Learning the shape of market risk with a compact neural network

A flexible distribution inside a structured model

In Neural Score-Driven Volatility and Risk Forecasting, Qianli Zhao, Chao Wang, Minh-Ngoc Tran and Richard Gerlach propose GAS-NN, a model for forecasting changing financial volatility and downside risk. Version 1 was submitted to arXiv on 4 November 2026. It is a preprint; the supplied metadata does not establish peer-reviewed publication.

The approach builds on generalized autoregressive score, or GAS, models. These update volatility using the likelihood score: a derivative describing how the model’s assessment of an observed return changes with its volatility parameter. Conventional versions assume a distribution such as Gaussian or Student-t in advance. That choice shapes both the predicted returns and the response to new shocks.

The authors instead use a neural network to learn the negative derivative of the innovation log density. Here, innovations are returns after removing their mean and scaling by the model’s changing volatility. The selected network has two hidden layers of eight neurons each, with 97 trainable neural parameters. It learns a function within a statistical model rather than directly predicting prices.

Turning a learned score into probabilities

The network output is integrated and normalized to reconstruct a probability density. That density is then shifted and rescaled to have zero mean and unit variance. This identification lets the model interpret its changing scale as conditional volatility. The same reconstructed distribution supplies the likelihood, volatility updates and risk forecasts.

Flexibility does not mean an absence of assumptions. Innovations are assumed independent of past information, with a distribution fixed within each estimation window. Beyond the reconstruction interval, the learned score is held constant at its endpoints. Provided those endpoints have the required signs, this produces integrable exponential tails with finite moments—not power-law tails.

The distinction also affects shock responses. Although the neural density score flattens in the outer tails, its volatility-updating score grows approximately linearly with shock magnitude. It is therefore not bounded like the Student-t benchmark’s update, although it grows more slowly than the Gaussian model’s quadratic response.

Two ways to estimate the model

The two-step procedure first learns the neural score through score matching, then holds the resulting density fixed while estimating volatility dynamics. Importantly, its first-stage inputs use one sample-wide scale. They still contain changing conditional volatility, so the learned distribution is a working density rather than a direct estimate from fully filtered innovations.

The one-step procedure starts from that fit and jointly refines the neural and volatility parameters through a regularized conditional likelihood. The experiments compare 30 or 200 initial score-matching epochs, with 30 additional joint updates for the one-step versions. Architecture selection uses pre-2020 S&P 500 training and validation data; parameters are subsequently estimated separately for each market.

What the forecasting exercise found

The study covers eight indices, including the S&P 500, FTSE, Hang Seng and Nikkei 225, using daily observations from January 2004 to May 2026. One-day-ahead forecasts run from January 2020 through 29 May 2026. Models are re-estimated in rolling windows using only observations preceding each target date. January–June 2020 is evaluated separately.

For complete predictive distributions, neural specifications achieve the lowest average predictive negative log-likelihood in seven of eight markets over the full forecast period. Student-t wins for Hang Seng. During early 2020, the jointly refined model initialized with 30 score-matching epochs achieves the lowest loss in all eight markets.

But downside-risk rankings differ. The study evaluates the 2.5% return quantile, or Value-at-Risk, and Expected Shortfall, the expected return below that threshold. The 30-epoch two-step model records the lowest joint VaR–ES loss in six markets over the full period and all eight during early 2020. Conversely, it is excluded from every full-period confidence set for predictive-density loss. Better overall distribution forecasts do not automatically mean better realized tail-risk scores.

Limits to the evidence

These are numerical rankings, not evidence of a universally superior method. The authors’ 75% model confidence sets retain multiple competitors, especially for tail-risk losses. Longer training and joint refinement do not improve every forecast target or market.

The paper also leaves stability and estimation theory for further work. Its numerical checks do not establish global optimization or filter stability. The supplied extraction includes the results tables, appendix and explanatory prose, but does not preserve their colour-based confidence-set shading; confidence-set statements here therefore follow the authors’ written descriptions. The evidence supports a flexible forecasting approach with target-dependent gains, not a general guarantee of improved risk prediction.