Quantile Mechanics 3: Series Representations and Approximation of some Quantile Functions appearing in Finance

TL;DR

This paper develops Taylor and asymptotic series expansions for quantile functions of various distributions relevant in finance, facilitating their approximation when direct inversion is challenging.

Contribution

It introduces new series expansion methods for quantile functions of Variance Gamma, GIG, Hyperbolic, and alpha-Stable distributions, aiding their approximation.

Findings

Derived Taylor series for quantile functions

Established asymptotic series expansions

Provided insights into quantile function approximation

Abstract

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this paper will be to develop Taylor and asymptotic series expansions for the quantile functions belonging to the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic and alpha-Stable. As a secondary matter, based on these analytic expressions we briefly investigate the problem of approximating the quantile function.