Failure of Fourier pricing techniques to approximate the Greeks

Tobias Behrens; Gero Junike; Wim Schoutens

doi:10.3934/fmf.2024022

Article Contents

March 2025, Volume 4, 102-113. Doi: 10.3934/fmf.2024022 open access

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Failure of Fourier pricing techniques to approximate the Greeks

1.
Carl von Ossietzky Universität, Institut für Mathematik, 26129 Oldenburg, Germany
2.
KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium

^*Corresponding author: Gero Junike
^*Corresponding author: Gero Junike

Received: September 2024

Revised: November 2024

Early access: December 2024

Published: March 2025

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Abstract

The Greeks Delta and Gamma of plain vanilla options play a fundamental role in finance, e.g., in hedging or risk management. These Greeks were approximated in many models such as the widely used Variance Gamma model by Fourier techniques such as the Carr-Madan formula, or the Fourier cosine expansion (COS) method method or the Lewis formula. However, for some realistic market parameters, we show empirically that these three Fourier methods completely fail to approximate the Greeks. As an application we show that the Delta-Gamma value at risk is severely underestimated in realistic market environments. As a solution, we propose to use finite differences instead to obtain the Greeks.

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Mathematics Subject Classification: 60E10 91G20 91G60.

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Figure 1. Prices and Greeks of a call option and a digital put option using the CM formula, the COS method, the Lewis formula, and finite differences under the ME and the VG models

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Table 1. Delta and Gamma of a digital put option in the ME model with parameters $ \eta = 1 $ and $ \lambda = 2 $ and with $ S_{0} = 0.75 $, $ r = 0 $, $ K = 0.75 $ and maturity $ T = \frac{1}{12} $

	Analy-tical	finite diff., $ h=0.01 $	finite diff., $ h=0.02 $	finite diff., $ h=0.005 $	COS	CM	Lewis

$ \pi(S_{0}) $	0.45	/	/	/	0.45	0.46	0.45
$ \Delta(S_{0}) $	-2.10	-2.10	-2.10	-2.10	-2.09	-2.09	-2.01
$ \Gamma(S_{0}) $	12.47	12.47	12.55	12.46	36.98	29.01	38.54

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Table 2. Delta-Gamma VaR in the VG model: The portfolio consists of a digital put option and a loan in the risk-free asset. A full Monte Carlo VaR is equal to $ 2.1\times10^{-3} $

	finite diff., $ h=0.01 $	finite diff., $ h=0.02 $	finite diff., $ h=0.005 $	COS	CM	Lewis

$ \pi(S_{0}) $	/	/	/	0.99	0.99	0.98
$ \Delta(S_{0}) $	-0.21	-0.21	-0.21	0.06	0.05	0.24
$ \Gamma(S_{0}) $	-6.45	-7.01	-6.62	1347.08	1647.57	217.87
$ \Delta $-$ \Gamma $-VaR	$ 2.5\times10^{-3} $	$ 2.6\times10^{-3} $	$ 2.5\times10^{-3} $	$ 1.0\times10^{-6} $	$ 1.0\times10^{-6} $	$ 1.0\times10^{-4} $

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