A Model for Stock Returns and Volatility

TL;DR

This paper demonstrates that Student's t-distribution and generalized inverse gamma distribution effectively model stock returns and volatility data, respectively, and explores the relationship between volatility and stock returns, revealing Brown noise in volatility series.

Contribution

It introduces a novel approach to modeling stock returns and volatility using specific distributions and analyzes the stochastic properties of volatility time series.

Findings

Student's t-distribution fits stock returns well

Generalized inverse gamma distribution models VIX and VXO volatility

Brown noise observed in VIX and VXO time series

Abstract

We prove that Student's t-distribution provides one of the better fits to returns of S&P component stocks and the generalized inverse gamma distribution best fits VIX and VXO volatility data. We further argue that a more accurate measure of the volatility may be possible based on the fact that stock returns can be understood as the product distribution of the volatility and normal distributions. We find Brown noise in VIX and VXO time series and explain the mean and the variance of the relaxation times on approach to the steady-state distribution.