An SFP–FCC method for pricing and hedging early-exercise options under Lévy processes

Abstract

This paper extends the singular Fourier–Padé (SFP) method proposed by Chan [Singular Fourier–Padé series expansion of European option prices. Quant. Finance, 2018, 18, 1149–1171] for pricing/hedging early-exercise options–Bermudan, American and discrete-monitored barrier options–under a Lévy process. The current SFP method is incorporated with the Filon–Clenshaw–Curtis (FCC) rules invented by Domínguez et al. [Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals. IMA J. Numer. Anal., 2011, 31, 1253–1280], 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 $\mathcal{O}\left(\right(L-1\left)\right(N+1\left)\right(\tilde{N}\log \tilde{N}\left)\right)$ 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 compare the accuracy and computational time of our method with those of existing techniques in numerical experiments.

Keywords:

• Singular Fourier–Padé
• Chebyshev series
• Filon–Clenshaw–Curtis rules
• Early-exercise options
• Discrete-monitored barrier options
• Lévy process

JEL classification:

• C6
• C63

Acknowledgments

We thank Professor Bengt Fornberg, Department of Applied Mathematics, University of Colorado, for teaching the singular Fourier–Padé method and Professor Victor Domínguez, Department of Mathematics, University of Navarra, for help and advice on using the Filon–Clenshaw–Curtis rules.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

† A function is called piecewise continuous on an interval if the function is made up of a finite number of ν times with differentiable continuous pieces.

‡ A pole occurs where the function f is not well-defined or tends to infinity in a complex plane.

† Markets are complete when the Lévy process is a Brownian motion - the classical Black and Scholes model - or if it is a Poisson process

4 A Toeplitz matrix or diagonal-constant matrix is an invertible matrix in which each descending diagonal from left to right is constant.