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

A Bayesian Long Short-Term Memory Model for Value at Risk and Expected Shortfall Joint Forecasting