Regime-Switching Stochastic Volatility

RSSV models contain two unobservable components: a continuous stochastic volatility process and a discrete Markov chain governing the regime. The volatility follows a mean-reverting diffusion within each regime, while the regime variable determines the long-run mean, volatility of volatility, and correlation parameters. This structure captures both gradual volatility fluctuations and sudden structural breaks. For example, a two-regime model might feature a low-volatility state with fast mean reversion and a crisis state with high baseline variance and strong negative leverage effects.

The correlation between asset returns and volatility innovations—the leverage effect—is allowed to differ across regimes. In normal markets, the correlation may be moderately negative, reflecting standard equity dynamics. During financial crises, this correlation often becomes strongly negative, as large price drops coincide with massive volatility spikes. RSSV models parameterize this asymmetry explicitly, enabling more accurate option pricing and tail risk assessment during market dislocations.

Estimating RSSV models requires Sequential Monte Carlo methods because the joint posterior of the continuous volatility and discrete regime is analytically intractable. Particle filters approximate the filtering distribution using weighted samples:

• Bootstrap filter: Propagates particles through the state equations and reweights by the observation density
• Auxiliary particle filter: Resamples before propagation to improve efficiency when volatility is highly persistent
• Rao-Blackwellized particle filter: Marginalizes out the continuous volatility analytically, reducing variance in regime probability estimates

Full Bayesian inference for RSSV models employs Markov Chain Monte Carlo (MCMC) with data augmentation. The algorithm iterates between:

• Sampling the regime sequence using forward-filtering backward-sampling
• Drawing volatility paths via blocked Gibbs sampling or Metropolis-Hastings steps
• Updating regime-switching parameters and transition probabilities from conjugate priors This approach provides full posterior distributions for all parameters, including the transition probability matrix and regime-specific volatility dynamics, rather than point estimates.

RSSV models produce regime-conditional risk-neutral measures for derivative pricing. The market price of volatility risk can switch with the regime, reflecting investors' changing risk appetites. Option prices are computed as probability-weighted averages across regimes:

• European options: Solved via Fourier transform methods or characteristic functions
• Path-dependent options: Require Monte Carlo simulation with regime-switching volatility paths
• Implied volatility surfaces: Naturally generate volatility smiles and skews that shift shape across different market regimes

RSSV models provide density forecasts that incorporate regime uncertainty. The predictive distribution of future volatility is a mixture of regime-conditional distributions, weighted by the predicted regime probabilities derived from the transition matrix. This captures the possibility of sudden volatility regime shifts that single-regime models miss. Empirical studies show RSSV models significantly outperform GARCH and standard SV models for Value-at-Risk and Expected Shortfall calculations during periods of market turbulence, such as the 2008 financial crisis or the 2020 COVID-19 crash.

The foundational statistical framework where the system's true state—the market regime—is unobservable but inferred from a sequence of observable data like returns. In a Regime-Switching Stochastic Volatility model, the HMM governs the latent regime process, dictating which set of volatility parameters is active at any given time. The model assumes the Markov property: the probability of transitioning to the next regime depends only on the current regime, not the full history.

The workhorse iterative optimization method for calibrating models with latent variables. It alternates between two steps:

• E-Step (Expectation): Estimate the probability of being in each regime at each time point given current parameter estimates.
• M-Step (Maximization): Update model parameters (volatility levels, transition probabilities) to maximize the expected log-likelihood. This algorithm is essential because the true regime sequence is never directly observed, making standard maximum likelihood intractable.

A stochastic matrix that quantifies the persistence and switching dynamics between regimes. Each entry P(i,j) represents the probability of moving from regime i to regime j in the next time step. Key properties:

• Diagonal dominance: High diagonal values (e.g., 0.98) indicate regime persistence—markets tend to stay in their current state.
• Ergodic probability: The long-run unconditional probability of each regime, derived from the matrix, tells you what fraction of time the market spends in high versus low volatility states.

A Sequential Monte Carlo method for real-time state inference in nonlinear, non-Gaussian models. Unlike the Kalman filter, particle filters can handle the complex dynamics of stochastic volatility. The algorithm:

• Maintains a cloud of weighted 'particles' representing possible hidden states.
• Propagates particles forward using the state transition model.
• Re-weights them based on how well they explain new observations. This enables online regime detection without waiting for batch processing, critical for live trading systems.

The empirical stylized fact that large price changes cluster together and small changes cluster together, creating persistent periods of high and low variance. This phenomenon directly motivates regime-switching volatility models: rather than treating volatility spikes as isolated outliers, the model captures the autocorrelated nature of risk. Mandelbrot first observed this in 1963, noting that 'large changes tend to be followed by large changes—of either sign—and small changes tend to be followed by small changes.'

A tail-risk measure that calculates expected loss conditional on exceeding the VaR threshold, with the loss distribution specifically modeled for the current inferred regime. Unlike static risk measures that average across all market conditions, Regime-CVaR:

• Applies a high-volatility loss distribution during turbulent regimes.
• Applies a normal-volatility distribution during calm periods. This prevents underestimation of risk during crises and overestimation during stable markets, providing more accurate capital allocation signals.