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It has long been agreed by academics that the inversion method is the methodof choice for generating random variates, given the availability of thequantile function. However for several probability distributions arising inpractice a satisfactory method of approximating these functions is notavailable. The main focus of this paper will be to develop Taylor andasymptotic series expansions for the quantile functions belonging to thefollowing probability distributions; Variance Gamma, Generalized InverseGaussian, Hyperbolic and alpha-Stable. As a secondary matter, based on theseanalytic expressions we briefly investigate the problem of approximating thequantile function.

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