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

Abstract

Stable laws have a wide sphere of application: probability theory, physics, electronics, economics, sociology. They also play an important role in financial 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.