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In this work, we discuss the numerical method for the solution of the Black-Scholes model. First of all, the asymptotic convergence for the solution of Black-Scholes model is proved. Second, we develop a linear, unconditionally stable, and second-order time-accurate numerical scheme for this model. By using the finite difference method and Legendre-Galerkin spectral method, we construct a time and space discrete scheme. Finally, we prove that the scheme has second-order accuracy and spectral accuracy in time and space, respectively. Several numerical experiments further verify the convergence rate and effectiveness of the developed scheme.