  This study proposes a versatile model for the dynamics of the best bid and
ask prices using an extended Hawkes process. The model incorporates the zero
intensities of the spread-narrowing processes at the minimum bid-ask spread,
spread-dependent intensities, possible negative excitement, and nonnegative
intensities. We apply the model to high-frequency best bid and ask price data
from US stock markets. The empirical findings demonstrate a spread-narrowing
tendency, excitations of the intensities caused by previous events, the impact
of flash crashes, characteristic trends in fast trading over time, and the
different features of market participants in the various exchanges.
