Recently, Kaur and Natesan (Numer Algorithm 94:1519–1549, 2023) proposed a uniform mesh method to approximate the solution of Caputo time-fractional Black-Scholes (TBS) equation. This method is not capable of handling the weak singularity at \(t=0.\) To overcome this bottleneck, this paper presents a robust graded mesh method for the same TBS model with the weakly singular solution, where a graded mesh scheme is designed to approximate the Caputo temporal fractional (CTF) derivative of order \(\alpha \in (0,1)\) and a high-order compact difference (HOCD) scheme is designed to approximate the spatial derivatives. Convergence and stability of the fully discrete scheme are analyzed in the discrete \(L_2\) -norm. The proposed method is shown to be of \(O(N^{-\min ({2-\alpha },r \alpha )},\Delta x^4)\) accuracy, where \(\Delta x\) represents the step size in space direction. Numerical experiment is carried out to show the efficiency and accuracy of the method and verify the theoretical estimates. Moreover, the suggested scheme is employed to solve a European-double-barrier knock-out option (EDBKO) problem. The elapsed computational time for the suggested method is supplemented. Numerical results obtained by the proposed graded mesh method are compared with the results obtained by the uniform mesh method proposed in Kaur and Natesan (Numer Algorithm 94:1519–1549, 2023).
