# Singular Fourier-Pad\'e Series Expansion of European Option Prices

## TL;DR

This paper introduces the singular Fourier-Padé (SFP) method to efficiently and accurately price European options under Lévy and affine processes, overcoming Gibbs phenomenon issues in traditional Fourier techniques.

## Contribution

The paper applies the SFP method to option pricing, providing a new approach that reduces errors caused by non-smooth densities and improves convergence speed.

## Key findings

- SFP method effectively resolves Gibbs phenomenon in option pricing.
- Requires fewer terms for fast error convergence.
- Accurately prices deep in/out of the money options with various maturities.

## Abstract

We apply a new numerical method, the singular Fourier-Pad\'e (SFP) method invented by Driscoll and Fornberg (2001, 2011), to price European-type options in L\'evy 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.