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.*******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.*********