Extrapolation of difference methods in option valuation, rounding error in numerical solution of stochastic differential equations, and shooting methods for stochastic boundary-value problems
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My dissertation involves three different projects. The first project involves numerical solution of option prices. In particular, the fully implicit and Crank-Nicolson difference schemes for solving option prices are analyzed. It is proved that the error expansions for the difference methods have the correct form for applying Richardson extrapolation to increase the order of accuracy of the approximations. The difference methods are applied to European, American, and down-and-out knock-out call options. Computational results indicate that Richardson extrapolation significantly decreases the amount of computational work (by as much as a factor of 16) in estimation of option prices. The second project involves an analysis of rounding errors in numerical solution of stochastic differential equations. A statistical rounding error analysis of Euler's method for numerically solving stochastic differential equations is performed. In particular, rounding errors associated with the mean square error and for functional expectations of the solutions are investigated. It is shown that rounding error is inversely proportional to the square root of the step size. An extrapolation technique provides second-order accuracy, and is one way to increase accuracy while avoiding rounding error. Several computational results are given, which support the theoretical results. The third project is a development of shooting methods for numerical solution of Stratonovich boundary-value problems. In particular, shooting methods are examined for numerically solving systems of Stratonovich boundary-value problems. It is proved that these methods accurately approximate the solutions of stochastic boundary-value problems. An error analysis of these methods is performed. Computational simulations are given.