Inverting Analytic Characteristic Functions and Financial Applications

Abstract

This paper presents a set of schemes for the fast and accurate inversion of analytic characteristic functions. The schemes are based on sinc expansion approximation of functions that are analytic in a horizontal strip in the complex plane. A function in this class can be reconstructed highly accurately from its values on a uniform grid along a horizontal line in the strip. The discretization error decays exponentially in terms of $1/h$, where $h$ is the step size of the grid. Consequently, transforms and integrals of such functions can be approximated using very simple schemes with remarkable accuracy. These schemes lead to high performance numerical inversion of analytic characteristic functions. Probability densities are approximated by evaluating the corresponding inverse Fourier transform integrals. The trapezoidal rule is highly accurate with exponentially decaying discretization errors. Cumulative distribution functions and expectations involving indicator functions can be represented by Hilbert transforms, which can again be evaluated highly accurately. Numerical results exhibit that the proposed schemes are fast, accurate, and robust to extreme inputs. The schemes we present can be used in statistics, applied probability, engineering, economics, and finance, where the inversion of analytic characteristic functions often arises.

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