Year
2026
Season
Spring
Paper Type
Master's Thesis
College
College of Arts and Sciences
Degree Name
Master of Science in Mathematical Sciences (MS)
Department
Mathematics & Statistics
NACO controlled Corporate Body
University of North Florida. Department of Mathematics and Statistics
Committee Chairperson
Dr. Michelle R. DeDeo
Second Advisor
Dr. Daniela Genova
Rights Statement
http://rightsstatements.org/vocab/InC/1.0/
Third Advisor
Dr. Fei Heng
Fourth Advisor
Dr. Kening Wang
Department Chair
Dr. Richard Patterson
College Dean
Dr. Kaveri Subrahmanyam
Abstract
We compare five numerical approaches for approximating solutions to the Black–Scholes partial differential equation for pricing European call options: FTCS, BTCS, Crank– Nicolson, Monte Carlo simulation, and a physics–informed neural network (PINN). These methods span finite difference techniques, probabilistic simulation, and machine learning. Performance is evaluated based on computational efficiency and accuracy relative to the analytical Black–Scholes solution.
Among the methods, Crank–Nicolson and the PINN demonstrated the strongest overall performance. Crank–Nicolson achieved the highest accuracy but exhibited increased runtime as the number of underlying stock price grid points grew. In contrast, the PINN produced slightly less accurate results but with significantly lower computational cost and the advantage of a continuous approximation over the domain.
These results highlight a trade–off between accuracy and efficiency. Crank–Nicolson is preferred when high precision at fixed grid points is required, while the PINN offers a flexible and computationally efficient alternative. Overall, the PINN provides the best balance between speed and accuracy in this study.
Suggested Citation
Williams, Scott Cameron, "Performance of numerical methods applied to the Black–Scholes Model" (2026). UNF Graduate Theses and Dissertations. 1408.
https://digitalcommons.unf.edu/etd/1408
Included in
Applied Statistics Commons, Numerical Analysis and Computation Commons, Numerical Analysis and Scientific Computing Commons, Partial Differential Equations Commons
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