In this paper, we propose a collocation scheme for efficiently solving the mixed time-fractional Black-Scholes (MTF-BS) model and obtaining the option price. Our approach involves deriving the mixed fractional Black-Scholes (MF-BS) partial differential equation (PDE) considering the delta hedging strategy and the mixed fractional Geometric Brownian motion (MFGBM) model. To simplify the problem, we transform the MTF-BS PDE into a modified Riemann-Liouville derivative form. Subsequently, a collocation method is employed to numerically solve the transformed equation, where the solution is represented as a series of fractional Jaiswal functions with unknown coefficients. By utilizing operational matrices and collocation points, we convert the problem into a linear system of equations, allowing for the examination of convergence and stability in the Sobolev spaces. Finally, we present four examples to demonstrate the method’s effectiveness and accuracy.

在本文中，我们提出了一种有效求解混合时间分数布莱克-斯科尔斯（MTF-BS）模型并获得期权价格的配置方案。我们的方法包括考虑 Delta 对冲策略和混合分数几何布朗运动 (MFGBM) 模型，推导混合分数 Black-Scholes (MF-BS) 偏微分方程 (PDE)。为了简化问题，我们将 MTF-BS PDE 转换为修正的 Riemann-Liouville 导数形式。随后，采用配置方法对变换方程进行数值求解，其中解表示为一系列系数未知的分数 Jaiswal 函数。通过利用运算矩阵和配置点，我们将问题转换为线性方程组，从而可以检查 Sobolev 空间中的收敛性和稳定性。最后，我们提出了四个例子来证明该方法的有效性和准确性。