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

TL;DR

This paper introduces the SFP--FCC method, combining Singular Fourier--Padé and Filon--Clenshaw--Curtis rules, for efficient and accurate pricing and hedging of early-exercise options under Lévy processes, demonstrating spectral convergence and computational efficiency.

Contribution

The paper develops the SFP--FCC method that achieves fast convergence and formulates entire option curves, improving upon existing techniques for pricing and hedging early-exercise options under Lévy processes.

Findings

Retains spectral convergence rate in option pricing and hedging.

Requires fewer terms for fast error convergence.

Outperforms existing methods in numerical experiments.

Abstract

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.