Finite difference methods are commonly used in the pricing of discretely monitored exotic options in the Black-Scholes framework, but they tend to converge slowly due to discontinuities contained in terminal conditions. We present an effective analytical modification to existing finite difference methods which greatly enhances their performance on discretely monitored options with non-smooth terminal conditions. We apply this modification to the popular Crank-Nicolson method and obtain highly accurate option pricing results with significantly reduced CPU cost. We also introduce an adaptive mesh refinement technique which further improves the computational speed of the modified finite difference method. The proposed method is especially useful for options with high monitoring frequencies, which are difficult to price using other existing methods.