Boltzmann Distribution and Temperature of Stock Markets

Abstract

The minute fluctuations of of S&P 500 and NASDAQ 100 indices display Boltzmann statistics over a wide range of positive as well as negative returns, thus allowing us to define a {\em market temperature} for either sign. With increasing time the sharp Boltzmann peak broadens into a Gaussian whose volatility $ \sigma $ measured in $1/ \sqrt{{\rm min}}$ is related to the temperature $T$ by $T= \sigma / \sqrt{2}$. Plots over the years 1990--2006 show that the arrival of the 2000 crash was preceded by an increase in market temperature, suggesting that this increase can be used as a warning signal for crashes. A plot of the Dow Jones temperature over 78 years reveals a remarkable stability through many historical turmoils, interrupted only by short heat bursts near the crashes.