Value-at-Risk (VaR) is a widely used statistical measure in financial risk management for quantifying the level of risk associated with a specific investment portfolio. It is well-known that historical return data exhibit non-normal features, such as heavy tails and skewness. Current analytical (parameteric) calculation of VaR typically assumes the distribution of the portfolio return to be a n...

We examine certain Generalized Hyperbolic (GH) distributions for modeling equity returns, compared to usual Normal distributions. We describe these GH distributions and some of their properties, and test them against six years of daily S&P500 index prices. We estimate Value-at-Risk from calibrated distributions, and show that the Normal distribution leads to V aR estimates that significantly un...

The introduced three parameter (position μ, scale Σ and shape γ) multivariate generalized Normal distribution (γ-GND) is based on a strong theoretical background and emerged from Logarithmic Sobolev Inequalities. It includes a number of well known distributions such as the multivariate Uniform, Normal, Laplace and the degenerated Dirac distributions. In this paper, the cumulative distribution, ...

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeomet-ric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E 7 (ellip-tic, hyperbolic) and of type E 6...

Following Bowditch [5], we study representations of the free group on two generators into SL(2, C), and the connection with generalized Markoff maps. We show that Bowditch’s Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations. These conditions are very close to being necessary as well, and a large class of r...

Preface Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have a simple structure in comparison with general semimartingales. Thus stochastic models ba...

This paper discusses hyperbolic behavior of Jacobi fields along billiard flows on multidimensional Reimannian manifolds. A class of generalized differential operators associated with the impulsive equations (generalized Jacobi equations) are defined using a new Radon measure. We investigate the hyperbolic behavior of functions in the null-space of the operator by applying operator theory.

This paper proposes a multivariate generalization of the generalized Poisson distribution. Its definition and main properties are given. The parameters are estimated by the method of moments.