As presented in many papers, we have various methods to obtain option pricing formula, such as Carr and Madan (1999) approach and Lipton’s (2001). The primary conclusions have been reached with the help of Fourier transformation and complex analysis in these papers, and Fast Fourier Transform (FFT) algorithm makes them possible to be the desired numerical results. In this thesis Adaptive Fourier Decomposition (AFD) is applied to achieve the result instead of the direct use of FFT algorithm. In addition, AFD is better than Fourier Decomposition in aspect of convergence and frequency. We intend to implement AFD to decompose integrand function given by Lewis (2001) and reach some explicit formula solutions that can be used in theory. As an indispensable part, we give a more rigorous formulation of option pricing based on harmonic analysis.