High-performance applications of the nonuniform fast Fourier transform to option pricing

Leif Andersen and Mark Lake

Journal of Computational Finance

Abstract: Many sophisticated option pricing models involve random variables whose transition density functions are only tractable in Fourier space. For such models, fast Fourier transform algorithms are key for accelerating the repeated Fourier-space integrations that are common in applications. While simple algorithms have been devised in special cases, these methods tend to be both delicate and strongly model- and payoff-specific. Aiming for much broader applicability, this paper demonstrates how the type 3 nonuniform fast Fourier transform can be combined with modern quadrature methods to construct generic, robust and highly performant computing schemes for Fourier-based option pricing. Two approaches are considered: one (broadly applicable) based on density or cumulative distribution recovery, and one (more specialized) based on the Plancherel theorem. Both are handled by a single algorithm designed to fully use the node placement freedom of the nonuniform fast Fourier transform (NUFFT) and to efficiently accommodate adaptive integration strategies. Numerical tests in the paper include the pricing of European options in a variance gamma Lévy-jump model with a singular density, and in a constant elasticity of variance model extended with stochastic volatility. For both cases, we can achieve high-accuracy results (with errors of around 10-13 to 10-16, say) while pricing tens of thousands – and sometimes even millions – of options per second on a single core.

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