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In this dissertation, we discuss the generation of low discrepancy sequences, randomization of these sequences, and the transformation methods to generate normally distributed random variables. Two well known methods for generating normally distributed numbers are considered, namely; Box-Muller and inverse transformation methods. Some researchers and financial engineers have claimed that it is incorrect to use the Box-Muller method with low-discrepancy sequences, and instead, the inverse transformation method should be used. We investigate the sensitivity of various computational finance problems with respect to different normal transformation methods. Box-Muller transformation method is theoretically justified in the context of the quasi-Monte Carlo by showing that the same error bounds apply for Box-Muller transformed point sets. Furthermore, new error bounds are derived for financial derivative pricing problems and for an isotropic integration problem where the integrand is a function of the Euclidean norm. Theoretical results are derived for financial derivative pricing problems; such as European call, Asian geometric, and Binary options with a convergence rate of 1/N. A stratified Box-Muller algorithm is introduced as an alternative to Box-Muller and inverse transformation methods, and new numerical evidence is presented in favor of this method. Finally, a statistical test for pseudo-random numbers is adapted for measuring the uniformity of transformed low discrepancy sequences.
A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Includes bibliographical references.
Giray Okten, Professor Directing Dissertation; Fred Huffer, Outside Committee Member; Brian Ewald, Committee Member; Alec N. Kercheval, Committee Member; Michael Mascagni, Committee Member.
Florida State University
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