Numerical Simulations for Time-Fractional Black-Scholes Equations

This paper implements an efficient numerical algorithm for the time-fractional Black-Scholes model governing European options. The proposed method comprises the Crank-Nicolson approach to discretize the time variable and exponential B-spline approximation for the space variable. The implemented method is unconditionally stable. We present few numerical examples to confirm the theory. Numerical simulations with comparisons exhibit the supremacy of the proposed approach.

💡 Research Summary

The manuscript presents a novel numerical scheme for solving the time‑fractional Black‑Scholes equation (TFBSM), which extends the classical Black‑Scholes model by incorporating a Caputo‑type fractional derivative of order (0<\mu\le 1) in time. This fractional term captures memory effects and anomalous diffusion that are observed in real market data, especially for short‑term option dynamics.

Model formulation – Starting from the standard Black‑Scholes PDE, the authors perform a logarithmic transformation of the asset price ((\zeta=e^{y})) and a time reversal ((\tau = T-t)), obtaining a PDE with a modified Riemann‑Liouville fractional derivative (,{0}D^{\mu}{\tau}u(y,\tau)). The resulting equation (Eq. (4) in the paper) contains diffusion, convection, and reaction terms with constant coefficients (\kappa_{1}= \sigma^{2}/2), (\kappa_{2}= r-D-\kappa_{1}), (\kappa_{3}=r), plus an optional source term (g(y,\tau)) used for validation. The infinite spatial domain is truncated to (