The work contemplates a brief numerical study of the time-fractional generalized Black-Scholes model arising in mathematical finance. The fractional derivative is considered in Caputo sense of order \(\varvec{\alpha \in }\mathbf {(0,1)}\) . A typical solution to such type of problem exhibits mild singularity in a narrow region near the initial time, consequently affecting the accuracy of numerical schemes on a uniform mesh. Present manuscript comes to fill the gap of providing an advantageous scheme over the work of Cen et al. (Comput. Math. Appl., 2018); which is the only work solving such type of singular problem with a global first-order accuracy. In order to capture the layer behavior in the temporal direction, the L2- \(\varvec{1_\sigma }\) technique is applied on a non-homogeneous graded mesh. Further, the resulting semi-discrete problem is handled using a cubic B-spline collocation technique. A detailed analysis of the proposed scheme claims it to have a quadratic rate of convergence under a suitable choice of grading parameter. The theoretical claims are corroborated in practice with valid numerical tests. Further, as a practical implication, different European options presided over the Black-Scholes equations are priced using the discussed method.
