  We study historical correlations and lead-lag relationships between
individual stock risk (volatility of daily stock returns) and market risk
(volatility of daily returns of a market-representative portfolio) in the US
stock market. We consider the cross-correlation functions averaged over all
stocks, using 71 stock prices from the Standard \& Poor's 500 index for
1994--2013. We focus on the behavior of the cross-correlations at the times of
financial crises with significant jumps of market volatility. The observed
historical dynamics showed that the dependence between the risks was almost
linear during the US stock market downturn of 2002 and after the US housing
bubble in 2007, remaining on that level until 2013. Moreover, the averaged
cross-correlation function often had an asymmetric shape with respect to zero
lag in the periods of high correlation. We develop the analysis by the
application of the linear response formalism to study underlying causal
relations. The calculated response functions suggest the presence of
characteristic regimes near financial crashes, when the volatility of an
individual stock follows the market volatility and vice versa.
