Better volatility forecasts still need honest uncertainty ranges

A forecast can be close to the eventual outcome on average while still understating how uncertain that outcome is. In a November 2026 arXiv preprint, Kyle Parrella and Kejin Wu investigate that distinction for stock-market volatility. Their results suggest that improving a model’s central prediction does not necessarily produce more dependable prediction intervals.

The study combines established statistical models with a long short-term memory network, or LSTM: a neural network designed to process sequences. It then tests two ways to make uncertainty ranges more realistic. Both help in the reported experiments, but neither eliminates the tendency to miss more outcomes than intended.

What is being forecast?

Here, realized volatility means the standard deviation of daily logarithmic returns across the next five or 20 trading days, calculated around those future returns’ own average. It measures the size of market fluctuations, not whether prices rise or fall. Despite the terminology, this study does not use intraday observations to construct its volatility measure.

A point forecast supplies one predicted value. A 95% prediction interval instead aims to contain the future realized value in 95% of repeated forecasting cases. Coverage measures how often that actually happens; interval width measures how broad the range must be.

Combining financial structure with a neural network

The hybrids feed an LSTM information from three classical models: GARCH, which tracks changing volatility through past shocks and variances; EGARCH, which allows different responses to positive and negative shocks; and an exponentially weighted moving average, which discounts older observations.

The researchers compare two approaches to the expected return around which fluctuations occur. One uses a rolling historical average. The other uses an autoregressive moving-average model, or ARMA, incorporating past returns and past errors. In the ARMA hybrid, the expected-return component and neural-network parameters are fitted jointly using a likelihood-based objective.

To generate prediction intervals, the fitted models simulate many possible future return sequences. Each sequence produces a future volatility value; the distribution of those values supplies a point forecast and interval bounds. This forward-bootstrap procedure uses 2,000 simulated paths per interval.

The first proposed improvement changes the simulated shocks. Instead of drawing them from a normal distribution, the method resamples centered, standardized historical residuals—the errors left after fitting the model. This aims to better reflect financial data’s heavy tails, where unusually large movements occur more often than a normal distribution suggests.

The second improvement widens intervals according to historical forecasting misses, scaled by interval width. Inspired by conformal prediction, it expands both ends and prevents the lower bound from falling below zero.

Different winners for different tasks

The simulations use heavy-tailed returns with either fixed or changing volatility-model coefficients. Each setting is evaluated over 500 independent repetitions, with training samples of 250 or 500 observations and forecast horizons of five or 20 days.

The ARMA hybrid often improves point forecasts, but the tables do not support universal superiority over classical models. For example, in the changing-coefficient simulation with 500 training observations and a 20-day horizon, both GARCH and EGARCH have lower root mean squared errors than the ARMA hybrid.

The real-data analysis uses daily closing prices for the S&P 500, NASDAQ Composite, NYSE Composite and Dow Jones Industrial Average from 2020 through 2025. Rolling training windows and forecast horizons create 16 evaluation scenarios.

Across those scenarios, the ARMA hybrid’s geometric-mean relative root mean squared error is 0.952 against GARCH’s benchmark of 1—about 4.8% lower. EGARCH is close behind at 0.956. The standalone LSTM performs worse than GARCH on all three aggregate point-error measures.

For intervals, the rolling-mean hybrid with both residual resampling and widening performs best on the reported coverage measures. Its mean coverage reaches 92.5%, versus 88.5% for GARCH, but remains below the intended 95%. Its lowest scenario coverage is 89.8%. Its geometric-mean relative interval length is 1.437: roughly 44% wider than GARCH’s. Better coverage therefore comes with substantially broader ranges.

Improvement without a guarantee

The conformal-inspired adjustment does not carry the usual finite-sample coverage guarantee. Historical scores are dependent because forecast windows overlap, and the model used to calculate them is fitted using the full training sample rather than refitted at each historical forecast origin. The authors explicitly acknowledge these limitations and defer theoretical analysis.

They also note that uncertainty from estimating the model remains incompletely captured. Repeatedly refitting models on bootstrap datasets could address that problem more directly, but would increase computation.

This article is grounded in the supplied text extraction of the preprint, not an independent replication. Peer review is not established by the supplied evidence; extraction omits figures and may damage equations, tables or layout. The results support improved empirical coverage in these tests—not guaranteed protection against future market turbulence.