  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.
