To price vanilla European and American options via the fractional Black–Scholes model, first a (2− α)-order discretization scheme for the Caputo fractional derivative based upon graded meshes along time is presented. This is fruitful for problems having nonsmooth data at the initial time. Second, a nonuniform discretization of space is also considered and the coefficient weights based upon meshless radial basis function generated finite difference (RBF-FD) methodology are contributed under the inverse quadratic function. The fully space–time nonuniform solution method is then constructed via the presented discretization formulas. Numerical tests are given to show effectiveness and accuracy of our numerical scheme.