Continuous Hidden Markov Models for Equity Returns: Heavy-Tail Emission Families and Regime-Conditional Value-at-Risk

Abstract: Synthetic generators of daily equity returns let practitioners stress test, backtest, and design scenarios that a single realized market history cannot supply, but only if the generator reproduces the stylized facts of real returns: heavy tails, negligible linear autocorrelation, and slow decay of the absolute-return autocorrelation. Hidden Markov models with few Gaussian states were long thought unable to reproduce that slow decay, and the standard fix was to abandon them for more complex hidden semi-Markov models. We revisit this issue with a continuous hidden Markov model whose regime chain governs the autocorrelation while per-regime densities govern the marginal, separating the temporal and distributional sides of the original failure. A unified expectation-maximization framework fits Gaussian, Student-t, Laplace, and generalized-error emissions under shared forward-backward recursions and quantile-based initialization, and a spectral identity bounds the number of decay modes by the rank of the centred transition matrix. Across SPY walk-forward folds, a sector-balanced 30-ticker panel, a CRSP cross-decade transfer, and a six-asset basket, that bound was not binding once a few states were used: heavy-tailed marginals, not additional decay modes, closed most of the fit gap, recovering volatility clustering above the i.i.d. baseline and narrowing the kurtosis gap without a tuning hyperparameter. The original failure is therefore distributional, not temporal. On daily US equities, a simple, interpretable Markov model suffices, and unlike a bootstrap or semi-Markov fit that wins only on a single-window fit, the fitted model also yields a regime-conditional Value-at-Risk that passes a joint conditional-coverage test and a copula that reproduces cross-asset correlations: one interpretable generator serving both path simulation and downstream risk and portfolio tasks.

• The paper introduces a unified EM estimation framework for continuous HMMs that incorporates heavy-tailed emission families to better capture equity return features.
• It shows that spectral analysis of regime transitions reveals temporal constraints are non-binding beyond three states, shifting the focus to marginal flexibility.
• Empirical results indicate that the CHMM framework outperforms traditional GARCH and deep-learning models in both risk quantification and multi-asset simulation.

Continuous Hidden Markov Models for Equity Returns with Heavy-Tailed Emissions and Regime-Conditional VaR

Introduction and Motivation

This work focuses on the use of continuous Hidden Markov Models (CHMMs) for modeling daily equity returns, emphasizing the necessity to capture the "stylized facts" observed in real-world financial time series: heavy-tailed marginal distributions, volatility clustering (persistent autocorrelations of absolute returns), and minimal linear autocorrelation of returns. Traditional regime-switching models with Gaussian emissions and few latent states have been shown to inadequately reproduce the slow autocorrelation decay exhibited by absolute returns, a core empirical feature (2606.23492). This apparent limitation, often attributed to the geometric sojourn-time structure of finite-state Markov chains, was originally interpreted as a fundamental obstacle to CHMMs matching observed data, leading to the adoption of more complex alternatives such as semi-Markov models.

The authors revisit this hypothesis by leveraging CHMM frameworks with flexible emission distributions, particularly those admitting heavy tails (Student-$t$, Laplace, and Generalized Error Distributions), and by decoupling the marginal and temporal aspects of the model fit. They contribute a unified EM-based estimation approach that supports all emission families under a common forward-backward recursion, allowing rigorous empirical diagnostics as to whether temporal or marginal flexibility is the true binding constraint in regime-switching models of equity returns.

Model Specification and Estimation Framework

The central model is a $K$-state CHMM, where the latent Markov regime transitions control autocorrelation structure, and each regime emits observed growth rates according to an associated parametric probability density. The emission families considered are Gaussian (CHMM-N), Student-$t$ (CHMM-t), Laplace (CHMM-L), and Generalized Error Distribution (CHMM-GED). EM estimation leverages quantile-based initialization to mitigate state overlap at larger $K$, and simultaneous parameter updates are performed via either closed-form M-steps or block-coordinate conditional maximization (as for scale/shape parameters in Student-$t$ and GED cases).

A key technical tool is the spectral decomposition of the regime transition matrix, which bounds the number of distinct autocorrelation decay modes and frames the long-standing question of temporal persistence in terms of the numerical rank of the deflated transition matrix $\mathbf{T} - \mathbf{1}\bar{\pi}^\top$. This enables precise diagnostics regarding which aspect (temporal vs. distributional) limits the fit of CHMMs under varying regimes and emission families.

Stylized-Fact Reproduction and Spectral Analysis

Empirically, the CHMM framework accurately reproduces the three symmetric Cont stylized facts for daily US equity returns: pronounced non-Gaussianity with high excess kurtosis, negligible raw-return autocorrelation, and pronounced, slowly decaying autocorrelation in absolute returns. Investigation across different $K$ values revealed that beyond $K=3$, increasing the number of latent states did not improve the fit on the temporal (autocorrelation) axis; a dominant non-unit eigenmode of the transition matrix captured most of the absolute-return ACF for SPY (Figure 1).

Figure 1: Per-state degrees-of-freedom $\nu_k$ for the CHMM-t fit at $K = 18$ on SPY IS, showing the distribution of heavy-tail indices across regimes and the role of constraint bracketing.

This indicates that the temporal limitation ('too rapid ACF decay') attributed to geometric sojourns in low-$K$0 HMMs is not the bottleneck on modern daily equity data at moderate $K$1. Instead, the empirical limitation becomes the marginal flexibility of the emission mixture, especially for capturing the observed heavy tails. Simulation analysis confirmed that Student-$K$2 emissions (especially with per-state or learned global degrees of freedom) narrow the kurtosis gap left by Gaussian mixtures without introducing additional tuning parameters, as shown by the movement of regime-conditional kurtosis into conformity with observed excess kurtosis figures.

Figure 2: Marginal density of simulated and empirical daily returns, highlighting the improved heavy-tail fit under non-Gaussian emission families.

The spectral mechanism is thus primarily responsible for the persistent volatility clustering once a minimal regime count is reached, while heavy-tail emission families are critical for matching marginal distributional properties.

Regime-Conditional Value-at-Risk and Multi-Asset Extensions

The regime-conditional VaR framework leverages the CHMM's closed-form predictive distribution: the one-step-ahead forecast is a mixture over regimes, weighted by the forward-filtered latent-state probabilities, each contributing according to its emission density. This construction passes strict unconditional and conditional coverage tests (Kupiec and Christoffersen), with empirical breach rates near nominal and no significant evidence of violation in pooled-archive backtesting at $K$3. The regime-conditional approach yields improved coverage over static VaR models and is robust to the volatility regimes evidenced in out-of-sample testing.

Figure 3: IS VaR at $K$4, showing close envelope bracketing between observed extreme losses and simulated predictive quantiles.

For multi-asset joint simulation, the authors decouple dependence structure from single-asset margins using rank-based copula construction. Fitted CHMMs per asset are composed via a Student-$K$5 copula estimated on empirical Kendall's $K$6; joint simulation preserves marginal distributions exactly and reproduces observed cross-asset correlation structure within observed in-sample and out-of-sample error, with Student-$K$7 outperforming Gaussian and truncated vine alternatives in off-diagonal correlation MAE.

Figure 4: Profile log-likelihood of the Student-t copula for the multi-asset (six-asset SPY cross-section), showing the optimal degrees-of-freedom grid search.

Ablation Results and Benchmarks

Extensive comparative analysis against alternative generative models—empirical i.i.d.\ bootstrap, GARCH(1,1) variants, Markov-switching GARCH, deep-generative TimeGAN/QuantGAN—demonstrates the empirical advantages of the CHMM heavy-tail/mixture approach. Notably, while i.i.d.\ bootstraps can match marginal distributions, they lack temporal persistence and support for regime-conditional risk forecasting, and while GARCH-type models capture volatility clustering, they fail on marginal kurtosis or cannot provide regime semantics essential for conditional VaR calculation.

Figure 5: Histogram of fitted $K$8 parameters in the CHMM-GED, illustrating data-driven partitioning between Gaussian-bulk and Laplace-tail regime states.

Additionally, CHMMs offer transparent regime interpretability with relatively few parameters compared to deep-learning alternatives, enabling economic regime characterization and live monitoring.

Practical and Theoretical Implications

Contradicting the long-standing interpretation, the study empirically demonstrates that on contemporary moderate-$K$9 CHMMs, temporal autocorrelation constraints are not binding on daily US equity returns, and the limiting factor for fit is the emission family’s marginal flexibility, not the regime chain. Furthermore, the unified EM estimation framework allows readily extensible benchmarking across emission families and joint modeling scenarios, including portfolio-level tail-risk budgeting, multi-asset synthetic path generation, and scenario-based stress testing.

From a theoretical perspective, the findings underscore the need to reconsider regime count as a primary control for ACF matching and recommend emission-family selection and per-state heavy-tail modeling as key levers for fidelity to stylized facts. The analysis clarifies that the finite-mode mixture constraint intervenes predominantly at very low $t$0, with heavy-tail misspecification persisting across all $t$1 when emission flexibility is lacking.

Failure Modes and Future Directions

The study identifies genuine scope limitations: static-fit CHMMs fail under regime introductions not present in-sample, such as new volatility states, cross-asset structural breaks, or asset-class-specific shocks (e.g., non-equity GLD). While periodic refit can partially mitigate non-stationarity in evolving equity panels, it does not resolve fundamental distributional shifts outside the observed regime library. Importantly, regime-conditional VaR and multi-asset composition can inherit these failures if the per-asset marginals become miscalibrated.

For further extension, the authors propose exploring:

• Skew/asymmetric emission families for leverage-effect modeling.
• State-dependent or time-varying transition matrices to more effectively track regime shifts (potentially via online EM or particle filtering).
• Richer multi-asset dependence structures (high-dimensional copulas or vine models) to scale joint path simulation beyond the single-index cluster regime.

Conclusion

Continuous HMMs with flexible heavy-tailed emissions, jointly estimated via a unified EM framework and initialized using quantile-based partitioning, are sufficient to reproduce the three symmetric Cont stylized facts for daily US-equity returns when moderate state counts are used. Temporal autocorrelation constraints, interpreted via the spectral rank of the regime transition matrix, become non-binding beyond $t$2, and emission-family flexibility becomes the dominant factor limiting empirical fit. The approach enables regime-conditional risk quantification, supports multi-asset synthetic data generation via copula composition, and outperforms conventional GARCH and deep-learning time-series models in both interpretability and targeted statistical fidelity, provided the observed data remain within the regime scope of the trained model.

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Figure 6: Observed cross-asset correlation matrix, serving as the ground truth reference for multi-asset synthetic generation benchmarks.

In sum, CHMMs with heavy-tailed emissions and quantile-based initialization, evaluated via comprehensive empirical and spectral diagnostics, are an effective and extensible framework for time-series synthesis and risk management in financial applications. Periodic refit is essential for maintaining fidelity under non-stationary conditions, and future development should focus on integrating online adaptation and asymmetric risk modeling.

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Reference: "Continuous Hidden Markov Models for Equity Returns: Heavy-Tail Emission Families and Regime-Conditional Value-at-Risk" (2606.23492)