  Many studies assume stock prices follow a random process known as geometric
Brownian motion. Although approximately correct, this model fails to explain
the frequent occurrence of extreme price movements, such as stock market
crashes. Using a large collection of data from three different stock markets,
we present evidence that a modification to the random model -- adding a slow,
but significant, fluctuation to the standard deviation of the process --
accurately explains the probability of different-sized price changes, including
the relative high frequency of extreme movements. Furthermore, we show that
this process is similar across stocks so that their price fluctuations can be
characterized by a single curve. Because the behavior of price fluctuations is
rooted in the characteristics of volatility, we expect our results to bring
increased interest to stochastic volatility models, and especially to those
that can produce the properties of volatility reported here.
