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We study the cascading dynamics immediately before and immediately after 219market shocks. We define the time of a market shock T_{c} to be the time forwhich the market volatility V(T_{c}) has a peak that exceeds a predeterminedthreshold. The cascade of high volatility "aftershocks" triggered by the "mainshock" is quantitatively similar to earthquakes and solar flares, which havebeen 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 duringthe two-year period 2001-2002 at the 1-minute time resolution. We findquantitative relations between (i) the "main shock" magnitude M \equiv \logV(T_{c}) occurring at the time T_{c} of each of the 219 "volatility quakes"analyzed, and (ii) the parameters quantifying the decay of volatilityaftershocks as well as the volatility preshocks. We also find that stocks withlarger trading activity react more strongly and more quickly to market shocksthan stocks with smaller trading activity. Our findings characterize thetypical volatility response conditional on M, both at the market and theindividual stock scale. We argue that there is potential utility in these threestatistical quantitative relations with applications in option pricing andvolatility trading.

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