In this paper, we develop a hybrid approach to forecasting the volatility and risk of financial instruments by combining econometric GARCH models with deep learning networks. For the latter, we employ Gated Recurrent Unit (GRU) networks, whereas four different specifications are used for GARCH: standard GARCH, EGARCH, GJR-GARCH and APARCH. Models are tested using daily returns on the S&P 500 index and Bitcoin prices. As the main volatility estimator, also the target function of our hybrid models, we use the modified Garman-Klass estimator. Volatility forecasts resulting from the hybrid models are employed to evaluate the assets’ risk using the Value-at-Risk (VaR) and Expected Shortfall (ES). Gains from combining the GARCH and GRU approaches are discussed in the contexts of both the volatility and risk forecasts. It can be concluded that the hybrid solutions produce more accurate point volatility forecasts, although it does not necessarily translate into superior risk forecasts.