<p>We present a convergent fourth-order finite difference method (FDM) for the Black–Scholes (BS) equation. The proposed numerical scheme is constructed based on an implicit Euler method and a fourth-order accurate FDM. Many options used in the financial market have payoff functions with discontinuous differentials. Because the second and first derivatives in the BS equation cause high errors and low convergence rates, we propose the numerical method to overcome this difficulty by applying a fine mesh at the early stage of the numerical method. For a sufficiently smooth option price function, a coarse mesh is employed. We can achieve a fourth-order convergence rate applying the proposed algorithm. The effectiveness and capability of the proposed algorithm are validated through various computational results using a European call option.</p>