Inverse Problems in Mathematical Finance: Exploring Bayesian Model Selection Algorithms

数学金融中的反问题：探索贝叶斯模型选择算法

• 批准号: 9973226
• 负责人: Marco Avellaneda
• 金额: $ 22.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973226&HistoricalAwards=false
• 关键词: Inverse; Problems; Mathematical; Finance; Exploring

Technical DescriptionThis project explores information-theoretic ideas for solving modelselection problems in Mathematical Finance. Typically, these problemsconsist in the specification of a diffusion measure, or more generally, ameasure on path-space, which describes the future states of the market.The data used in the inversion consists of expected values of functionals,which correspond to observed prices of "benchmark" securities. Thisproblem is studied from the point of view of partial differentialequations and Monte-Carlo simulation. We focus on model selectioncriteria based on minimizing the Kullback-Leibler entropy distance betweenthe unknown probability and a Bayesian prior. In the special case ofdiffusion processes, this leads to a constrained stochastic controlproblem that can be solved via Lagrange multipliers. In the case of MonteCarlo simulation, one must construct appropriate weighted measures, in atechnique which is reminiscent of "importance sampling." These problemsand their generalizations to non-linear constraints will be studied in aunified way using methods of Applied Mathematics and Numerical Analysis.The goal is to achieve a better understanding of the question of modelselection in Financial Economics, which is crucial for the management offinancial risk by quantitative methods.Non-Technical DescriptionThis research proposal deals with the pricing and hedging of complexfinancial instruments called derivative securities. The technology derivedhere, which is based on the mathematical fields of probability, statisticsand numerical analysis, is used to develop accurate tools for pricing andmanaging complex financial instruments. The motivation for this researchcomes from the fact that quantitative finance offers many challengingmathematical and computer-related problems. This is a consequence of theso-called "globalization'' phenomenon that links different financialmarkets and economies throughout the world. The current proposal dealswith new mathematical methods for fine-tuning these models. Our aim is tobetter understand how they work and how they represent financial risk. Bybringing to bear robust statistical methods and powerful mathematicaltechniques, we expect to shed light on pricing and risk-management systemsand to develop better models that can be shared with the financialindustry. This is an important application of Mathematics to a new areaof research: quantitative finance. The proposal is part of a greatereffort at New York University's Courant Institute in the field of financeand markets. So far, we have been successful in training young scientiststhat enter the business arena with a unique set of professional skills.This suggests that such research is both directly and indirectly suitablein terms of the larger picture of the national economy.

技术描述本项目探讨解决金融数学模型选择问题的信息理论思想。 通常，这些问题包括在一个扩散措施的规格，或更一般地说，在路径空间上的测量，它描述了市场的未来状态，在反演中使用的数据由期望值的泛函，这对应于观察到的价格的“基准”证券。 从偏微分方程和蒙特-卡罗模拟的角度研究了这个问题。 我们专注于模型选择标准的基础上最小化Kullback-Leibler熵距离之间的未知概率和贝叶斯先验。 在扩散过程的特殊情况下，这导致了一个约束随机控制问题，可以通过拉格朗日乘子解决。 在蒙特卡洛模拟的情况下，必须采用一种让人想起“重要性抽样”的技术来构建适当的加权度量。“这些问题及其对非线性约束的推广将使用应用数学和数值分析的方法以统一的方式进行研究。目标是更好地理解金融经济学中的模型选择问题，这对于通过定量方法管理金融风险至关重要。技术说明本研究计划涉及称为衍生证券的复杂金融工具的定价和对冲。这里所衍生的技术，是基于概率论、几何学和数值分析的数学领域，用于开发精确的工具，为复杂的金融工具定价和管理。 这项研究的动机来自于这样一个事实，即定量金融提供了许多具有挑战性的数学和计算机相关的问题。这是所谓的“全球化”现象的结果，这种现象将世界各地不同的金融市场和经济联系在一起。 目前的建议涉及新的数学方法微调这些模型。我们的目标是更好地了解它们是如何工作的，以及它们如何代表金融风险。通过引入稳健的统计方法和强大的分析技术，我们希望能够揭示定价和风险管理系统，并开发出更好的模型，与金融行业共享。 这是一个重要的应用数学的一个新的研究领域：定量金融。这项提议是纽约大学柯朗研究所在金融和市场领域的重大改革的一部分。到目前为止，我们已经成功地培养了具有独特专业技能的年轻科学家进入商业竞技场。这表明，这种研究直接和间接地适合于国民经济的大局。