Fast Option Pricing Using Non Uniform Discrete Fourier Transform on Gaussian Discretization Grids

Abstract

The aim of this work is to offer for the first time an application in finance of a new tool that appears to have a great potential in terms of derivative pricing. Non Uniform Discrete Fourier Transforms are innovative, precious tools in the fields of Signals Theory and Images Reconstruction where they are often useful when the sampling frequencies of the signals are non uniformly sampled in space and time. In the context of derivative pricing via Fourier Transform methods, the need of a non uniform discretization grid for the risk neutral characteristic function of the transformed spot price arises naturally trying to improve the accuracy of option pricing formulas at their best. Gaussian quadrature schemes are a powerful and elegant solution to the problem of maximizing both accuracy and stability of the pricing formulas, but unfortunately they don’t allow a direct and immediate fast implementation as it happens for the more rudimental Newton - Cotes schemes (Carr - Madan, 1999). Non Uniform DFT’s can be used to restate the discretized pricing formulas as convergence limits of properly rescaled Fourier transforms. An interesting interpolation tecnique (Gaussian gridding), properly improved to exalt speed of computation, is then applied to allow fast computation of Non Uniform DFT’s. The computational framework is impressive: perfect accuracy and precise error control are the key features of this very fast implementation, that is valid under the hyphotesis of general processes for the underlying (Sepp, 2003) and easily extendable to any type of option pricing formulas.