  We study the cascading dynamics immediately before and immediately after 219
market shocks. We define the time of a market shock T_{c} to be the time for
which the market volatility V(T_{c}) has a peak that exceeds a predetermined
threshold. The cascade of high volatility "aftershocks" triggered by the "main
shock" is quantitatively similar to earthquakes and solar flares, which have
been described by three empirical laws --- the Omori law, the productivity law,
and the Bath law. We analyze the most traded 531 stocks in U.S. markets during
the two-year period 2001-2002 at the 1-minute time resolution. We find
quantitative relations between (i) the "main shock" magnitude M \equiv \log
V(T_{c}) occurring at the time T_{c} of each of the 219 "volatility quakes"
analyzed, and (ii) the parameters quantifying the decay of volatility
aftershocks as well as the volatility preshocks. We also find that stocks with
larger trading activity react more strongly and more quickly to market shocks
than stocks with smaller trading activity. Our findings characterize the
typical volatility response conditional on M, both at the market and the
individual stock scale. We argue that there is potential utility in these three
statistical quantitative relations with applications in option pricing and
volatility trading.
