Abstract :

In this research, we explore the nonlinear generalized distributed-order time-fractional Black-Scholes equation using an implicit numerical approach. Finite difference techniques are employed to approximate the time and spatial derivatives. Our numerical results exhibit high accuracy, underscoring the method’s robustness in addressing financial models. Additionally, our approach offers significant advantages in computational efficiency and stability. By using the implicit method, we ensure solution stability even with larger time steps, which is particularly beneficial for long-term financial modeling. The implications of this study extend beyond financial engineering. The methodologies developed can be adapted to solve various fractional differential equations in different scientific and engineering fields. The successful application of these techniques to the Black-Scholes equation suggests their potential utility in other areas requiring precise and efficient numerical solutions.

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Keywords :

AMS, Black-Scholes model, Generalized Fractional distributed-order derivatives, Trust region-dogleg algorithm. Subject Classification: 26A33; 35G25; 65D07.

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