Topics in stochastic processes and mathematical finance: counterparty risk valuation and hedging, Markov consistency and Markov copulae, and dynamic performance assessment indices

随机过程和数学金融主题：交易对手风险评估和对冲、马尔可夫一致性和马尔可夫联结函数以及动态绩效评估指数

• 批准号: 1211256
• 负责人: Tomasz Bielecki
• 金额: $ 34.43万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211256&HistoricalAwards=false
• 关键词: Topics; stochastic; processes; mathematical; finance

BieleckiDMS-1211256 The investigator and his colleague continue development of new mathematical methods for risk management in complex stochastic dynamical systems, such as financial markets. They consider problems in three areas. The first is mathematical modeling for financial risk management, with application to hedging, valuation, and management of counterparty credit risk (CCR). Here they develop mathematical tools for evaluating and managing counterparty risk embedded in a large variety of over-the-counter contracts. The valuation and hedging of CCR in credit default swaps and interest rate swaps, which are essential for the financial industry, are studied and new analytical tools are developed for this purpose. The second area involves applications of stochastic analysis to investigation of dependence between Feller-Markov processes. The investigators study dynamic "copula" problems with regard to Markov processes: given N one-dimensional Markov processes, construct a multivariate Markov process such that each component is also Markovian in its own filtration, and such that its law agrees with the law of the original Markov process. The third area deals with mathematical modeling of dynamic performance assessment indices with applications to conic finance. New applications of convex analysis, probability, and L0-module theory are developed to study dynamic Performance Assessment Indices. Dynamic performance assessment indices are measures of performances of a given activity in a random environment. They apply this theory to conic finance for the purpose of computing acceptable bounds for arbitrage-free bid and ask prices in illiquid markets. The investigator and his colleague develop methods to analyze and manage risk in financial markets. By viewing markets as stochastic systems whose behavior changes in time, they can bring to bear concepts from dynamical systems, probability, and stochastic systems. They aim to take into account the provisions of recent legislation, such as the Dodd-Frank act of 2011, and the provisions of the Basel III regulations. Outcomes of the project contribute to increasing the efficiency and competitiveness of financial institutions (both government and private), by effectively controlling the counterparty risk and, by extension, the systemic risk in financial markets. The study of dynamic performance measures and assessment indices in particular provides new tools beyond the classical Value at Risk or Sharpe Ratio for measuring the risk and performance of a given financial institution (or portfolio). Results are useful for market participants, including regulators and government agencies. The project includes the training of graduate students in stochastic analysis and its application to financial markets.

BieleckiDMS-1211256这位研究员和他的同事继续为复杂随机动力系统(如金融市场)的风险管理开发新的数学方法。他们从三个方面考虑问题。第一个是金融风险管理的数学模型，应用于对冲、评估和管理交易对手信用风险。在这里，他们开发了数学工具，用于评估和管理嵌入在大量场外合约中的交易对手风险。研究了信用违约互换和利率互换中信用违约互换和利率互换中CCR的估值和套期保值，并为此开发了新的分析工具。第二个领域涉及随机分析在研究费勒-马尔可夫过程之间的相关性方面的应用。研究了关于马尔可夫过程的动态Copula问题：给定N个一维马尔可夫过程，构造一个多元马尔可夫过程，使得每个分量在其自身的过滤中也是马尔可夫的，并且它的定律与原始的马尔可夫过程的定律一致。第三个领域涉及动态绩效评估指标的数学建模及其在圆锥金融中的应用。拓展了凸分析、概率论和L0模理论在动态绩效评价指标研究中的新应用。动态绩效考核指标是对某一特定活动在随机环境中的绩效的衡量。他们将这一理论应用于圆锥金融，目的是计算非流动性市场中无套利买卖价格的可接受界限。这位调查员和他的同事开发了分析和管理金融市场风险的方法。通过将市场视为行为随时间变化的随机系统，它们可以应用动力系统、概率和随机系统的概念。它们旨在考虑最近立法的条款，如2011年的多德-弗兰克法案和巴塞尔III法规的条款。该项目的成果有助于提高金融机构(政府和私营部门)的效率和竞争力，有效控制交易对手风险，进而控制金融市场的系统性风险。尤其是对动态业绩衡量和评估指标的研究，为衡量特定金融机构(或投资组合)的风险和业绩提供了超越传统风险价值或夏普比率的新工具。结果对包括监管机构和政府机构在内的市场参与者是有用的。该项目包括对研究生进行随机分析及其在金融市场中的应用的培训。