Computation of Multivariate Barrier Crossing Probability, and Its Applications in Finance
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In this thesis, we consider computational methods of finding exit probabilities for a class of multivariate stochastic processes. While there is an abundance of results for one-dimensional processes, for multivariate processes one has to rely on approximations or simulation methods. We adopt a Large Deviations approach in order to estimate barrier crossing probabilities of a multivariate Brownian Bridge. We use this approach in conjunction with numerical techniques to propose an efficient method of obtaining barrier crossing probabilities of a multivariate Brownian motion. Using numerical examples, we demonstrate that our method works better than other existing methods. We present applications of the proposed method in addressing problems in finance such as estimating default probabilities of several credit risky entities and pricing credit default swaps. We also extend our computational method to efficiently estimate a barrier crossing probability of a sum of Geometric Brownian motions. This allows us to perform a portfolio selection by maximizing a path-dependent utility function.