The Omega function of a univariate probability distribution is, like the probability density function, characteristic function or moment generating function (when one exists), another mathematically equivalent way of encoding the information in the distribution. It is particularly well suited to the analysis of properties of tails and asymmetries of distributions. As a result, it provides significant new insights into those risk/reward characteristics of financial instruments which are ignored by conventional mean/variance tools and only partially captured by skew and kurtosis.
Unlike the alternative descriptions of a distribution listed above, the Omega function was only recently discovered, although it requires no mathematical technology which was not available to Gauss. I will give an introduction to Omega functions and some of their intriguing mathematical properties, followed by examples of the impact of applying this new approach real financial data. These will be accompanied by some graphical illustrations which require rather more computational power than even Gauss would likely have been able to muster.
(Joint work with Ana Cascon, Con Keating and Brad Shadwick)