New Efficient Versions of Fourier Transform Method in Applications to Option Pricing

64 Pages Posted: 24 May 2011 Last revised: 6 Jun 2011

See all articles by Svetlana Boyarchenko

Svetlana Boyarchenko

University of Texas at Austin - Department of Economics

Sergei Levendorskii

Calico Science Consulting

Date Written: May 19, 2011

Abstract

We review popular methods for pricing European options based on the Fourier expansion of the payoff function (iFT method) and the trapezoid rule, and suggest several new efficient variations. The first variation is a group of PMwFT methods (Payoff Modification with Fourier Transform), which contains COS as a subclass. In some cases, PMwFT methods may need a smaller number of terms N in the summation formula than iFT; however, N can be made smaller by a factor less than two, at best. Summation by parts in the truncated part of the infinite sum, and, especially, contour deformations with subsequent conformal transformations (conformal parabolic and hyperbolic iFT methods) allow one to decrease N several times, if iFT requires several dozen of points, and dozens and hundreds times in cases, when iFT may need thousands and millions of terms. The wonderful performance of the simple trapezoid rule and its variations follows from the properties of the Whittaker cardinal series (Sinc expansion) for functions analytic in a strip around the real axis, introduced to finance by Feng and Linetsky (Mathematical Finance, 2008). We give general recommendations for an (approximately) optimal choice of parameters for each numerical scheme, and produce numerical examples to illustrate the relative performance of different methods for calculation of prices, sensitivities and pdf.

Keywords: European Options, Greeks, FFT, Inverse Fourier Transform, COS, CONV, KoBoL, CGMY, VG Model, Hyperbolic iFT Method, Parabolic iFT Method

JEL Classification: C63, G13

Suggested Citation

Boyarchenko, Svetlana I. and Levendorskii, Sergei Z., New Efficient Versions of Fourier Transform Method in Applications to Option Pricing (May 19, 2011). Available at SSRN: https://ssrn.com/abstract=1846633 or http://dx.doi.org/10.2139/ssrn.1846633

Svetlana I. Boyarchenko

University of Texas at Austin - Department of Economics ( email )

Austin, TX 78712
United States

Sergei Z. Levendorskii (Contact Author)

Calico Science Consulting ( email )

Austin, TX
United States

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