🤖 AI Summary
Traditional Gaussian hidden Markov models struggle to simultaneously capture the heavy-tailed nature of stock returns, their weak linear autocorrelation, and the slow decay of absolute return autocorrelations. This work proposes a continuous hidden Markov model that decouples temporal dynamics from marginal distribution modeling: temporal dependence is governed by a state transition chain, while each state employs heavy-tailed emission distributions—such as the Student-t or generalized error distribution—to accurately represent marginal characteristics. A unified EM framework is developed for parameter estimation. Theoretical analysis reveals that the inadequacy of conventional models stems from restrictive distributional assumptions rather than temporal structure, and that volatility clustering and high kurtosis can be reproduced with only a few latent states. Empirical results demonstrate that the proposed model significantly reduces fitting gaps across multiple U.S. equity datasets, generates paths that pass joint conditional coverage tests, and faithfully captures cross-asset dependencies, thereby supporting robust risk measurement and portfolio optimization.
📝 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.