Before revision:  We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the affine case .
After revision:  We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices .