Singular Fourier–Padé series expansion of European option prices

Abstract

We apply a new numerical method, the singular Fourier–Padé (SFP) method invented by Driscoll and Fornberg [Numer. Algorithms, 2001, 26, 77–92; The Gibbs Phenomenon in Various Representations and Applications, 2011], to price European-type options in Lévy 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 probability 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. Moreover, 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.

Keywords:

• Singular Fourier–Padé series
• European-type options
• Lévy processes
• Affine processes

Acknowledgements

We thank Professor Toby Driscoll, Department of Mathematical Sciences, University of Delaware and Professor Bengt Fornberg, Department of Applied Mathematics, University of Colorado for comments that greatly improved the manuscript.

Notes

No potential conflict of interest was reported by the author.

1 A vector space in which functions are continuously differentiable.

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

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

4 The Dirac Delta function is a generalized function or distribution on the real number line that is zero everywhere except at zero.