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。

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.