Numerical analysis of fractional order Black-Scholes option pricing model with band equation method

In recent years, there has been a surge of interest in fractional calculus, primarily due to its enhanced capability in accurately modeling complex systems compared to classical methods. This study delves into the fractional Black-Scholes option pricing model, which adeptly captures the intricate dynamics and memory effects inherent in financial markets. The novelty of this research lies in introducing a novel approach employing band matrix equations for numerically solving the fractional order Black-Scholes option pricing model. We explore the numerical solutions for fractional European options using three distinct methods: Laplace Transform, Hilbert Transform, and Monte Carlo simulation. Each method's convergence is rigorously proven, and their respective convergence orders are derived. Furthermore, we conduct a comprehensive analysis to discern the factors influencing the convergence rates for each method. Additionally, we observe the effectiveness and efficiency of the proposed methods in solving fractional European options through numerical examples. Our findings offer an indepth insight into the comparative performance of Laplace transform, Hilbert transform, and Monte Carlo simulation in terms of accuracy, computational complexity, and stability. This research contributes significantly to the comprehension of fractional European option pricing and provides valuable guidance for both practitioners and researchers in selecting the most suitable method for pricing these intricate financial instruments.