Structural Intraday-volatility Prediction

• Structural Intraday-volatility Prediction (SIP) is a high-frequency forecasting approach that models intraday volatility as a structured object rather than a single daily scalar.
• SIP employs diverse methodologies—including low-rank matrix completion, causal neural sequence aggregation, latent-state probabilistic modeling, and state-heterogeneous diffusion—to capture the geometry and causality of intraday dynamics.
• Empirical studies show that SIP methods can outperform traditional volatility predictors, though challenges remain in model specification, scalability, and robustness under structural breaks.

Searching arXiv for recent and foundational papers on Structural Intraday-volatility Prediction and closely related intraday volatility forecasting methods. Structural Intraday-volatility Prediction (SIP) is a line of high-frequency volatility research in which the prediction target is constructed from intraday data and the forecasting rule imposes explicit structure on how intraday information is organized, denoised, and propagated across time. In the narrow sense, the term is introduced for a low-rank, nonparametric method that predicts the remaining instantaneous volatility path within the current trading day from previous days’ data and the current day’s observed partial path (Choi et al., 29 Jul 2025). In a broader sense, closely related work treats intraday volatility as a structured object rather than a single daily scalar, whether the object is next-day daily realized variance from raw intraday returns, a next-day intraday instantaneous volatility vector, a 24-hour latent spot-volatility path, or a state-dependent continuous-time volatility process (Moreno-Pino et al., 2022, Choi et al., 2024, Stroud et al., 2012).

1. Scope and prediction targets

SIP departs from daily-volatility forecasting by changing both the target and the representation. Instead of compressing intraday observations into a fixed realized-volatility statistic before modeling, SIP methods either retain the ordered intraday return sequence, organize daily intraday volatility curves into matrices, or embed intraday volatility in a state-space or diffusion system. The central distinction is therefore structural: intraday information is not merely an input source, but an object whose geometry, causality, or factorization is modeled explicitly.

The literature supports several distinct prediction targets. DeepVol forecasts day-ahead realized volatility, more precisely the next-day daily volatility proxy denoted $\sigma_{t+1}^2$, with the practical target given by daily realized variance $RV_t=\sum_{i=1}^{I} r_{i,t}^2$, using raw intraday returns from the previous $T$ days as inputs (Moreno-Pino et al., 2022). TIP-PCA forecasts the full next-day intraday instantaneous volatility vector $(c_{D+1,1},\dots,c_{D+1,n})$, where $c_{i,j}=\sigma_{i,t_j}^2$, by treating the historical sample as an interday-by-intraday matrix (Choi et al., 2024). SIP in the strict 2025 sense predicts the future segment of the current day’s spot-variance path, $E[(c_{D,n_1+1},\ldots,c_{D,n})\mid F_{D,n_1}]$, from a partially observed current row and a low-rank historical volatility surface (Choi et al., 29 Jul 2025). The Bayesian 24-hour stochastic-volatility framework instead models 5-minute futures returns directly and evaluates forecasts mainly through future realized variance or realized volatility over hourly and daily windows (Stroud et al., 2012).

Framework	Prediction object	Structural device
DeepVol (Moreno-Pino et al., 2022)	Next-day daily realized variance	Dilated causal convolutions on raw intraday returns
TIP-PCA (Choi et al., 2024)	Next-day intraday instantaneous volatility vector	Low-rank interday-by-intraday matrix with two-sided projected-PCA
SIP (Choi et al., 29 Jul 2025)	Remaining current-day instantaneous volatility path	Low-rank matrix completion using current-day partial observations
Bayesian 24-hour SV (Stroud et al., 2012)	Future latent 5-minute volatility and aggregated realized volatility	State-space stochastic-volatility model with latent factors, seasonality, announcements, and jumps

This variation in targets is substantive. A scalar next-day realized-variance forecast, a full next-day intraday path forecast, and a current-day path-completion problem are not interchangeable tasks. Much of the SIP literature can be read as a progressive refinement from scalar volatility prediction toward vector- and function-valued intraday objects.

2. Structural representations

A first structural family is based on low-rank matrix representations. TIP-PCA organizes historical instantaneous variances into $\mathbf{C}_{D,n}=(c_{i,j})_{D\times n}$ and assumes a low-rank signal-plus-noise decomposition

$\mathbf{C}_{D,n}=U\Lambda V^\prime + E,$

where the left singular structure captures interday dependence and the right singular structure captures intraday deterministic shape, such as the U-shaped volatility pattern. In the semiparametric specification, $u_{i,k}=g_k(X_i)$ and $v_{j,k}=h_k(W_j)$, so the row scores depend on observable daily covariates while the column loadings depend on intraday clock-time covariates (Choi et al., 2024). The later SIP method keeps the low-rank surface view but reformulates the problem as structured completion of the missing future block of the current day, using the oracle identity

$RV_t=\sum_{i=1}^{I} r_{i,t}^2$0

and its feasible estimator

$RV_t=\sum_{i=1}^{I} r_{i,t}^2$1

The structural prior is that intraday volatility lives near a low-dimensional manifold whose row and column spaces are learnable from recent days and the currently observed segment of the day (Choi et al., 29 Jul 2025).

A second family is neural and sequential rather than low-rank. DeepVol uses Dilated Causal Convolutions inspired by WaveNet-style architectures to map raw intraday return sequences to next-day daily realized variance. Its core structural claim is that ex ante econometric compression of intraday information into realized measures loses predictive content, whereas a causal multi-scale temporal architecture can learn the appropriate aggregation rule end to end. The architecture imposes causality, translation-invariant local pattern extraction, exponentially growing dilation, residual connections, and ReLU nonlinearities (Moreno-Pino et al., 2022). The structural content lies less in explicit factorization than in a constrained temporal operator.

A third family is latent-state and probabilistic. The 24-hour Bayesian stochastic-volatility model specifies

$RV_t=\sum_{i=1}^{I} r_{i,t}^2$2

where $RV_t=\sum_{i=1}^{I} r_{i,t}^2$3 is a slow-moving interday volatility factor, $RV_t=\sum_{i=1}^{I} r_{i,t}^2$4 a fast intraday volatility factor, $RV_t=\sum_{i=1}^{I} r_{i,t}^2$5 a deterministic 288-point time-of-day seasonal component, and $RV_t=\sum_{i=1}^{I} r_{i,t}^2$6 a deterministic announcement effect. The full system also includes price jumps, volatility jumps, heavy-tailed innovations, and leverage through correlation between return and volatility shocks (Stroud et al., 2012). Here SIP is structural in the strongest generative sense: each source of intraday volatility has an explicit role in the return-generating process.

A fourth family embeds exogenous state heterogeneity into a continuous-time intraday diffusion. The SG-Ito model sets

$RV_t=\sum_{i=1}^{I} r_{i,t}^2$7

so that a daily state variable $RV_t=\sum_{i=1}^{I} r_{i,t}^2$8 selects the intraday volatility law for day $RV_t=\sum_{i=1}^{I} r_{i,t}^2$9. The integrated-volatility recursion is state dependent, and the paper develops both QMLE and Wald testing for whether the state changes $T$0-type volatility dynamics (Chun et al., 2021). This places SIP at the intersection of high-frequency measurement and low-frequency economic conditioning.

A fifth, more economic, line derives structural intraday volatility from equilibrium order imbalances and benchmark-following behavior. In the TWAP/VWAP model, predictable and stochastic benchmark trajectories generate intraday price pressure and contribute directly to conditional quadratic variation. In the stochastic-target case,

$T$1

so benchmark randomness and hedging-demand randomness become explicit volatility channels (Choi et al., 2018). This suggests that some SIP predictors should be interpreted as proxies for latent benchmark pressure, risk-sharing capacity, or stochastic hedging demand rather than as reduced-form lags alone.

3. Measurement layer and data representation

SIP is inseparable from its measurement layer because intraday volatility is latent and microstructure contaminated. The literature therefore differs sharply on whether structure is imposed before or after denoising. DeepVol minimizes preprocessing by transforming raw prices into intraday log returns

$T$2

using within-day return sequences sampled at 1, 5, 15, 30, or 60 minutes, and forecasting the next day’s realized variance proxy directly from those raw sequences rather than from precomputed realized measures (Moreno-Pino et al., 2022). This is a deliberate rejection of handcrafted daily aggregation as an input representation.

Matrix-based SIP methods instead make the spot-volatility estimator explicit. Both TIP-PCA and the 2025 SIP paper assume noisy high-frequency log-prices $T$3 and construct $T$4 at intraday grid points using the jump-robust pre-averaging kernel spot-volatility estimator of Figueroa-López and Wu (2024). The resulting error decomposition

$T$5

is then carried through the low-rank estimation theory, so denoising and structural prediction are analytically linked rather than treated as unrelated stages (Choi et al., 2024, Choi et al., 29 Jul 2025).

A more microstructure-specific measurement layer appears when observed prices are not centered-noise transaction prices but one-sided noisy order prices. In the stochastic boundary model for ask prices,

$T$6

the relevant information about the latent efficient price is concentrated in local minima rather than in averaged returns. The paper proposes the left-sided spot-variance estimator

$T$7

with jump-robust truncation and feasible bias correction through $T$8 (Bibinger, 2023). For SIP, this paper functions as a structural measurement module: if the data source is the order book, conventional centered-noise realized-volatility estimators are misspecified.

Another branch treats the representation problem as a cross-sectional panel task. The commonality paper defines intraday realized volatility as

$T$9

with $(c_{D+1,1},\dots,c_{D+1,n})$0 based on 1-minute mid-price returns, and constructs a market volatility proxy

$(c_{D+1,1},\dots,c_{D+1,n})$1

Pooling stock-level observations turns SIP into a panel-learning problem in which a market-wide latent volatility state is represented by cross-sectional averaging (Zhang et al., 2022).

At the finest event-time scale, the intraday GARCH model for discrete price changes takes observations as irregularly spaced transactions, with durations $(c_{D+1,1},\dots,c_{D+1,n})$2 and discrete price increments $(c_{D+1,1},\dots,c_{D+1,n})$3. It explicitly models zero durations, simultaneous transactions, many zero price changes, and excess zeros through a zero-inflated Skellam law, while spline-estimated $(c_{D+1,1},\dots,c_{D+1,n})$4, $(c_{D+1,1},\dots,c_{D+1,n})$5, and $(c_{D+1,1},\dots,c_{D+1,n})$6 provide structural adjustments for duration seasonality, volatility seasonality, and the duration-volatility relation (Holý, 2022). This event-time formulation shows that SIP need not be clock-time and need not assume continuous return support.

4. Estimation, loss functions, and evaluation

The most common forecast loss in the SIP literature is QLIKE. DeepVol uses QLIKE both as an evaluation metric and as the main training loss,

$(c_{D+1,1},\dots,c_{D+1,n})$7

and emphasizes its relevance because QLIKE is known to be relatively robust to noise in the volatility proxy (Moreno-Pino et al., 2022). Matrix-based next-day and current-day path predictors also evaluate by MSPE and QLIKE on vector-valued spot-variance forecasts, making pointwise path accuracy and volatility-loss robustness the dominant criteria (Choi et al., 2024, Choi et al., 29 Jul 2025).

Panel machine-learning work on intraday realized volatility retains the same emphasis. The commonality paper evaluates MSE and QLIKE, applies Diebold-Mariano tests on cross-sectionally averaged loss differences, and uses Model Confidence Set procedures. Its design makes QLIKE the principal accuracy measure for pooled, nonlinear intraday RV forecasters (Zhang et al., 2022). The 24-hour Bayesian literature instead evaluates through bias, MAE, Mincer-Zarnowitz $(c_{D+1,1},\dots,c_{D+1,n})$8, predictive likelihoods, VaR exceedance rates, and ordered predictive quantile fit, because the model generates full predictive distributions rather than only point estimates (Stroud et al., 2012).

Several SIP methods replace single-objective estimation with composite objectives. WamOL calibrates an intraday implied-volatility surface by combining market-fit loss, PDE residual loss, and no-arbitrage derivative-inequality penalties: $(c_{D+1,1},\dots,c_{D+1,n})$9 Its distinctive feature is not a different volatility target but adaptive balancing across objective categories and across collocation points during online intraday updates (Hoshisashi et al., 2024). This broadens SIP from return-volatility prediction to structurally constrained intraday surface calibration.

Upstream structural monitoring has its own asymptotic apparatus. The functional-data break-detection paper develops separate tests for shape changes, magnitude changes, and arbitrary changes in intraday volatility curves, proves asymptotically correct size, and derives consistency rates that depend on both the number of trading days $c_{i,j}=\sigma_{i,t_j}^2$0 and the intraday grid size $c_{i,j}=\sigma_{i,t_j}^2$1. The shape test is effectively $c_{i,j}=\sigma_{i,t_j}^2$2-consistent, while the magnitude test is $c_{i,j}=\sigma_{i,t_j}^2$3-consistent (Kokoszka et al., 2024). In SIP terms, evaluation is not only forecast accuracy but also structural stability of the object being forecast.

5. Empirical findings and applications

The empirical record of SIP is heterogeneous but broadly favorable to structural approaches. DeepVol’s main NASDAQ-100 out-of-sample experiment, using 5-minute data and a 1-day receptive field, reports that DeepVol outperforms HEAVY on MAE, RMSE, SMAPE, QLIKE, and MedAE, with relative improvements of 14.502% in MAE, 17.404% in RMSE, 4.452% in SMAPE, 0.789% in QLIKE, and 21.097% in MedAE. The paper also finds that adding realized variance as an extra input to DeepVol degrades performance, and that the best configuration is 5-minute sampling with a 1-day receptive field rather than a longer raw lookback (Moreno-Pino et al., 2022). The structural conclusion is that learned intraday aggregation can outperform fixed realized-measure compression, but only when the sampling frequency respects the usual microstructure-noise trade-off.

TIP-PCA extends the empirical scope from a scalar next-day forecast to the entire next-day intraday instantaneous volatility vector. In the empirical study on SPY and 11 sector ETFs, it delivers the lowest MSPE for all 12 ETFs, while QLIKE is more mixed against PC in some individual series; it also passes the most 10-minute VaR backtests overall, especially at lower quantiles (Choi et al., 2024). The 2025 SIP method further shifts the task from next-day prediction to current-day path completion and reports the lowest MSPE and QLIKE in nearly all cases across assets and observed fractions $c_{i,j}=\sigma_{i,t_j}^2$4, with the gains strongest when only a small early-day segment is observed. In downstream 5-minute VaR forecasting, SIP records the largest number of non-rejection cases after Benjamini-Hochberg adjustment (Choi et al., 29 Jul 2025).

Probabilistic state-space SIP also performs strongly. The Bayesian 24-hour stochastic-volatility models improve realized-volatility forecasts by up to 50% relative to standard benchmarks; daily-horizon $c_{i,j}=\sigma_{i,t_j}^2$5 is around 73%, versus 47%–57% for intraday GARCH variants, and hourly $c_{i,j}=\sigma_{i,t_j}^2$6 improves to about 66% from roughly 56%–60% (Stroud et al., 2012). The gain is attributed primarily to multiple volatility factors, time-of-day seasonality, and jump or heavy-tail handling rather than to any single component.

Cross-sectional pooling and commonality are another recurring empirical theme. In the pooled intraday RV paper, intraday commonality is high and stable, rising through the day and peaking near the close. The average commonality is reported around 74.3% for 65-minute data, versus 35.5% for daily data. Neural networks dominate linear regressions and tree-based models for intraday horizons, and trained models transfer successfully to unseen stocks, which the paper interprets as evidence for a universal volatility mechanism among stocks (Zhang et al., 2022). This places SIP in a panel-structure tradition as much as in a univariate time-series one.

Structural instability is also pervasive. The functional break-detection study finds at least one global break in 7168 of 7293 US stocks, or 98.3% of the sample, with first-break dates clustering around the 2008 subprime crisis, the European debt crisis period, and post-COVID 2020 (Kokoszka et al., 2024). This is directly consequential for SIP: any model that assumes a stable intraday shape or stable volatility surface across those regimes is structurally exposed to misspecification.

Theoretical market-structure work complements these forecasting results by identifying economic channels behind intraday volatility. The benchmark-trading equilibrium model shows that TWAP and VWAP trading both reduce market liquidity and increase price volatility relative to just terminal trading targets alone, while randomness in VWAP target trajectories adds additional randomness to intraday price-pressure patterns (Choi et al., 2018). This suggests that some of the “intraday structure” captured statistically in SIP models may proxy benchmark-following order flow and endogenous liquidity rather than purely exogenous information arrival.

6. Limitations, misconceptions, and open problems

A common misconception is that SIP names a single algorithm. In fact, the literature contains at least four distinct structural logics: low-rank matrix completion, causal neural sequence aggregation, latent-state probabilistic decomposition, and state-heterogeneous diffusion or equilibrium models. Another misconception is that “structural” means purely econometric. DeepVol shows that a neural architecture can be structural when it imposes causality, multi-scale dilation, and end-to-end temporal aggregation rather than accepting arbitrary vectorized inputs (Moreno-Pino et al., 2022).

The literature also has clear limitations. DeepVol is conceptually clear but under-specified for exact reproduction: it does not report the exact number of layers, kernel size, channel dimensions, dilation schedule beyond exponential growth, dropout, or learning rate. It also compares mainly against traditional econometric benchmarks, not against LSTMs, Transformers, HAR/HAR-RV, or Realized GARCH (Moreno-Pino et al., 2022). Low-rank matrix methods gain interpretability and denoising, but rank choice matters, the true volatility surface may not be low-rank, and extending the framework to many assets introduces an additional cross-sectional curse of dimensionality (Choi et al., 2024). WamOL is structurally rich, yet its PDE and no-arbitrage conditions are enforced by soft penalties rather than exact constraints, so the method empirically reduces violations without rigorously guaranteeing an arbitrage-free surface (Hoshisashi et al., 2024).

Some state-aware models are structurally appealing but operationally incomplete. SG-Ito forecasts daily integrated volatility rather than explicit minute-ahead volatility, allows that the relevant state may not be observed at the start of day $c_{i,j}=\sigma_{i,t_j}^2$7, and develops the theory mainly for binary exogenous states (Chun et al., 2021). Functional break-detection methods are highly useful for regime segmentation, but they are retrospective rather than sequential online surveillance and therefore serve SIP most naturally as an upstream retraining and diagnostics layer rather than as a forecaster (Kokoszka et al., 2024).

The principal open problems are already visible inside the literature. DeepVol explicitly identifies multivariate SIP, time-of-day-aware structure, limit order book inputs, probabilistic forecasts, hybrid econometric-neural SIP, cross-resolution architectures, and noise-aware layers as natural extensions (Moreno-Pino et al., 2022). A plausible implication is that future SIP systems will combine three layers that are currently studied mostly separately: a microstructure-aware measurement layer, a regime-detection layer, and a structured forecasting layer that can switch between low-rank, neural, or state-space regimes as market conditions change. The accumulated evidence suggests that intraday volatility is neither an unstructured high-dimensional sequence nor merely a daily realized-variance scalar. It is a structured object whose representation, noise model, and temporal conditioning jointly determine predictability.