2017/06/21 by Tat Lung Chan, Chan, Tat Lung
Economics, Econometrics and Finance · Mathematics · #Applied mathematics #Computational Finance (q-fin.CP) #Convergence (economics) #Econometrics #Economics #FOS: Economics and business #Financial Markets and Investment Strategies #Financial Risk and Volatility Modeling #Fourier series #Fourier transform #Hedge #Mathematical analysis #Mathematical optimization #Mathematics #Piecewise #Stochastic processes and financial applications #Valuation of options #q-fin.CP
paper · pdf · doi:10.48550/arxiv.1706.06709
published in arXiv (Cornell University) (Cornell University) · 37 pages, 14 figures and 14 tables
openalex publication_date 2017/06/21 · arxiv created 2017/11/13 · arxiv updated 2017/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We apply a new numerical method, the singular Fourier-Padé (SFP) method invented by Driscoll and Fornberg (2001, 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.