  We apply a new numerical method, the singular Fourier-Pad\'e (SFP) method
invented by Driscoll and Fornberg (2001, 2011), to price European-type options
in L\'evy and affine processes. The motivation behind this application is to
reduce the inefficiency of current Fourier techniques when they are used to
approximate piecewise continuous (non-smooth) probability density functions.
When techniques such as fast Fourier transforms and Fourier series are applied
to price and hedge options with non-smooth probability density functions, they
cause the Gibbs phenomenon, accordingly, the techniques converge slowly for
density functions with jumps in value or derivatives. This seriously adversely
affects the efficiency and accuracy of these techniques. In this paper, we
derive pricing formulae and their option Greeks using the SFP method to resolve
the Gibbs phenomenon and restore the global spectral convergence rate.
Moreover, we show that our method requires a small number of terms to yield
fast error convergence, and it is able to accurately price any European-type
option deep in/out of the money and with very long/short maturities.
Furthermore, we conduct an error-bound analysis of the SFP method in option
pricing. This new method performs favourably in numerical experiments compared
with existing techniques.
