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A Spectral Element Framework for Option Pricing Under General Exponential Lévy Processes

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Abstract

We derive a spectral element framework to compute the price of vanilla derivatives when the dynamic of the underlying follows a general exponential Lévy process. The representation of the solution with Legendre polynomials allows one to naturally approximate the convolution integral with high order quadratures. The method is spectrally accurate in space for the solution and the greeks, and third order accurate in time. The spectral element framework does not require an approximation of the Lévy measure nor the lower truncation of the convolution integral as commonly seen in finite difference approximations. We show that the spectral element method is ten times faster than Fast Fourier Transform methods for the same accuracy at strike, and two hundred times faster if one reconstructs the greeks from the solution obtained by FFT. We use the SEM approximation to derive the \(\Delta \) and \(\Gamma \) in a variance gamma model, for which there is no closed form solution.

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Acknowledgments

We would like to thank Prof. Alec Kercheval for his helpful suggestions and motivating discussion. We are grateful for the excellent comments of an anonymous referee which significantly improved the quality of this paper.

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  1. Department of Mathematics, Florida State University, 1017 Academic Way, 208 Love Building, Tallahassee, FL, 32306, USA

    Pierre Garreau

  2. Department of Mathematics, Florida State University, 1017 Academic Way, 219 Love Building, Tallahassee, FL, 32306, USA

    David Kopriva

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  1. Pierre Garreau
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  2. David Kopriva
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Correspondence to Pierre Garreau.

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Garreau, P., Kopriva, D. A Spectral Element Framework for Option Pricing Under General Exponential Lévy Processes. J Sci Comput 57, 390–413 (2013). https://doi.org/10.1007/s10915-013-9713-0

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  • DOI: https://doi.org/10.1007/s10915-013-9713-0

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