<p>In this article, we develop and analyze a higher-order numerical scheme specifically designed to solve the generalized Black-Scholes model, a key mathematical model for the valuation of European options. Firstly, we employ the implicit Euler method in the time dimension and then apply the exponential B-spline collocation approach in the spatial dimension. Both the time and space discretization techniques are applied to the equidistant meshes, which leads to a tridiagonal system and is solved by the Thomas algorithm. A rigorous theoretical analysis is conducted to estimate the stability and convergence of the initial scheme. The resultant system is shown to be uniquely solvable and unconditionally stable using the Von Neumann stability analysis technique. Further, we demonstrate that the initial scheme achieves second-order spatial convergence and first-order temporal accuracy. To further improve the time accuracy of our scheme, we incorporate the Richardson extrapolation technique in the temporal domain. This technique combines solutions calculated at different time steps to obtain a more accurate solution. Using the Richardson extrapolation scheme, the refined numerical scheme is proven to be more efficient and achieve second-order convergence in both space and time domains. Some numerical experiments are employed to validate our theoretical findings. By comparing our results with those of other methods, we demonstrate the efficacy of our approach for practical applications.</p>