A Jump-Diffusion Model for Option Pricing
Management Science
A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models
SIAM Journal on Numerical Analysis
A Spectral Element Method to Price European Options. I. Single Asset with and without Jump Diffusion
Journal of Scientific Computing
Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers
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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 $$\Delta $$ Δ and $$\Gamma $$ Γ in a variance gamma model, for which there is no closed form solution.