The Quotient of Normal Random Variables And Application to Asset Price Fat Tails

Abstract

The quotient of random variables with normal distributions is examined and proven to have have power law decay, with density $f\left( x\right) \simeq f_{0}x^{-2}$, with the coefficient depending on the means and variances of the numerator and denominator and their correlation. We also obtain the conditional probability densities for each of the four quadrants given by the signs of the numerator and denominator for arbitrary correlation $\rho \in\lbrack-1,1).$ For $\rho=-1$ we obtain a particularly simple closed form solution for all $x\in$ $\mathbb{R}$. The results are applied to a basic issue in economics and finance, namely the density of relative price changes. Classical finance stipulates a normal distribution of relative price changes, though empirical studies suggest a power law at the tail end. By considering the supply and demand in a basic price change model, we prove that the relative price change has density that decays with an $x^{-2}$ power law. Various parameter limits are established.