Numerical Simulations for Time-Fractional Black-Scholes Equations
Neetu Garg, A. S. V. Ravi Kanth
TL;DR
This paper addresses European option pricing under a time-fractional Black-Scholes framework that captures memory effects via the fractional derivative $_{0}D_{ au}^{\mu}$. It proposes a numerical solver that couples Crank-Nicolson time discretization with exponential B-spline spatial approximation on a truncated domain, achieving unconditional stability with second-order spatial accuracy and $2-\mu$ temporal accuracy. Numerical experiments with a manufactured solution and comparisons to existing methods demonstrate high accuracy and computational efficiency, and illustrate how the fractional order $\mu$ influences option prices. The work provides a robust, fast tool for pricing European options in fractional models and highlights the practical relevance of memory effects in financial pricing.
Abstract
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.