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

TL;DR

This paper analyzes the distribution of the quotient of two normal variables, showing it has a power law tail with decay rate x^{-2}, and applies this to explain the heavy tails observed in asset price changes.

Contribution

It provides a rigorous derivation of the quotient distribution's power law tail and applies it to model the heavy tails in financial relative price changes.

Findings

Quotient of normal variables exhibits power law decay with x^{-2} tail.

Conditional densities for different sign quadrants are derived.

Application to financial data explains heavy tails in asset price changes.

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.