This thesis is devoted to the study of the Hilbert transform and its applications in computationalfinance. We will show in this thesis that under some mild conditions, the Hilbert transform can be approximated by the discrete Hilbert transforms with exponentially decaying errors in both one dimensional and two dimensional cases. The resulting discrete Hilbert transform can be efficiently implemented using fast Fourier transform. Based on this theory, many effective numerical schemes are developed to price European and American type vanilla and exotic options under various financial assets models.
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