Jacquier-Polson-Rossi (2004) Bayesian Analysis of Stochastic Volatility Models with Fat-Tails and Correlated Errors

Summary

JPR (2004) extends the original Jacquier-Polson-Rossi (1994) Bayesian Markov chain Monte Carlo (MCMC) framework for discrete-time stochastic volatility (SV) models to accommodate (i) fat-tailed return distributions via a Student-t innovation and (ii) a leverage effect via same-period correlation $\rho$ between return and volatility shocks (the ASV2 specification). Bayesian model comparison via Bayes Factors strongly favors both extensions for U.S. equity data, with correlated errors providing more decisive evidence than fat tails alone; foreign-exchange (FX) series (CAD/USD) show no strong preference for either extension.

Key Claims

• Four nested models: Basic SVOL (Gaussian, independent); FC (fat-tailed: $u_t \sim t(\nu)$); ASV2 (correlated: $\text{corr}(u_t, v_t) = \rho$, same period); Full (FC + ASV2).
• Scale-mixing augmentation for fat tails: $t(\nu)$ innovations represented as $u_t = w_t^{-1/2} z_t$ with $w_t \sim \chi^2(\nu)/\nu$ and $z_t \sim \mathcal{N}(0,1)$; augmenting with latent weights $\{w_t\}$ makes the full conditional for each $h_t$ log-concave, enabling adaptive rejection sampling (Wild-Gilks 1993) rather than the inverse-gamma Metropolis-Hastings (M-H) step of JPR (1994).
• Leverage (ASV2): rewrite $\log h_t = \alpha + \delta \log h_{t-1} + \sigma_v(\rho u_t + \sqrt{1-\rho^2}\, v_t^*)$; $\rho$ drawn via a linearization-based Gaussian M-H proposal with high acceptance rates.
• Bayes Factors via importance sampling: BF strongly favors fat tails and correlated errors for S&P 500 and Center for Research in Security Prices (CRSP) equity; correlated model wins more decisively than fat tails; FX (CAD/USD): $\mathrm{BF} \approx 1$ — no evidence for either extension.
• Equity estimates: $\rho \approx -0.43$ to $-0.48$ (strong leverage); $\nu \approx 25$–$30$ for daily data; stronger fat-tail evidence at weekly frequency (smaller $\nu$).
• Fat-tailed model smooths volatility: extreme returns absorbed into heavy tails rather than volatility spikes → smoother $h_t^{1/2}$ path than basic SVOL.
• Option pricing: fat-tailed SV reduces option pricing root mean squared error (RMSE) by ${\sim}15\%$ vs. basic SVOL; leverage creates skew (put implied volatility, IV > call IV).
• Simulation performance: low bias and RMSE in all models; fat-tailed model correctly identifies near-normality when data are Gaussian ($\nu$ posterior $\to \infty$). MCMC chains for $\rho$ have autocorrelation function $\mathrm{ACF} \approx 0.9$ at lag 1 — recommend $25{,}000+$ draws.

Concepts Introduced or Extended

• Stochastic Volatility — fat-tailed and correlated SV extensions; Bayes Factors for model selection
• Stylized Facts of Asset Returns — leverage effect documented for equity, absent for FX

Entities Mentioned

• Eric Jacquier
• Nicholas Polson
• Peter E. Rossi

Quotes

"…the model with fat tails substantially smooths out peaks in the volatility path, attributing extreme returns to the fat tails rather than the volatility process."

"The Bayes Factor strongly favors the correlated model over the basic SVOL for all equity series, with less decisive evidence for FX data."

My Take

JPR (2004) cleanly extends the JPR (1994) framework and fills a practical gap: the basic SV model forces volatility spikes at outlier observations, which fat-tailed innovations resolve by competing with the volatility mechanism. The leverage finding ($\rho < 0$ for equity, $\rho \approx 0$ for FX) is consistent with Black (1976) and aligns with Brandt-Kang (2004) and Yu (2005). The main limitation is the single-move MCMC algorithm — known to mix slowly for high-persistence series — and the paper does not compare against multi-move alternatives such as Kim-Shephard-Chib (1998). Bayes Factor computation via importance sampling is also potentially sensitive to proposal quality.