Statistical and computational topics from modern finance and insurance

现代金融和保险的统计和计算主题

• 批准号: RGPIN-2015-04059
• 负责人: Kolkiewicz, Adam
• 金额: $ 1.02万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=655829
• 关键词: Statistical; computational; topics; modern; finance

While many mathematical models used in modern finance are formulated in continuous time, in practice they are used in discrete time only. An important example of such an approach is delta hedging, where sensitivity to the market risk obtained from a continuous-time model is used to create a locally static hedge portfolio. Due to the importance of the problem, numerous studies have been aimed at describing different properties of the induced hedging error. The majority of the results, however, deal with European options, despite the fact that most of the traded options depend on the price-path of the underlying security. I recently demonstrated that delta hedging for Asian options is significantly less efficient than optimal risk-minimizing hedging strategies. This result has important practical implications, and I plan to broaden and intensify my research in this area. One direction is to provide similar analysis for other path-dependent options, including American and barrier options. Second direction is to consider models suitable in other areas of applications, such as insurance.*******I also plan to develop new statistical procedures for risk measurement. It is known that the two steps in measuring the risk, estimating the loss distribution and computing the risk measure, are in practice intertwined. In particular, it has been shown that a risk measurement procedure will have desirable properties only if the estimation procedure is robust. I plan to study a particular class of such methods, which minimize a distance between the data and the model. They have many attractive properties, including robustness and the ability to produce goodness-of-fit tests. My research will focus on methods based on Kolmogorov-Levy metrics, which have the desirable property of generating large neighborhoods. Currently more complete results exist only for the location parameter, which is not sufficient for most applications.*******Another area of my research is related to the development of efficient integration methods for high-dimensional problems, which often arise in statistics, computational finance, and applied mathematics. For such problems, simulation methods are typically the only feasible approach, but to be practical techniques are required to enhance efficiency. Among such techniques, particularly successful are those that effectively reduce the dimension of the problem by identifying its important coordinates. Recently I developed a novel approach to dimension reduction for problems involving integrals of Brownian motion, and I plan to explore several important extensions of the method, one of which is a generalization to diffusion processes. The resulting techniques are potentially applicable to a range of problems in finance, including pricing of interest rate derivatives and risk management.*********

虽然现代金融中使用的许多数学模型都是在连续时间内制定的，但实际上它们只在离散时间内使用。这种方法的一个重要例子是delta对冲，其中从连续时间模型获得的市场风险的敏感性被用来创建一个局部静态对冲投资组合。由于该问题的重要性，许多研究旨在描述不同性质的诱导套期保值错误。大多数的结果，然而，处理欧式期权，尽管事实上，大多数的交易期权依赖于价格路径的基础证券。 我最近证明了亚式期权的Delta对冲比最优风险最小化对冲策略的效率低得多。这一结果具有重要的实际意义，我计划扩大和加强我在这一领域的研究。一个方向是为其他路径依赖期权提供类似的分析，包括美式期权和障碍期权。 第二个方向是考虑适用于其他应用领域的模型，例如保险。我还计划制定新的风险衡量统计程序。众所周知，风险度量的两个步骤，即估计损失分布和计算风险度量，在实践中是相互交织的。 特别是，它已被证明，风险测量程序将具有理想的属性，只有当估计程序是强大的。 我计划研究一类特殊的方法，它可以最小化数据和模型之间的距离。它们有许多吸引人的特性，包括鲁棒性和产生拟合优度测试的能力。我的研究将集中在基于Kolmogorov-Levy度量的方法上，这些方法具有生成大邻域的理想属性。目前，只有位置参数的结果更完整，这对大多数应用程序来说是不够的。**我的研究的另一个领域是与高维问题的有效集成方法的发展有关，这些问题经常出现在统计学，计算金融学和应用数学中。 对于这样的问题，模拟方法通常是唯一可行的方法，但要实用的技术，以提高效率。在这些技术中，特别成功的是那些通过确定问题的重要坐标而有效地减少问题的维度的技术。最近，我开发了一种新的方法来降低涉及布朗运动积分的问题的维数，我计划探索几个重要的扩展方法，其中之一是扩散过程的推广。 由此产生的技术可能适用于金融领域的一系列问题，包括利率衍生品的定价和风险管理。