  Conditional extreme value theory (EVT) methods promise enhanced forecasting
of the extreme tail events that often dominate systemic risk. We present an
improved two-tailed peaks-over-threshold (2T-POT) Hawkes model that is adapted
for conditional quantile forecasting in both the left and right tails of a
univariate time series. This is applied to the daily log-returns of six large
cap indices. We also take the unique step of fitting the model at multiple
exceedance thresholds (from the 1.25% to 25.00% mirrored quantiles).
Quantitatively similar asymmetries in Hawkes parameters are found across all
six indices, adding further empirical support to a temporal leverage effect in
financial price time series in which the impact of losses is not only larger
but also more immediate. Out-of-sample backtests find that our 2T-POT Hawkes
model is more reliably accurate than the GARCH-EVT model when forecasting
(mirrored) value-at-risk and expected shortfall at the 5% coverage level and
below. This suggests that asymmetric Hawkes-type arrival dynamics are a better
approximation of the true data generating process for extreme daily log-returns
than GARCH-type conditional volatility; our 2T-POT Hawkes model therefore
presents a better performing alternative for financial risk modelling.
