A compact quadratic spline collocation (QSC) method for the time-fractional Black–Scholes model governing European option pricing is presented. Firstly, after eliminating the convection term by an exponential transformation, the time-fractional Black–Scholes equation is transformed to a time-fractional sub-diffusion equation. Then applying \(L1 - 2\) formula for the Caputo time-fractional derivative and using a collocation method based on quadratic B-spline basic functions for the space discretization, we establish a higher accuracy numerical scheme which yields \(3-\alpha \) order convergence in time and fourth-order convergence in space. Furthermore, the uniqueness of the numerical solution and the convergence of the algorithm are investigated. Finally, numerical experiments are carried out to verify the theoretical order of accuracy and demonstrate the effectiveness of the new technique. Moreover, we also study the effect of different parameters on option price in time-fractional Black–Scholes model.

提出了一种紧凑的二次样条搭配（QSC）方法，用于控制欧洲期权定价的时间分数Black-Scholes模型。首先，在通过指数变换消除对流项后，将时间分数的Black-Scholes方程转换为时间分数的子扩散方程。然后将\（L1-2 \）公式用于Caputo时间分数导数，并使用基于二次B样条基本函数的搭配方法进行空间离散化，我们建立了一个更精确的数值方案，得出\（3- \ alpha \）时间的四阶收敛和空间的四阶收敛。此外，研究了数值解的唯一性和算法的收敛性。最后，通过数值实验验证了理论精度的顺序，并证明了该新技术的有效性。此外，我们还研究了时间分形Black-Scholes模型中不同参数对期权价格的影响。