The Importance of Being Normal

Analysts have used a variety of statistical distributions to model losses due to operational risks [Embrechts, Cruz] or insurance claims [Hogg and Klugman]. In contrast, the normal distribution is the undisputed queen of capital market models. This paper examines an alternative family of distributions, the conic moment family, for application to capital market models. We answer the following questions: l What is the empirical evidence for normality or for other distributions in financial data? l For what financial questions do non-normal distributions have a large impact on the answer? l What tools and diagnostics are available for model calibration? l How can I model many variables at once? l How can I incorporate alternative distributions into financial time series models? l What are the implications for prices of options and of corporate bonds? l What is the extra effort in deviating from normal assumptions, and is it worth the trouble? In our first section, we examine some empirical evidence, and fit some distributions. Our conclusions (section 7) summarise the business issues arising from choice of distributions. To get a brief overview of the paper, read just these two sections. The interior covers the details. This is the meat that backs up our conclusions. We assume that you are already familiar with the normal theory, including multi dimensional problems, matrix manipulations and the logic behind Black and Scholes’ formula. We continually make use of parallels and contrasts, between the familiar normal theory and the newer non-normal distributions. 1. Flypaper Models A long and tall building has a west wall entirely covered with flypaper. A swarm of flies is released, and each fly buzzes according to a three dimensional Brownian motion. A breeze blows the flies gently in the direction of the building. Eventually every fly is stuck somewhere on the flypaper. The picture might look like this: This is an example of a bivariate conic moment distribution. Let us suppose in 3dimensional coordinates that the flies are released at the origin, and the flypaper corresponds to the plane x = 1. Let us suppose also that the flies’ movements are multivariate normal with mean m and variance-covariance matrix V per unit time. Arrivals on the flypaper are described by the bivariate density function: ( ) w V w V m V m w V w w V m w V w m V m V m z y f T T T T