Modified Cubic B-spline Based Differential Quadrature Methods for Time-fractional Black-Scholes Equation

TL;DR

This paper introduces a modified cubic B-spline differential quadrature method to efficiently solve the time-fractional Black-Scholes equation, achieving high accuracy and stability for option pricing models with long-term memory effects.

Contribution

It develops a novel numerical algorithm combining fractional time discretization and modified cubic B-spline-based DQM with proven stability and high convergence order.

Findings

Achieves fourth-order spatial convergence.

Attains order 2−α in time, improving as α approaches 0.

Demonstrates superior accuracy compared to existing methods.

Abstract

The time-fractional Black-Scholes equation (TFBSE) is intended to price the options for which the underlying price fluctuates within a correlated fractal transmission system. Although the TFBSE is an influential approach for grasping the long-term memory traits of financial markets, the non-local nature of fractional derivatives makes significant challenges in finding an accurate solution. We perform an efficient use of the differential quadrature method (DQM) based on modified cubic B-splines to solve the TFBSE governing European options. This paper constructs an algorithm by the combination of time fractional discretization using the finite difference method $L1$ and space discretization using the modified cubic B-spline-based differential quadrature method. Uniform meshes are considered for the discretization of both temporal and spatial domains. Theoretical stability has been established by finding an estimate for the maximum norm of the inverse operator regardless of the involvement of mesh parameters. We trigger the Neumann series theorem to obtain a uniform bound for the inverse operator under reasonable conditions on the mesh parameters. The numerical illustrations show that this implicit numerical method exhibits a fourth-order convergence in the space direction and the order $2-\alpha$ in time. Moreover, we observe an enhancement in order of spatial convergence whenever $\alpha$ tends to $0$. The results obtained are then compared with existing popular techniques to demonstrate the accuracy of modified cubic B-spline-based DQM.