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，但实际上它们仅在离散时间内使用，这种方法的一个重要例子是德尔塔对冲，其中使用从连续时间模型获得的对市场风险的敏感性。由于该问题的重要性，许多研究都旨在描述引起的对冲误差的不同属性，然而，大多数结果都涉及欧式期权，尽管大多数研究都是针对欧式期权的。交易的期权取决于标的资产的价格路径我最近证明了亚洲期权的 Delta 对冲策略的效率明显低于最优风险最小化对冲策略，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。对于其他路径依赖期权，包括美国期权和障碍期权，第二个方向是考虑适用于其他应用领域的模型，例如保险。********我还计划开发新的风险衡量统计程序。已知衡量风险的两个步骤，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，事实证明，只有当估计过程是稳健的时，风险测量过程才会具有理想的属性，我计划研究此类方法的特定类别。它们具有许多吸引人的特性，包括稳健性和产生拟合优度检验的能力，这些方法具有以下理想特性。目前生成大型社区较为完整。结果仅存在于位置参数，这对于大多数应用来说是不够的。********我研究的另一个领域与高维问题的有效积分方法的开发有关，这些问题经常出现在统计中对于此类问题，模拟方法通常是唯一可行的方法，但需要实用技术来提高效率，其中特别成功的是那些通过识别来有效降低问题维度的技术。最近我开发了一种新的方法来解决涉及布朗运动积分的问题，我计划探索该方法的几个重要扩展，其中之一是对扩散过程的推广，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。**** *****