An SFP--FCC Method for Pricing and Hedging Early-exercise Options under L\'evy Processes

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.