Singular Fourier-Padé Series Expansion of European Option Prices

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 prob-
ability 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. More-
over, 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.