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。 由于目前全球市场动荡，并考虑到引发信贷紧缩的已知原因，将特别强调信贷衍生工具的动态估值和对冲，重点是信贷违约掉期和一揽子信贷衍生工具，以及其他复杂的金融工具，如通货膨胀指数掉期。本研究将架构以下两个主要的研究领域：金融风险管理的数学建模，应用，除其他外，对冲，估值和管理的投资组合信用风险，和随机分析的应用，研究随机过程之间的依赖关系。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，预计它将导致新的和实用的工具，最终成为广泛使用的金融行业和其他应用学科，以减轻和适当地控制所涉及的风险。这项研究将具有根本性的重要性有几个原因。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，用于评估和管理篮子信用衍生品，如篮子互换，现金抵押债务，资产支持证券，自2007年春季以来，该行业遭受了巨大损失，主要是因为人们对结构性信用衍生品等复杂混合衍生品的性质并没有真正了解，特别是在风险管理层面（风险敞口对冲）。要做的研究将导致开发新的强大的动态模型，这些证券，并提供可靠的动态对冲策略与这些模型，从而将提供一个更好的和更广泛的理解现代金融市场的一个重要组成部分。其次，将开发随机分析的新应用，特别是在动态可接受性指标方面。 动态可接受性指标是一个给定的现金流的性能，这将从抽象的概率的角度来研究的无单位的措施。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，特别强调金融业中必不可少的信用违约互换（CDS）的估价和对冲，以及信用违约互换的估价和对冲，并开发新的分析工具。第四，如果是真的，那么一个类似于斯克拉尔定理的概率测度的随机过程的典范空间将是一个重要的扩展经典斯克拉尔定理。也许，在这个过程中会推导出一个类似于Sklar定理的关于某些一般向量空间（如波兰空间）上概率测度的定理。第五，研究人员将研究关于马尔可夫过程的动态“copula”问题：对于一个给定的多维马尔可夫过程，每个坐标也是马尔可夫的，什么条件需要满足的伪微分算子对应的多维过程的（扩展的）无穷小生成元，假定对应于每个坐标的（扩展的）无穷小生成元的伪微分算子是已知的。第六，从篮子衍生产品（一篮子股票期权、一篮子信用衍生产品等）的估价和套期保值等应用的角度来看，研究上述问题具有重要的现实意义。

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.).