Theses and Dissertations

Advisor

Razzaghi, Mohsen

Committee Member

Yarahmadian, Shantia

Committee Member

Lim, Hyeona

Committee Member

Diegel, Amanda

Date of Degree

12-12-2025

Original embargo terms

Immediate Worldwide Access

Document Type

Dissertation - Open Access

Major

Mathematical Science

Degree Name

Doctor of Philosophy (Ph.D.)

College

College of Arts and Sciences

Department

Department of Mathematics and Statistics

Abstract

This dissertation leverages advanced spectral methods and wavelet techniques to address complex quantitative finance models, enhancing the computation of financial derivatives and risk assessments. Building on foundational studies, this research extends these methods to broader, intricate financial contexts. The first section explores fractional-order generalized Chebyshev wavelets (FOCW) applied to fractional advection equations, relevant in both mathematics and physics. Using a regularized beta function to compute the Riemann-Liouville fractional integral operator, this study introduces a novel numerical scheme with robust accuracy, confirmed through error analysis and empirical tests. The second part examines the fractional Black-Scholes equations for option pricing under subdiffusive dynamics, using fractional-order generalized Taylor wavelets (FGTW). This approach accurately approximates the Greeks of financial derivatives, showcasing precision in financial computation through rigorous error analysis and extensive testing, demonstrating its value for industry applications. Finally, inspired by work on credit risk, this research generalizes the Lévy model to incorporate tempered stable processes, a recent financial innovation. Using radial basis function (RBF) collocation methods, we address the singular nature of partial integro-differential operators in structural credit risk models. This approach enhances both the desingularization and computational efficiency of default probability estimations for public companies. Overall, this dissertation synthesizes and extends current methodologies, introducing new computational techniques that advance quantitative finance. The integration of spectral methods and wavelet techniques provides a powerful framework for tackling challenging problems in financial mathematics.

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