  This paper extends the Singular Fourier--Pad\'e (SFP) method proposed by Chan
(2018) to pricing/hedging early-exercise options--Bermudan, American and
discrete-monitored barrier options--under a L\'evy process. The current SFP
method is incorporated with the Filon--Clenshaw--Curtis (FCC) rules invented by
Dom\'inguez et al. (2011), and we call the new method SFP--FCC. The main
purpose of using the SFP--FCC method is to require a small number of terms to
yield fast error convergence and to formulate option pricing and option Greek
curves rather than individual prices/Greek values. We also numerically show
that the SFP--FCC method can retain a global spectral convergence rate in
option pricing and hedging when the risk-free probability density function is
piecewise smooth. Moreover, the computational complexity of the method is
$\mathcal{O}((L-1)(N+1)(\tilde{N} \log \tilde{N}) )$ with $N$ a (small) number
of complex Fourier series terms, $\tilde{N}$ a number of Chebyshev series terms
and $L$, the number of early-exercise/monitoring dates. Finally, we show that
our method is more favourable than existing techniques in numerical
experiments.
