Hybrid Hidden Markov Model for Modeling Equity Excess Growth Rate Dynamics: A Discrete-State Approach with Jump-Diffusion

Generating synthetic financial time series that preserve statistical properties of real market data is essential for stress testing, risk model validation, and scenario design. Existing approaches, from parametric models to deep generative networks, struggle to simultaneously reproduce heavy-tailed distributions, negligible linear autocorrelation, and persistent volatility clustering. We propose a hybrid hidden Markov framework that discretizes continuous excess growth rates into Laplace quantile-defined market states and augments regime switching with a Poisson-driven jump-duration mechanism to enforce realistic tail-state dwell times. Parameters are estimated by direct transition counting, bypassing the Baum-Welch EM algorithm. Synthetic data quality is evaluated using Kolmogorov-Smirnov and Anderson-Darling pass rates for distributional fidelity, and ACF mean absolute error for temporal structure. Applied to ten years of SPY data across 1,000 simulated paths, the framework achieves KS and AD pass rates exceeding 97% and 91% in-sample and 94% out-of-sample (calendar year 2025), partially reproducing the ARCH effect that standard regime-switching models miss. No single model dominates all quality dimensions: GARCH(1,1) reproduces volatility clustering more accurately but fails distributional tests (5.5% KS pass rate), while the standard HMM without jumps achieves higher distributional fidelity but cannot generate persistent high-volatility regimes. The proposed framework offers the best joint quality profile across distributional, temporal, and tail-coverage metrics. A Single-Index Model extension propagates the SPY factor path to a 424-asset universe, enabling scalable correlated synthetic path generation while preserving cross-sectional correlation structure.