On the computation of the probability density function of a-stable distributions

Abstract

Stable laws have a wide sphere of application: probability theory, physics, electronics, economics, sociology. They also play an important role in - nancial mathematics, since the classical models of nancial market based on the hypothesis of normality often become inadequate. However, the practical implementation of stable models is a nontrivial task, because (with few exceptions) probability density functions of -stable distributions have no analytical representation. In our previous works Zolotarev representation of the probability density function and direct numerical integration method (with 96-point Gaussian quadrature) were used, but it seems that the application of another probability density function integral representation (with Gaussian quadrature and Laguerre quadrature combination) is more e cient in computational aspect, fairly precise and easily implementable. The suggested method is compared with other algorithms.