  We construct a two-tailed peak-over-threshold Hawkes model that captures
asymmetric self- and cross-excitation in and between left- and right-tail
extreme values within a time series. We demonstrate its applicability by
investigating extreme gains and losses within the daily log-returns of the S&P
500 equity index. We find that the arrivals of extreme losses and gains are
described by a common conditional intensity to which losses contribute twice as
much as gains. However, the contribution of the former decays almost five times
more quickly than that of the latter. We attribute these asymmetries to the
different reactions of market traders to extreme upward and downward movements
of asset prices: an example of negativity bias, wherein trauma is more salient
than euphoria.
