Abstract
We develop a spectral element method to price single factor European options with and without jump diffusion. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element, which allows an exact representation of the non-smooth payoff function. The convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The method is spectrally accurate (exponentially convergent) in space for the solution and Greeks, and second-order accurate in time. The spectral element solution to the Black-Scholes equation is ten to one hundred times faster than commonly used second order finite difference methods.
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Zhu, W., Kopriva, D.A. A Spectral Element Method to Price European Options. I. Single Asset with and without Jump Diffusion. J Sci Comput 39, 222–243 (2009). https://doi.org/10.1007/s10915-008-9267-8
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DOI: https://doi.org/10.1007/s10915-008-9267-8