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

Many sophisticated option pricing models involve random variables whose transition density functions are only tractable in Fourier space. For such models, Fast Fourier Transform (FFT) algorithms are key to 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 payout-specific. Aiming here for much broader applicability, this paper demonstrates how the Type 3 Nonuniform Fast Fourier method 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 density recovery and one (more specialized) based on the Plancherel Theorem. Both are handled by a single algorithm designed to fully utilize the node placement freedom of the 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 CEV model extended with stochastic volatility. For both cases, we can achieve high-accuracy results (with errors around 10−13 to 10−16, say) while pricing tens of thousands– and sometimes even millions– of options per second on a single core.