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Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including Lévy processes and the Hestondoi:10.1137/080718061 fatcat:nwtfrv6e5bbvthldnegcjqlkpu