Learning Volatility: A Bayesian Neural Stochastic Framework

Forecasting volatility is a central task in financial econometrics, and accuracy is crucial for risk management, portfolio allocation, and asset pricing. This paper develops a class of Bayesian neural network stochastic volatility (NN–SV) models that combine the flexibility of machine learning with the probabilistic structure of stochastic volatility frameworks. The models incorporate high-frequency information through realised variance, jump variation, and semivariance, and use Bayesian stacking to combine predictive distributions across specifications, explicitly targeting out-of-sample forecast performance. Using data from the DAX, FTSE-100, and S\&P 500 indices, we compare the proposed models with classical SV benchmarks. The empirical results show that the predictive value of high-frequency measures is both market- and horizon-specific. Realised variance provides substantial short-horizon gains in European markets, while model combination and semivariance asymmetry improve medium-term forecasts, particularly for the FTSE. In the US, volatility is predominantly persistence-driven at short horizons, but realised and asymmetric components become more informative as the horizon lengthens. In general, the Bayesian NN–SV framework produces more accurate and robust forecasts, offering new insights into how the market structure and forecast horizon jointly shape the volatility dynamics.