From characteristic functions to multivariate distribution functions and European option prices by the damped COS method

TL;DR

This paper introduces the damped COS method, an extension of the classical Fourier-cosine expansion technique, enabling efficient and accurate computation of distribution functions and option prices in multiple dimensions, with proven convergence properties.

Contribution

The paper develops the damped COS method for multivariate integrals, providing explicit formulas for parameters and demonstrating exponential convergence under certain conditions.

Findings

Method converges exponentially for characteristic functions with exponential decay.

Numerical experiments up to five dimensions confirm theoretical convergence rates.

Explicit formulas for truncation range and number of terms improve practical implementation.

Abstract

We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine expansion (COS) method. The classical COS method is numerically very efficient in one-dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a cosine series. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.