Rough Path Mathematics Meets Graph Theory to Sharpen Global Volatility Forecasts

Volatility is the invisible force that shapes nearly every decision in modern finance, from the price of an option to the capital a bank must hold against a market crash. Forecasting it well has challenged economists for decades, and a new study published in Applied Intelligence by Zeyu Guo, Andi Han, Chao Wang, and Junbin Gao of the University of Sydney claims a meaningful step forward. Their model, called Sig-GSPHAR, blends three previously separate mathematical traditions: the classical econometrics of heterogeneous autoregressive (HAR) models, graph signal processing on directed networks of market spillovers, and path signatures, a tool from rough path theory that captures the fine geometric shape of a time series. Tested on 29 global stock markets, the framework beats both econometric benchmarks and recent deep learning competitors across most forecasting scenarios.

The starting point for the research is the HAR model, one of the most influential tools in realized volatility forecasting. Realized volatility is estimated from intraday price data, typically by summing squared five-minute returns, and is far more informative than the squared daily returns used in older GARCH-style models. HAR summarizes the recent past with a handful of interpretable averages, typically over one, five, and twenty-two days, mirroring the daily, weekly, and monthly rhythms of market participants. Its simplicity makes it remarkably robust, but it has two structural blind spots. It is essentially univariate, ignoring how volatility transmits from one market to another, and its linear averages can miss nonlinear information hiding inside the look-back window, such as the particular shape of a volatility path that rises slowly and then collapses.

To handle cross-market transmission, the authors build on a 2025 framework called GSPHAR, which embeds a directed spillover network directly into the forecasting architecture. The network is constructed using the Diebold-Yilmaz connectedness framework: a vector autoregression with a lag order of 22 is fitted to standardized realized volatility, and a generalized forecast error variance decomposition with a 22-day horizon quantifies how much shocks in one market drive forecast uncertainty in another. Crucially, this graph is estimated only on the training sample to avoid look-ahead bias, then sparsified by keeping only the strongest quarter of links. Because the graph is directed, with volatility flowing asymmetrically between markets, the model uses the magnetic Laplacian, a spectral operator that encodes direction through complex phases, allowing a graph Fourier transform that preserves which way information travels.

The genuine innovation of Sig-GSPHAR lies in what it adds to this spectral backbone. For each market, the model takes the raw 22-day window of realized volatility, augments it with a normalized time channel to form a two-dimensional path, and computes a truncated path signature of depth four. Path signatures, rooted in the rough path theory developed by Terry Lyons and building on work by Kuo-Tsai Chen from the 1950s, convert a sequential path into a structured collection of iterated integrals. The first level captures simple increments, the second captures interactions between pairs of increments, and higher levels encode progressively richer temporal structure. A key theoretical property guarantees that linear functions of signatures can approximate any continuous nonlinear functional of the path, meaning the model can learn complex within-window dynamics without resorting to massive black-box recurrent networks.

The mathematics also dictates a careful trade-off. The magnitude of the depth-k signature terms scales with the k-th power of the path’s total variation, divided by k factorial. A single large jump inside the look-back window therefore inflates the higher-order coordinates, making them noisy and potentially misleading descriptions of what follows, since volatility typically mean-reverts after extreme spikes. This factorial bound explains why the authors cap the truncation depth at four: deeper signatures grow geometrically in dimension and inject noise rather than signal. An ablation study confirms the prediction, with depth four consistently delivering the best performance across all five volatility proxies, while depth two is too shallow to capture nonlinear interactions and depth five degrades generalization. A learnable sigmoid gate, shared across markets and time, further suppresses uninformative signature channels.

The empirical evaluation covers 29 stock market indices from June 2009 to May 2022, spanning 3,395 daily observations per market and five realized volatility proxies: five-minute realized variance, bipower variation, median realized volatility, realized kernel variance with a Parzen kernel, and realized semivolatility. All models were trained under an identical protocol, with the same chronological 70/30 split, the same 22-day look-back, and the same tuning budget, ensuring that performance differences reflect modeling choices rather than experimental design. For one-step-ahead forecasts, Sig-GSPHAR achieved the lowest error in 114 of 145 market-proxy pairs for mean squared error and 112 of 145 for mean absolute error. Paired one-sided Wilcoxon signed-rank tests with Holm correction for multiple comparisons confirmed that the improvements over GSPHAR are statistically significant for most proxies, with p-values as low as 0.006 for mean squared error on realized semivolatility.

The advantage becomes even more pronounced at the 22-day horizon, where the signature block, computed from the full look-back segment aligned with the long target, can encode nonlinear interactions that linear lag aggregation cannot. Sig-GSPHAR improved on GSPHAR in 26 of 29 markets for realized variance and in all 29 markets for four of the five proxies under mean absolute error. Holm-adjusted p-values fell to the order of 10 to the minus 6 for mean squared error and 10 to the minus 8 for mean absolute error, among the strongest statistical evidence reported in this literature. The result suggests that the longer the forecasting horizon, the more valuable it becomes to represent the shape of the volatility path rather than just its coarse averages.

Perhaps the most scientifically interesting part of the paper is its honest diagnosis of when signatures fail. The aggregate advantage of the new model comes almost entirely from calm days: on the 95 percent of test days with ordinary volatility levels, Sig-GSPHAR is more accurate, while on the top 5 percent of extreme days it slightly under-predicts true spikes, with a median forecast-to-realized ratio of 0.62 versus 0.68 for GSPHAR. The two markets where the new model loses to its predecessor, Finland’s OMXHPI and Denmark’s OMXC20, are precisely the most heavy-tailed in the panel, with sample kurtosis of 259 and 89 respectively. For OMXHPI, just ten days account for 43 percent of the entire out-of-sample squared error, so a small difference on a handful of high-leverage days determines the headline number. The authors trace this behavior to the graph-HAR backbone rather than the signature branch itself, and suggest jump-aware loss functions or variance-stabilized log domains as remedies.

Two robustness analyses strengthen the practical case. First, a rolling-window re-estimation of the spillover network shows that the dominant transmission structure is highly persistent: in normal periods the Spearman rank correlation between rolling and static net-spillover vectors is about 0.81, and even during the COVID-19 shock, when the correlation drops to roughly 0.56, total connectedness stays high and the set of top net-transmitters is largely preserved. Because the model uses the graph only through the magnetic-Laplacian spectral basis, which depends on the dominant structure rather than fragile individual edges, a static graph estimated once on the training data remains adequate. Second, the authors equip their point forecasts with distribution-free conformal prediction intervals using a volatility-normalized score. Both Sig-GSPHAR and GSPHAR achieve near-nominal coverage, but at matched coverage the new model produces intervals that are 7 to 11 percent narrower across all five proxies, translating its point-forecast gains into sharper, more informative uncertainty bands that widen during turbulence and tighten in calm markets.

The broader significance of the work lies in demonstrating that ideas from stochastic analysis can be fused productively with graph machine learning in a domain as noisy and adversarial as financial volatility. Rather than replacing interpretable econometrics with opaque deep networks, Sig-GSPHAR keeps the HAR skeleton, adds a mathematically principled nonlinear feature layer, and aligns everything in a single spectral representation governed by the physics of market spillovers. The authors point to time-varying spillover graphs and robust training losses as natural next steps, but the core message stands: when the goal is anticipating how turbulence propagates through a globally connected financial system, the shape of the recent past matters as much as its average, and rough path theory provides exactly the language needed to describe that shape.

Subject of Research: Multivariate realized volatility forecasting using path signatures and directed graph spectral methods

Article Title: Signature-enhanced graph spectral HAR for multivariate realized volatility forecasting

Article References: Guo, Z., Han, A., Wang, C., & Gao, J. (2026). Signature-enhanced graph spectral HAR for multivariate realized volatility forecasting. Applied Intelligence, 56(14), Article 406. https://doi.org/10.1007/s10489-026-07445-1

Image Credits: AI Generated

DOI: 10.1007/s10489-026-07445-1

Keywords: realized volatility, path signatures, rough path theory, graph signal processing, HAR model, magnetic Laplacian, Diebold-Yilmaz spillover, volatility forecasting, conformal prediction, financial econometrics, machine learning, spillover networks