Learning Volatility:A Bayesian Neural Stochastic Framework

We propose a Bayesian neural stochastic volatility (NN-SV) framework that embeds a neural network in the latent-state transition of a stochastic volatility state-space model. The approach preserves coherent predictive distributions while learning nonlinear volatility dynamics. The framework incorporates high-frequency information via realised variance and realised upside and downside semivariances, and applies Bayesian stacking to combine predictive distributions across nested specifications to improve out-of-sample accuracy and robustness to specification uncertainty. Using daily returns for the DAX, FTSE-100 and S\&P~500 alongside realised measures, we evaluated one- and ten-day-ahead volatility forecasts against standard stochastic-volatility and realised-volatility benchmarks. The predictive value of high-frequency measures is market and horizon-dependent: within the NN-SV class, realised variance delivers the largest short-horizon improvements in European markets; semivariance-based asymmetries and stacking deliver the most reliable medium-horizon performance, particularly for the FTSE-100; and for the US market, short-horizon predictability is largely persistence-driven, with realised and asymmetric components contributing more at longer horizons. Overall, the NN-SV framework yields accurate and stable forecasts across markets and horizons, providing a practical bridge between machine-learning flexibility and probabilistic time-series structure.