Statistical and computational topics from modern finance and insurance

虽然现代金融中使用的许多数学模型都是在连续时间内建立的，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增量套期保值，即利用从连续时间模型获得的对市场风险的敏感性来创建局部静态对冲组合。由于这一问题的重要性，许多研究旨在描述诱致性套期保值错误的不同性质。然而，大多数结果都涉及欧式期权，尽管大多数交易的期权取决于标的证券的价格路径。我最近证明，针对亚洲期权的增量对冲比最优风险最小化对冲策略的效率要低得多。这一结果具有重要的现实意义，我计划拓宽和深化我在这一领域的研究。一个方向是为其他路径依赖型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式，例如保险。 我还计划开发新的风险衡量统计程序。众所周知，风险度量的两个步骤--估计损失分布和计算风险度量--实际上是相互交织的。特别是，已经证明，只有当估计过程是稳健的时，风险度量过程才具有所希望的性质。我计划研究一类特殊的此类方法，这种方法可以最小化数据和模型之间的距离。它们具有许多吸引人的特性，包括健壮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性质。目前只有Location参数有更完整的结果，这对大多数应用程序来说是不够的。 我的另一个研究领域是关于高维问题的有效积分方法的开发，这些问题经常出现在统计学、计算金融和应用数学中。对于这类问题，模拟方法通常是唯一可行的方法，但为了实用，需要技术来提高效率。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来有效降低问题维度的技术。最近，我发展了一种新的方法来解决涉及布朗运动积分的降维问题，并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。

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.