  We consider the tail probabilities of stock returns for a general class of
stochastic volatility models. In these models, the stochastic differential
equation for volatility is autonomous, time-homogeneous and dependent on only a
finite number of dimensional parameters. Three bounds on the high-volatility
limits of the drift and diffusion coefficients of volatility ensure that
volatility is mean-reverting, has long memory and is as volatile as the stock
price. Dimensional analysis then provides leading-order approximations to the
drift and diffusion coefficients of volatility for the high-volatility limit.
Thereby, using the Kolmogorov forward equation for the transition probability
of volatility, we find that the tail probability for short-term returns falls
off like an inverse cubic. Our analysis then provides a possible explanation
for the inverse cubic fall off that Gopikrishnan et al. (1998) report for
returns over 5-120 minutes intervals. We find, moreover, that the tail
probability scales like the length of the interval, over which the return is
measured, to the power 3/2. There do not seem to be any empirical results in
the literature with which to compare this last prediction.
