Market dynamics immediately before and after financial shocks: quantifying the Omori, productivity and Bath laws

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.