  Value-at-Risk (VaR) and Expected Shortfall (ES) are widely used in the
financial sector to measure the market risk and manage the extreme market
movement. The recent link between the quantile score function and the
Asymmetric Laplace density has led to a flexible likelihood-based framework for
joint modelling of VaR and ES. It is of high interest in financial applications
to be able to capture the underlying joint dynamics of these two quantities. We
address this problem by developing a hybrid model that is based on the
Asymmetric Laplace quasi-likelihood and employs the Long Short-Term Memory
(LSTM) time series modelling technique from Machine Learning to capture
efficiently the underlying dynamics of VaR and ES. We refer to this model as
LSTM-AL. We adopt the adaptive Markov chain Monte Carlo (MCMC) algorithm for
Bayesian inference in the LSTM-AL model. Empirical results show that the
proposed LSTM-AL model can improve the VaR and ES forecasting accuracy over a
range of well-established competing models.
