A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices

Abstract

There has been a great deal of discussion about the statistical distribution of rates of return on common stocks. At an early stage the prevalent belief was that distributions of rates of return on common stocks were adequately characterized by the normal distribution. This belief seemed to be consistent with the pioneering work of Bachelier.1 It was also observed, however, that empirical distributions of such returns had more kurtosis (i.e., fatter tails) than that predicted by the normal distribution. The evidence provided by Mandelbrot and Fama suggested that one could explicitly account for the observed fat tails by using the symmetric-stable distribution.2 This article considers another family of symmetric distributions that can also account for the observed fat tails of returns distribution. This alternative is the Student (or t) distribution. It will be indicated that this alternative model has implications for empirical and theoretical work that are quite different from those of the symmetric-stable model. The descriptive validity of the Student model, relative to that of the symmetric-stable model, will be assessed using actual daily rates of return. This article is organized as follows: Section II describes the prop-