When Do Path Signatures Improve Volatility-Regime Forecasts?

Signatures describe how volatility evolves, not how high it is

Path signatures summarize a return path through its iterated integrals, making them model-free features of the path's shape. The paper's central claim is that this shape information is not the same as the level information that realized-volatility predictors already carry. Regimes that differ mainly in variance level are already separated by realized volatility, so accuracy in that setting says little about what signatures contribute. Signatures matter more when regimes share one stationary variance law but differ in mean-reversion speed, because that difference lives in the temporal ordering of the path rather than in its average variance.

The authors build the path from three channels over a 50-day window: time, cumulative return, and cumulative absolute return. The time channel makes the representation sensitive to when variation occurs inside the window. For example, the level-2 signature includes the Lévy area between time and cumulative absolute return, which records whether large absolute returns arrive early or late. That is a statement about path geometry, not about how volatile the window was overall.

Forecasting next-month volatility terciles in US and developed markets

The main evidence is an out-of-sample forecasting study. The target is the tercile of annualized realized volatility over the next 21 trading days, with cut-points estimated on training data only. Forecasts are issued daily from information available that day, using daily value-weighted market returns for the United States and for developed markets excluding the United States, July 1990 to December 2025, with an out-of-sample period of 2005 to 2025 giving 21 annual test years.

Predictors come in two conventional sets. The compact Statistics set has 11 features, including full- and short-window volatility, mean absolute return, downside and upside semivariance, skewness, excess kurtosis, lag-one autocorrelations of returns, absolute returns and squared returns, and maximum log drawdown. The rich B3 set adds 29 more features such as return quantiles, autocorrelations at lags 2, 5 and 10, variance ratios, spectral entropy, frequency-band powers, realized variance and quarticity, bipower variation, a jump fraction, and sign-change and turning-point rates. Both sets also include averages of squared returns over 1, 5 and 22 days, as in the HAR model, giving 14 and 43 conventional features.

Each predictor set is combined with three learners: ℓ2-regularized logistic regression, random forests, and histogram gradient boosting, each with a prespecified three-setting hyperparameter grid. Three learner-free benchmarks are also used: persistence of the current tercile, a transition benchmark that turns persistence into a probability forecast, and a HAR regression of log next-month realized volatility on the logs of realized volatility over the last 1, 5 and 22 days, whose Gaussian predictive distribution is converted into tercile probabilities. Models are refitted each year on an expanding window with a 21-day gap before each test year, and no test observation is used for any selection.

Machine-learning classifiers do not beat the HAR benchmark

The first finding is that the conventional benchmarks are hard to beat. The HAR regression forecasts next-month terciles as accurately as any classifier, with or without signatures, and persistence of the current tercile is not far behind. This matters because it sets the bar that signatures must clear: adding a flexible learner on top of conventional features does not by itself improve on a simple realized-volatility regression.

Signatures do improve the classifiers built on the compact Statistics predictor set in the United States, in both balanced accuracy and the ranked probability score. But that improvement is relative to the classifier's own baseline, not to HAR. The paper reports that classifiers, with or without signatures, do not beat the HAR regression, so the classifier gains are real but do not change the ranking of the best forecaster.

Raw signatures improve the HAR regression in the United States

The more decision-relevant result is that signatures improve the HAR regression itself. In the United States, adding raw signatures to the HAR regression raises balanced accuracy by 2.7 percentage points and lowers the ranked probability score, and both differences are resolved after Holm correction. Adding the conventional Statistics features to the same regression changes nothing. Outside the United States, no gain survives correction for multiplicity.

The augmentation is implemented by adding standardized extra predictors to the HAR regression: the 14 log-signature coordinates, the 39 raw signature coordinates, or, as a conventional control, the 11 Statistics features. Only the extra coefficients are ridge-penalized, with the penalty chosen on validation years by the ranked probability score from a grid that includes an infinite penalty returning the plain HAR forecast. The authors note that this comparison was designed after the benchmark results were known but fixed before any augmented forecast was scored.

Controlled Heston designs separate scale from speed

To explain the pattern, the authors run two controlled Heston designs. In Design A, regimes differ in variance scale: all states share the mean-reversion speed κ and the scale coefficient, while the long-run variance θ differs across states. In Design B, regimes share the same stationary variance law but differ in mean-reversion speed: the long-run variance θ and the variance ratio q are common, while κ differs across states. Each path receives a fresh parameter draw, with 5,000 observations per path and a minimum dwell of 50 observations.

The simulated classification results show the mechanism. In Design A, balanced accuracy is already high for the conventional Statistics block, around 73%, and signatures add only modestly. In Design B, the conventional Statistics block is much weaker, around 39.8% for logistic regression, and signatures add substantially more: raw signatures raise logistic-regression balanced accuracy by 4.53 percentage points, and log-signatures by 1.88 points. Across the twelve augmentation contrasts, the gain is larger when regimes differ in speed than in scale in eleven of them, and it is largest when raw signatures feed a linear learner.

The authors also audit the discretization. In Design B, one Euler step per observation inflates the stationary variance of the faster states by up to 3.2%, a spread of about 3.0 percentage points across states, so Design B uses four substeps, which reduces the spread to about 1.0 point. Design A uses one step because all states share κ and the scale-only control holds exactly.

Better regime identification does not deliver earlier signals or portfolio gains

Two further results limit the practical reach. First, better identification of established regimes does not produce earlier regime-change signals. In the detection experiment, at a target of one false alarm per 1,000 observations, CUSUM detects 22.8% of changes in Design A, compared with 18.0% for the hidden Markov filter and 18.2% for Statistics plus log-signatures. In Design B, the best representation, Statistics plus raw signatures, detects 12.4% of changes, versus 12.2% for Statistics, 10.8% for the hidden Markov filter, and 8.8% for CUSUM. Across all designs, budgets and augmentations, only five of 32 detection contrasts are nominally resolved and none survives Holm correction. A time-since-change diagnostic shows no resolved gain during the first 25 observations after a switch, with the clearest gains at ages of 100 observations or more in Design A and 50 to 99 in Design B.

Second, no accuracy gain translates into a resolved Sharpe-ratio improvement in a volatility-managed portfolio. The forecasts are translated into a volatility-managed portfolio following Moreira and Muir, and the paper reports that the accuracy gains found here need not change portfolio outcomes. The authors therefore suggest judging any accuracy gain by the decision it is meant to inform, and evaluating the onset of regimes separately at explicit false-alarm budgets with multiplicity-aware inference.

Add signatures to a strong benchmark, not instead of one

The paper's practical recommendation follows from the comparison structure rather than from a generic preference for richer features. Because classifiers with or without signatures do not beat HAR, the useful move is to add signature coordinates to a strong realized-volatility benchmark rather than to replace that benchmark with a classifier. Because the gain appears with raw signatures and not with the conventional Statistics control, the choice of signature coordinate system should be made together with the model. Because the gain survives correction in the United States but not outside it, the result should be checked in the market sample at hand rather than assumed to transfer.

The authors also flag the conditions of their conclusions: the simulated parameter families, numerical discretization, window length, learners, forecast horizon and market samples all bound what the results support. The signature-augmented HAR comparison was designed after the benchmark results were known, and its economic evaluation was added after the forecasts had been scored. The most useful reading is therefore conditional: signatures are worth testing when the forecasting problem is about how volatility evolves, when the benchmark is a realized-volatility regression, and when the decision at stake is sensitive to regime dynamics rather than to regime level alone.

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