High-Performance Applications of the Non-Uniform Fast Fourier Transform to Option Pricing

Many sophisticated option pricing models involve random variables whose probability density functions are only tractable in Fourier space. Moreover, popular choices for these variables often lead to numerical challenges, due to, for instance, singular densities or slowly decaying characteristic functions. Although Fourier-based option pricing 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 Non-Uniform Fast Fourier Transform (NUFFT) method can be combined with modern quadrature methods to construct generic, robust and highly performant computing schemes for Fourier-based option pricing. Two separate algorithms are studied: one (broadly applicable) based on density recovery and one (more specialized) based on the Plancherel Theorem. Both are designed to fully utilize the node placement freedom granted by NUFFT, and, in particular, can efficiently accommodate automatic or adaptive integration strategies. Numerical tests in the paper include the pricing of European options in a Variance Gamma Levy-jump model with a singular density, and in a CEV model extended with stochastic volatility. For both cases, we can achieve high-precision results (with average errors around 1e−12 – 1e−15, say) while pricing tens of thousands, and sometimes hundreds of thousands, of options per second on a single PC