Statistical and computational topics from modern finance and insurance

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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.