Chib-Nardari-Shephard (2002) MCMC Methods for Stochastic Volatility Models

Summary

Chib, Nardari, and Shephard (2002) develop efficient Markov chain Monte Carlo (MCMC) algorithms for two stochastic volatility (SV) extensions: SVt (Student-t observation errors via scale mixing, level effect, covariates) and SVJt (SVt plus a Bernoulli jump component). The key technical device is the seven-component Gaussian mixture of Kim-Shephard-Chib (1998) that approximates the $\log\chi^2(1)$ distribution, enabling the de Jong-Shephard simulation smoother to draw the entire latent log-variance path in one block. Model comparison via the Chib (1995)/Chib-Jeliazkov (2001) marginal likelihood identity is validated against an auxiliary particle filter. Applied to S&P 500 returns (1962–1997), SVt decisively outperforms both the basic Gaussian model and the Gaussian-plus-jumps model.

Key Claims

• SVt beats SVJ (SV with jumps): The excess kurtosis in S&P 500 daily returns is better captured by fat-tailed innovations (Student-t, $\nu \approx 12.5$) than by Gaussian errors plus a Bernoulli jump component. Bayes factor ($\log_{10}$): SVt vs. SV0 (basic Gaussian SV) = 10.75; SVt vs. SVJ = +5.57.
• SVJt ≈ SVt: Adding jumps to the fat-tailed model (SVJt) does not decisively improve fit (Bayes factor SVJt vs. SVt = −0.99). Jump intensity in SVJt falls to ≈0.00195/day (1 per 500 days) because SVt absorbs most tail realizations.
• SV0 inflates volatility: The Gaussian model attributes large returns entirely to volatility spikes; SVt/SVJ/SVJt smooth these into tails or jumps, systematically lowering volatility level $\mu$ and vol-of-vol $\sigma$.
• Scale-mixture + 7-component Gaussian mixture: This combination reduces MCMC inefficiency factors to 1–5, making 5,000 Gibbs sweeps sufficient (confirmed by 50,000-sweep robustness check).
• Prior robustness: Model rankings are invariant to seven alternative prior specifications, including priors that point decisively toward Gaussian or toward very fat-tailed distributions.

Concepts Introduced or Extended

• Stochastic Volatility — SVt and SVJt extensions; 7-component mixture enabling simulation smoother; auxiliary particle filter; marginal likelihood via Chib identity

Entities Mentioned

• Siddhartha Chib
• Federico Nardari
• Neil Shephard
• Eric Jacquier
• Nicholas Polson

Quotes

"The numerical optimization used to find the location and spread of the proposal density appears to be crucial in this context, especially for the μ and ρ parameters. Extensive experimentation shows that, without such careful implementation of the M-H algorithm, the inefficiency factors are much higher."

"Big movements in returns in the basic Gaussian model are attributed almost exclusively to sharp changes in the volatility level whereas they are filtered differently, either as jumps (SVJ) or as tail realizations (SVt), or both (SVJt), by other models."

My Take

The paper's main contribution is methodological: showing that the scale-mixture representation + 7-component Gaussian mixture + simulation smoother pipeline is fast enough for practical use with low inefficiency factors. The empirical finding — fat tails beat Gaussian+jumps — is robust and has important implications for option pricing and risk management. The result that SVt and SVJt are empirically indistinguishable suggests that, for S&P 500 data of this length, the jump component adds little beyond what heavy-tailed errors already provide.