将特别强调对金融行业至关重要的信用违约互换（CDS）的估值和对冲以及信用违约互换的估值和对冲，并为此开发新的分析工具。第四，如果成立，那么关于随机过程正则空间上概率测度的Sklar定理的类比将是经典Sklar定理的重要扩展。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。第五，研究马尔可夫过程的动态“copula”问题：对于给定的多维马尔可夫过程，每个坐标都是马尔可夫的，假设每个坐标的（扩展）无穷小生成器对应的伪微分算子已知，该多维过程的（扩展）无穷小生成器对应的伪微分算子需要满足什么条件。第六，鉴于潜在的应用，如一篮子衍生品（一篮子股票期权、一篮子信用衍生品等）的估值和套期保值，对上述问题的研究将具有现实意义。

The aim of the research to be done is to develop new mathematical methodologies and models in the areas of stochastic analysis and financial engineering, for the propose of solving complex and important problems related to financial risk management and to financial decision making. Due to the current world-wide market turbulence, and taking into account the known causes that triggered the credit crunch, particular emphasis will be put on dynamic valuation and hedging of credit derivatives, with focus on credit default swaps and basket credit derivatives, as well as other complex financial instruments such as inflation-indexed swaps. This research will frame the following two main research areas: mathematical modeling for the purpose of financial risk management, with applications, among others, to hedging, valuation, and management of portfolio credit risk, and applications of stochastic analysis to studying of dependence between stochastic processes. Although the forthcoming research will yield fundamental advances within the areas of stochastic analysis and financial mathematics, it is anticipated that it will lead to new and practical tools that eventually become widely used in the financial industry and other applied disciplines, so to mitigate and to appropriately control the involved risks.This research will be of fundamental importance for several reasons. First, the troubled credit derivative industry will benefit directly from it, as new methods and techniques based on solid mathematical and financial advances will be developed for the purpose of valuing and managing basket credit derivatives, such as basket swaps, cash collateral debt obligations, asset backed securities, etc. This industry suffered great losses since the spring of 2007 mainly because the nature of such complex hybrid derivatives as structured credit derivatives was not really well understood, especially on the level of risk management (hedging of risk exposure). The research to be done will result in developing new robust dynamic models for these securities and provide reliable dynamic hedging strategies associated to these models, which consequently will give a better and broader understanding of an important part of the modern financial markets. Second, new applications of stochastic analysis will be developed, in particular with regard to dynamic acceptability indices. Dynamic acceptability indices are unitless measures of performances of a given cashflow, which will be studied from abstract probability point of view. The obtained results will be beneficiary for all market participants including regulators and government agencies, and will lead to a better understanding of market efficiency. Third, valuation and hedging of credit default swaps (CDS) as well as valuation and hedging of credit default swaptions, which are essential for the financial industry, will be specifically emphasized and new analytical tools will be developed for this purpose. Fourth, if true, then an analog of Sklar's theorem for probability measures on canonical spaces of stochastic processes will be an important extension of the classical Sklar's theorem. 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. Fifth, the researchers will study the dynamic "copula" problem with regard to Markov processes: for a given multidimensional Markov process, with each coordinate being also Markovian, what conditions need to be satisfied by the pseudo-differential operator corresponding to the (extended) infinitesimal generator of the multidimensional process, given that the pseudo-differential operator corresponding to the (extended) infinitesimal generator of each coordinate are known. Sixth, 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 equity options, basket credit derivatives, etc.).