  Econophysics and econometrics agree that there is a correlation between
volume and volatility in a time series. Using empirical data and their
distributions, we further investigate this correlation and discover new ways
that volatility and volume interact, particularly when the levels of both are
high. We find that the distribution of the volume-conditional volatility is
well fit by a power-law function with an exponential cutoff. We find that the
volume-conditional volatility distribution scales with volume, and collapses
these distributions to a single curve. We exploit the characteristics of the
volume-volatility scatter plot to find a strong correlation between logarithmic
volume and a quantity we define as local maximum volatility (LMV), which
indicates the largest volatility observed in a given range of trading volumes.
This finding supports our empirical analysis showing that volume is an
excellent predictor of the maximum value of volatility for both same-day and
near-future time periods. We also use a joint conditional probability that
includes both volatility and volume to demonstrate that invoking both allows us
to better predict the largest next-day volatility than invoking either one
alone.
