Stochastic Volatility

Definition

A stochastic volatility (SV) model specifies the log-variance of a return series as a latent autoregressive process, rather than as a deterministic function of past observations. Unlike generalized autoregressive conditional heteroskedasticity (GARCH), the conditional variance is not observable given past returns — it is a hidden state that must be inferred alongside the model parameters.

Key Ideas

GARCH variance $h_t$ is determined exactly by past returns: $h_t = f(\varepsilon_{t-1}, \varepsilon_{t-2}, \ldots)$ . SV log-variance $\ln h_t$ is instead driven by an independent noise process $\eta_t$ , making $h_t$ genuinely latent.
The latent structure introduces a second source of randomness, which gives SV models more flexibility in fitting the tails and autocorrelation structure of squared returns.
Estimation is more demanding: because the likelihood involves an integral over unobserved volatility paths, standard maximum likelihood (ML) is unavailable in closed form. Markov chain Monte Carlo (MCMC) (Kim, Shephard, and Chib 1998) and particle filters are the main approaches.
SV models are widely used in options pricing (Heston 1993 continuous-time SV) and in Bayesian macroeconometrics (Cogley-Sargent time-varying parameter VAR (TVP-VAR) with SV).

How It Works

Taylor (1982) Discrete-Time SV Model

The first-order autoregressive SV model (Taylor 1982) specifies:

$r_t = \sigma_t z_t, \qquad z_t \overset{iid}{\sim} (0,1) \tag{Obs}$

$\ln \sigma_t^2 = \alpha + \beta \ln \sigma_{t-1}^2 + \eta_t, \qquad \eta_t \overset{iid}{\sim} \mathcal{N}(0, \sigma_\eta^2) \tag{State}$

with $\eta_t \perp z_s$ for all $t, s$ . Parameters: $\alpha$ (unconditional log-variance level), $\beta \in (-1,1)$ (persistence), $\sigma_\eta^2$ (volatility of volatility).

Unconditional properties. The log-variance is Gaussian autoregressive (AR)(1) with stationary distribution $\ln \sigma_t^2 \sim \mathcal{N}(\alpha/(1-\beta),\; \sigma_\eta^2/(1-\beta^2))$ when $|\beta|<1$ .

Comparison with GARCH

Feature	GARCH(1,1)	SV (AR(1))
$h_t$ observable?	Yes (deterministic given past $\varepsilon$ )	No (latent state)
Likelihood	Closed form	Requires integration over $h_{1:T}$
Estimation	Numerical ML (fast)	MCMC or particle filter (slow)
Flexibility	Limited tail fit	More flexible via $\sigma_\eta$
Asymmetry	Via Exponential GARCH (EGARCH)/Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) add-ons	Via leverage effect: $\mathrm{Cor}(z_t, \eta_{t+1})<0$

Stein-Stein (1991) Continuous-Time OU Model

The continuous-time counterpart specifies:

$dP = \mu P\, dt + \sigma P\, dz_1 \tag{1}$ $d\sigma = -\delta(\sigma - \bar\sigma)\, dt + k\, dz_2 \tag{2}$

where $dz_1 \perp dz_2$ . Volatility $\sigma$ (not $\sigma^2$ ) follows an arithmetic Ornstein-Uhlenbeck (AR1) process with mean-reversion speed $\delta$ , long-run mean $\bar\sigma$ , and vol-of-vol $k$ .

Closed-form distribution. The stock price distribution $S(P,t)$ has an exact closed-form (Stein-Stein eq. 9–10) involving sinh/cosh terms and a single numerical integration. Key result from Appendix A–B: $S(P,t) = \int L(\sigma) m_t(\sigma)\, d\sigma$ , i.e., a mixture of lognormals, where $L(\sigma)$ is lognormal with RMS volatility $\sigma = (t^{-1}\int_0^t \sigma^2(s)\,ds)^{1/2}$ and $m_t(\sigma)$ is the mixing distribution over realized RMS vol. This representation holds for any diffusion SV model, not just OU.

Volatility smile. Stochastic volatility raises all option prices above Black-Scholes. Implied volatility is U-shaped in strike: lowest at-the-money (where Black-Scholes is near-linear in $\sigma$ , so only the mean of $m_t$ matters) and rising away-from-the-money (where convexity of Black-Scholes in $\sigma$ adds a dispersion-of-mixing-distribution premium).

Fat tails. $S(P,t) \sim P^{-\gamma}$ as $P \to \infty$ (power law, fatter than lognormal):

$\delta = 0$ : $\gamma \to 2$ as $t$ or $k \to \infty$ — variance can disappear at long horizons.
$\delta > 0$ : $\lim_{t\to\infty} \gamma > 2$ — variance always exists; stronger mean reversion → thinner tails.
$t, k \to 0$ : $\gamma \to \infty$ — approaches lognormal.

Matches Bookstaber-McDonald (1987) empirical finding: fat tails at 1–5 day returns, near-lognormal at 250 days — consistent with stationary (mean-reverting) volatility.

Limitation. $dz_1 \perp dz_2$ : no leverage effect. Correlated Wiener processes were analytically intractable; Heston (1993) solved the correlated case via characteristic functions.

Ball-Roma (1994): MGF of Average Variance Framework

Ball and Roma unify the Hull-White (H-W) and Stein-Stein (S-S) approaches through the moment-generating function (MGF) of average variance $K(\lambda) = E[\exp(-\lambda \int_0^T \sigma^2(s)\,ds)]$ :

H-W power series: differentiate $K$ at $\lambda=0$ to obtain moments $E[AV]$ , $\mathrm{Var}(AV)$ , $\mathrm{Skew}(AV)$ ; then $OP \approx BS(E[AV]) + \tfrac{1}{2}\frac{\partial^2 BS}{\partial AV^2}\mathrm{Var}(AV) + \cdots$ . Second-order truncation accurate to the penny.
S-S Fourier inversion: invert $K$ as a Fourier transform to get the exact price density.

Cox-Ingersoll-Ross (CIR)/Heston square-root model. For $dv = a(m-v)dt + \zeta\sqrt{v}\,dz$ , the MGF is available directly from CIR (1985) bond pricing: $I^*(\lambda) = \exp(N^*(T) + M^*(T)v_0)$ , where $\gamma = \sqrt{a^2 + 2\lambda\zeta^2/T}$ , $g(T) = 2\gamma + (a-\gamma)(1-e^{-\gamma T})$ , $N^*(T) = \frac{am}{\zeta^2}\ln\!\frac{2\gamma}{g(T)}$ , $M^*(T) = \frac{-2(1-e^{-\gamma T})}{g(T)}$ . Simpler than S-S's sinh/cosh formula; square-root process is always non-negative with gamma limiting distribution.

Volatility smile derivation. Implied variance under the (incorrect) constant-volatility assumption is $E[AV] + \varepsilon$ , where: $\varepsilon \approx \tfrac{1}{2}\frac{\mathrm{Var}(AV)}{E[AV]}\!\left(-\!\left(\tfrac{c_2}{2}\right)^2 + \left(\tfrac{c_1}{c_2}\right)^2 - 1\right),\quad c_1 = \ln(P/Ke^{-rT}),\; c_2 = \sqrt{E[AV]\cdot T}$ $\varepsilon$ is quadratic in log-moneyness $c_1$ , minimized at at-the-money (ATM) ( $c_1=0$ ) and rising for out-of-the-money (OTM)/in-the-money (ITM) — formal derivation of the U-shaped smile.

S-S error corrected. S-S compared their SV price to BS at variance $\bar\sigma^2$ , but the correct benchmark is $E[AV] = \bar\sigma^2 + k^2/2\delta + \text{transient} > \bar\sigma^2$ . The excess arises because S-S model $|\sigma(t)|$ (absolute value of OU) rather than a reflected OU process: $|\sigma|$ has long-run mean above $\bar\sigma$ by Jensen's inequality. With the correct benchmark, SV exerts a downward bias ATM (BS is concave in variance there) and upward bias away-from-the-money.

Zero-correlation constraint. The average-variance framework — both power series and Fourier — requires $dz_1 \perp dz_2$ . Nonzero correlation breaks the lognormality of $P$ conditional on $AV$ .

Amin-Ng (1993): Equilibrium Option Pricing with Systematic Stochastic Volatility

Amin and Ng (1993) provide an equilibrium foundation for the expected-Black-Scholes formula, working in a pure discrete-time constant proportional risk aversion (CPRA) framework without continuous-time mathematics.

Systematic variance decomposition (eq. 3): The asset's conditional variance decomposes as $h_{s,t} = \beta^2 h_{c,t} + h_{d,t}$ , where $h_{c,t}$ is the conditional variance of log consumption growth (systematic component) and $h_{d,t}$ is idiosyncratic; $\beta$ measures the stock's sensitivity to aggregate consumption shocks. This structure directly anticipates factor GARCH models.

Proposition 1 (Preference-dependent): Under CPRA utility $E_0[\sum \rho^t C_t^{1-b}/(1-b)]$ and bivariate conditional lognormality of stock-consumption returns, the call price is a preference-weighted expectation of payoffs. The weights depend on the covariance of the stochastic discount factor $\rho^T C_T^{-b}/E[\rho^T C_T^{-b}]$ with the cumulative asset return.

Proposition 2 (Preference-free): When the variance process $\{h_{s,t}\}$ is predictable (measurable in the investor's time-0 information set), the call price collapses to: $\pi(\text{Call}) = E_0\!\left[C_\text{BS}(\sigma^2_{0T})\right], \qquad \sigma^2_{0T} = \sum_{t=1}^T h_{s,t}$ No risk-aversion parameter $b$ appears — this is the same expected-Black-Scholes formula derived by Hull-White (1987) and Ball-Roma (1994), now grounded in discrete-time general equilibrium.

Endogenous riskless rate (eq. 11): $r_t = -\ln\rho + b\mu_{c,t+1} - \tfrac{1}{2}b(1+b)h_{c,t+1}$ . The interest rate declines with consumption variance — a flight-to-quality channel — distinguishing Amin-Ng from no-arbitrage SV models that treat the interest rate as exogenous.

Systematic jump formula (eq. 27): When consumption growth contains a compound Poisson jump component, the call price becomes a Merton (1976)-type series $\sum_{n=0}^\infty [e^{-\lambda T}(\lambda T)^n / n!] \cdot C_\text{BS}(\sigma^2_{0T} + n\sigma_J^2)$ , with jump risk priced through the CPRA kernel rather than under risk neutrality.

Bakshi-Cao-Chen (1997): Empirical Horse Race via Nested SVSI-J Model

Bakshi, Cao, and Chen (1997) embed continuous-time SV within a fully general nested model — SVSI-J — and evaluate four special cases on 38,749 S&P 500 call options (June 1988–May 1991) along three dimensions: internal consistency, out-of-sample pricing, and hedging.

General dynamics. The SVSI-J model has three state variables: stock price $S$ , instantaneous variance $V$ , and short rate $R$ :

$\frac{dS}{S} = [R(t) - \lambda\mu_J]\,dt + \sqrt{V(t)}\,d\omega_S(t) + J(t)\,dq(t)$ $dV(t) = [\theta_v - \kappa_v V(t)]\,dt + \sigma_v\sqrt{V(t)}\,d\omega_v(t), \quad \mathrm{Cov}[d\omega_S,d\omega_v]=\rho\,dt$ $dR(t) = [\theta_R - \kappa_R R(t)]\,dt + \sigma_R\sqrt{R(t)}\,d\omega_R(t)$

Jumps arrive as a Poisson process with intensity λ; log jump sizes are $\mathcal{N}(\ln[1+\mu_J]-\frac{1}{2}\sigma_J^2, \sigma_J^2)$ . Setting $\lambda=0$ gives SVSI; $\sigma_R=0$ gives SVJ; both gives SV; all three zero gives BS.

Pricing via characteristic function inversion. The European call price takes the Black-Scholes-Merton (BSM) form $C = S\Pi_1 - K\cdot B(t,\tau)\cdot\Pi_2$ , where $\Pi_1$ and $\Pi_2$ are recovered by inverting the characteristic function of $\ln S(T)$ . The characteristic function factors into closed-form exponential-affine contributions from $V$ , $R$ , and the jump process.

Implied parameter estimation. On each trading day, structural parameters are estimated by minimizing the sum of squared pricing errors across all available options. This daily cross-section sum of squared errors (SSE) minimization yields a time series of implied $(\kappa_v, \theta_v, \sigma_v, \rho, \lambda, \mu_J, \sigma_J)$ .

Three-yardstick results (Table III, All Options):

Model	$\kappa_v$	$\sigma_v$	$\rho$	$\lambda$	SSE
BS	—	—	—	—	69.60
SV	1.15	0.39	$-0.64$	—	10.63
SVSI	0.98	0.42	$-0.76$	—	10.68
SVJ	2.03	0.38	$-0.57$	0.59/yr	6.46

Internal consistency: All models are significantly misspecified. Implied $\rho \approx -0.64$ (SV) is 2–3 $\times$ larger than time-series ML estimate ( $\approx -0.28$ ); implied $\sigma_v \approx 0.39$ is $\approx 4\times$ its ML counterpart. Among SV-class models, SVJ is least misspecified.

Out-of-sample pricing: SVJ $\succ$ SVSI $\succ$ SV $\succ$ BS. SV reduces pricing errors 25–70% over BS; jumps add most value for short-term OTM options.

Hedging (single-instrument): SV $\succ$ SVJ $\succ$ SVSI $\succ$ BS. With $\lambda \approx 0.59$ /year (one jump per 1.7 years), jumps are essentially never realized at 1–5 day hedging intervals — so jump risk cannot be exploited in the delta-hedge.

BSDV control (delta-plus-vega-neutral BS): A BS hedge augmented with a vega-neutral option position (BSDV) nearly closes the hedging gap for non-ITM calls, confirming that most of the SV models' hedging advantage is mechanical (using a second option instrument) rather than structural.

See Jump-Diffusion Model for the full BCC nested model structure and empirical details.

Jacquier-Polson-Rossi (1994): Single-Move Independence Metropolis

The foundational MCMC approach (JPR 1994) uses data augmentation: augment $\theta$ with the full latent variance path $h = \{h_t\}_{t=1}^T$ and sample each $h_t$ one at a time from its full conditional, then draw parameters jointly.

Full conditional for $h_t$ : $p(h_t \mid h_{-t}, \theta, y_t) \propto \underbrace{h_t^{-1/2} \exp\!\left(-\tfrac{y_t^2}{2h_t}\right)}_{\text{return likelihood}} \cdot \underbrace{h_t^{-1} \exp\!\left(-\tfrac{(\ln h_t - \mu_t)^2}{2\sigma_t^2}\right)}_{\text{state (lognormal)}}$

where $\mu_t$ and $\sigma_t^2$ are the mean and variance of $\ln h_t$ conditional on its AR(1) neighbors. The return piece is inverse-gamma shaped; the state piece is lognormal. An IG blanketing density is formed by moment-matching the lognormal to an IG, and an independence Metropolis step accepts/rejects draws from this IG proposal. Acceptance rates 70–80%.

Parameter block: Given $\{h_t\}$ , the log-linear state equation is a standard Gaussian AR(1), so $(\alpha, \beta, \sigma_v^2)$ are drawn jointly from a Normal-inverse-gamma (IG) conjugate posterior.

Simulation results: Bayesian root mean squared error (RMSE) is 3.7–4.1 $\times$ smaller than method of moments (MM) and comparable or better than quasi-maximum likelihood (QML). MM suffers corner solutions (>50% of samples for high-persistence, low vol-of-vol regimes). Bayesian smoother $E[h_t \mid y]$ beats the approximate Kalman smoother even when given the true parameters (relative RMSE 1.11 median).

Precursor (Geweke 1993). The latent-variance data augmentation device in JPR (1994) has a static predecessor in Geweke (1993): augmenting a linear regression with independent and identically distributed (i.i.d.) Student-t errors with latent scale weights $\omega_i$ (where $\nu/\omega_i \sim \chi^2(\nu)$ ) renders all full conditionals conjugate and enables a direct Gibbs sampler. JPR's dynamic $h_t$ path is the time-series counterpart of Geweke's static $\omega_i$ — the device that makes Bayesian SV inference tractable originates in Geweke's cross-sectional regression context.

Kim-Shephard-Chib (1998): Multi-Move Sampler

KSC transform the observation equation by log-squaring:

$y_t^* = \ln r_t^2 = h_t + \xi_t, \qquad \xi_t = \ln \varepsilon_t^2 \sim \log\chi^2(1)$

The $\log\chi^2(1)$ noise has mean $-1.2704$ , variance $\pi^2/2 \approx 4.93$ , and severe left skewness — it is not Gaussian, so a Kalman filter applied directly would be misspecified (QML). KSC approximate it by a fixed mixture of $K=7$ Gaussians:

$\xi_t \approx \sum_{i=1}^7 p_i\, \mathcal{N}(m_i,\; v_i^2)$

with weights $\{p_i\}$ , means $\{m_i - 1.2704\}$ (centred), and variances $\{v_i^2\}$ chosen to match the first four moments plus the mode of $\log\chi^2(1)$ (Table 4 of KSC).

MIX Gibbs sampler. The three-block cycle is:

Indicator draw. For each $t$ , draw the discrete indicator $s_t \in \{1,\ldots,7\}$ from its multinomial full conditional: $\Pr(s_t = i \mid y_t^*, h_t) \propto p_i\, \phi(y_t^* - h_t;\; m_i,\; v_i^2)$
Path draw. Conditional on $\{s_t\}$ and parameters, the system is a linear Gaussian state space: $y_t^* - m_{s_t} \mid h_t, s_t \sim \mathcal{N}(h_t,\; v_{s_t}^2), \qquad h_{t+1} = \mu + \phi(h_t - \mu) + \sigma_\eta \eta_t$ Apply the Carter-Kohn (1994) forward filter / backward simulation smoother to draw the entire path $\{h_t\}_{t=1}^T$ jointly in $O(T)$ time.
Parameter draw. Given $\{h_t\}$ , the state equation is a Gaussian AR(1). Draw $(\mu, \phi)$ jointly Normal from the conjugate regression of $(h_{t+1} - h_t)$ on $(h_t - \bar{h})$ , and $\sigma_\eta^2$ from the conjugate Inverse-Gamma.

Importance reweighting. The MIX sampler targets a pseudo-posterior (under the mixture approximation). Exact inference is recovered by importance reweighting with: $w(h_{1:T}) \propto \prod_{t=1}^T \frac{f_{\log\chi^2}(y_t^* - h_t)}{\hat{f}_\text{mix}(y_t^* - h_t)}$ The effective sample size ratio exceeds 0.99 in simulations — the 7-component approximation is highly accurate.

Efficiency comparison. The inefficiency factor INEF $= 1 + 2\sum_{k\geq 1}\hat\rho_k$ measures sampling inefficiency relative to i.i.d. draws. Across the three samplers (SM: single-move; MIX: mixture multi-move; INT: integration particle filter):

Sampler	INEF ( $\phi$ )	INEF ( $\sigma_\eta$ )	Cost per draw
SM (single-move)	20–200	20–150	Low
MIX (mixture)	1–3	1–3	Low
INT (particle filter)	1–2	1–2	High

MIX achieves near-i.i.d. efficiency at the same computational cost as SM.

Bayes factors (Chib 1995 identity). Evaluated via $\log p(y) = \log p(y|\theta^*) + \log p(\theta^*) - \log p(\theta^*|y)$ at the posterior mode $\theta^*$ , with the log-likelihood computed by the Kalman filter under the mixture approximation. For daily S&P 500 (1981–1987): $\log_{10} BF(\text{SV vs. GARCH}) \approx +50$ ; SV also decisively beats $t$ -GARCH.

Estimation Methods and the Mixture Approximation (Mahieu-Schotman 1998)

Mahieu and Schotman (1998) provide the most systematic comparison of SV estimation techniques in an exchange-rate application. The log-squared-return transformation $y_t = \ln s_t^2 = h_t + \ln \varepsilon_t^2$ yields a linear state-space model, but the measurement error $\xi_t = \ln \varepsilon_t^2$ has a log chi-square(1) distribution — not Gaussian — with $E[\xi_t] \approx -1.27$ , $\text{Var}[\xi_t] = \pi^2/2 \approx 4.93$ , and severe left skewness. Applying the Kalman filter directly (QML) is therefore misspecified.

Variance decomposition. The total variance of $y_t$ decomposes as:

$\text{Var}(y_t) = \underbrace{\frac{\sigma_\eta^2}{1-\phi^2}}_{\text{true log-vol variation}} + \underbrace{\frac{\pi^2}{2}}_{\text{measurement noise} \approx 4.93}$

The log chi-square noise accounts for 60–80% of $\text{Var}(y_t)$ for weekly foreign exchange (FX) returns, leaving only 20–40% attributable to genuine log-volatility variation. This makes the latent path $h_t$ very difficult to estimate precisely.

Mixture-of-normals approximation. Kim, Shephard, and Chib (1998) approximate the log chi-square distribution by a fixed mixture of $K=7$ normals with pre-specified weights $\{p_i\}$ , means $\{m_i\}$ , and variances $\{v_i^2\}$ (chosen to match moments). Conditional on discrete mixture indicators $Z_t \in \{1,\ldots,7\}$ :

$y_t \mid Z_t = i, h_t \sim \mathcal{N}(h_t + m_i,\; v_i^2)$

which is Gaussian, allowing exact Kalman filtering and smoothing. Mahieu-Schotman extend to a flexible mixture: all $\{p_i, m_i, v_i^2\}$ are estimated from the data jointly with $(\phi, \sigma_\eta^2)$ via Simulation EM (SIEM), providing a better fit for fat-tailed exchange rate innovations.

Gibbs sampler for the flexible mixture (SIEM2/Bayesian). The sampler cycles through:

Draw latent path $\{h_t\}$ using the Carter-Kohn multimove smoother (conditional on $Z_t$ indicators and parameters).
Draw indicators $Z_t$ from their discrete full conditional: $\Pr(Z_t = i \mid y_t, h_t) \propto p_i\, \phi(y_t - h_t; m_i, v_i^2)$ .
Draw SV parameters $(\phi, \sigma_\eta^2)$ from Normal-IG conjugate posteriors.
Draw mixture parameters $\{p_i, m_i, v_i^2\}$ from their posteriors.

Empirical results (6 FX pairs, weekly 1975–1991):

Method	$\hat\phi$	$\hat\sigma_\eta$	Notes
QML	$\approx 0.92$	$\approx 0.10$	Biased downward on both
SIEM1 (fixed mixture)	$\approx 0.96$	$\approx 0.20$	Better
SIEM2 (flexible mixture)	$\approx 0.97$	$\approx 0.25$	Best ML-type
Bayesian	$\approx 0.97$	$\approx 0.25$	Similar to SIEM2

QML severely underestimates both $\phi$ and $\sigma_\eta$ . Standard errors of the smoothed volatility estimate $\hat h_t$ from QML are too small; the Bayesian/SIEM standard errors are 2–3 $\times$ larger, reflecting the true posterior uncertainty.

Option pricing implication. The option value under SV is $C = e^{-r\tau} E[\max(S_{t+\tau}-K, 0) \mid \mathcal{F}_t]$ , computed by averaging Black-Scholes values over simulated future volatility paths. Because QML underestimates $\sigma_\eta$ , it also underestimates the distribution of future volatility — producing option price confidence intervals that are too narrow.

Jacquier-Polson-Rossi (2004): Fat-Tailed and Correlated Errors

JPR (2004) extends the basic single-move MCMC framework to two richer SV specifications, then uses Bayes Factors to compare all four models.

Fat-tailed model (FC). Replace the Gaussian return innovation $u_t \sim \mathcal{N}(0,1)$ with $u_t \sim t(\nu)$ via scale mixing:

$u_t \overset{d}{=} w_t^{-1/2} z_t, \qquad w_t \sim \chi^2(\nu)/\nu \;(\text{i.e.}\; w_t \sim \mathrm{Gamma}(\nu/2,\,\nu/2)), \quad z_t \sim \mathcal{N}(0,1)$

Augmenting with latent precision weights $\{w_t\}$ restores Gaussianity conditional on $w_t$ : $y_t \mid h_t, w_t \sim \mathcal{N}(0, h_t/w_t)$ . The full conditional for $h_t$ is then log-concave, enabling adaptive rejection sampling (Wild and Gilks 1993) in place of the inverse-gamma M-H step of JPR (1994). The degrees-of-freedom parameter $\nu$ is drawn via a Metropolis step.

Correlated errors (ASV2). Introduce same-period correlation $\rho = \mathrm{corr}(u_t, v_t)$ by rewriting:

$\log h_t = \alpha + \delta \log h_{t-1} + \sigma_v\!\left(\rho\, u_t + \sqrt{1-\rho^2}\, v_t^*\right), \qquad v_t^* \perp u_t$

A fall in price (negative $u_t$ ) directly raises the current-period volatility shock when $\rho < 0$ — the leverage mechanism. Drawing $\rho$ uses a linearization of $\log h_t$ around its current value to construct a Gaussian M-H proposal with high acceptance rates.

Bayes Factors. All four models — basic SVOL, FC, ASV2, and Full (FC + ASV2) — are compared via importance-sampling BF computed from the MCMC posterior draws. For S&P 500 and CRSP equity indices, BF strongly favors both extensions; the correlated model wins more decisively than fat tails alone. For FX (CAD/USD), BF $\approx 1$ — no evidence for either extension.

Key empirical findings. For equity: $\hat\rho \approx -0.43$ to $-0.48$ (strong leverage); $\hat\nu \approx 25$ –30 for daily data (moderate fat tails), smaller at weekly frequency. Fat-tailed model produces smoother volatility paths: extreme returns are attributed to heavy tails rather than volatility spikes. Fat-tailed SV reduces option pricing RMSE by $\approx 15\%$ vs. basic SVOL; leverage creates put-call skew.

Leverage Effect in Discrete-Time SV (Yu 2005)

The leverage effect — negative correlation between return shocks and future volatility shocks — requires care in discrete-time SV. Two specifications exist:

	ASV1 (Harvey-Shephard 1996)	ASV2 (Jacquier et al. 2004)
Correlation	$\text{corr}(u_t, v_{t+1}) = \rho$	$\text{corr}(u_t, v_t) = \rho$
Timing	contemporaneous return → next-period vol shock	same-period return and vol shock
Efficient market hypothesis (EMH)	Martingale difference: $E(X_{t+1}\\|X_t) = 0$ ✓	$E(X_{t+1}\\|X_t) \neq 0$ — predictable returns ✗

ASV1 orthogonalised form (nonlinear state space, eq. 2.3): setting $w_{t+1} = (v_{t+1} - \rho u_t)/\sqrt{1-\rho^2}$ ,

$\ln\sigma^2_{t+1} = \alpha + \phi\ln\sigma^2_t + \underbrace{\rho\sigma_\eta\sigma_t^{-1}X_t}_{\text{leverage}} + \sigma_\eta\sqrt{1-\rho^2}\,w_{t+1}$

$E(\ln\sigma^2_{t+1}|X_t)$ is linear in $X_t$ : if $\rho<0$ , bad news raises future expected log-variance — unambiguous. The analogous ASV2 re-parameterisation is not analytically tractable.

Estimation: Multi-move Kim-Shephard-Chib (1998) algorithms use a log-squared transformation that destroys leverage information; BUGS (single-move MH) is applied directly on the nonlinear state-space form. Marginal likelihood via Chib (1995) identity + Kitagawa (1996) particle filter for log-likelihood evaluation.

Empirical verdict: Bayes factor ASV1/ASV2 $\approx 4{,}837$ (S&P500 1980–1987) — decisive. $\hat\rho$ (ASV1) $\approx -0.32$ to $-0.39$ ; ASV2 understates $|\rho|$ by $\approx 20\%$ . MCMC outperforms QML: relative efficiency 56–71%.

Omori-Chib-Shephard-Nakajima (2007): Mixture Sampler for Asymmetric SV

Omori, Chib, Shephard, and Nakajima (2007) extend the KSC (1998) multi-move MCMC sampler to the ASV model with correlated return and volatility shocks ( $\rho \neq 0$ ). The obstacle was that KSC's log-squaring transformation discards sign information and renders the joint density of return and volatility innovations non-separable. The resolution is to approximate the joint density of $(\varepsilon_t^*, \eta_t)$ — where $\varepsilon_t^* = \log\varepsilon_t^2$ — by a 10-component mixture of bivariate normals, exploiting the return sign $d_t = \text{sign}(y_t)$ to partially recover leverage information.

ASV model. Returns and log-volatility follow: $y_t = \varepsilon_t\exp(h_t/2), \qquad h_{t+1} = \mu + \phi(h_t - \mu) + \eta_t$ where $(\varepsilon_t, \eta_t) \sim N_2(0, R)$ , $R = \begin{pmatrix}1 & \rho\sigma\\ \rho\sigma & \sigma^2\end{pmatrix}$ . The parameter $\rho < 0$ captures the leverage effect; the difficulty is that standard KSC applies only when $\rho = 0$ .

Bivariate mixture approximation. Within each mixture component $j$ , the conditional mean of $\eta_t$ given $\varepsilon_t^*$ involves $\exp(\varepsilon_t^*/2)$ , which is approximated linearly: $\exp(\varepsilon_t^*/2) \approx \exp(m_j/2)\{a_j + b_j(\varepsilon_t^* - m_j)\}$ The mean-square-optimal closed-form coefficients are: $a_j = e^{v_j^2/8}, \qquad b_j = \tfrac{1}{2}e^{v_j^2/8}$ The resulting bivariate mixture approximation is: $f(\varepsilon_t^*, \eta_t \mid d_t) \approx \sum_{j=1}^{10} p_j\, \mathcal{N}(\varepsilon_t^* \mid m_j, v_j^2)\, \mathcal{N}\!\left(\eta_t \;\Big|\; d_t\rho\sigma e^{m_j/2}\{a_j + b_j(\varepsilon_t^* - m_j)\},\; \sigma^2(1-\rho^2)\right)$ Crucially, the quality of this approximation is independent of $\rho$ — the bivariate mixture remains accurate across all leverage values, including $\rho = -0.9$ .

Two-step MCMC. Conditional on mixture indicators $\{s_t\}$ , the model is a linear Gaussian state space, enabling the de Jong-Shephard (1995) simulation smoother. The sampler cycles through:

Indicator draw. For each $t$ , draw $s_t \in \{1,\ldots,10\}$ from its 10-point discrete full conditional given $(\varepsilon_t^*, \eta_t, d_t)$ .
Joint parameter and path draw. (a) Draw $\theta = (\phi, \rho, \sigma)$ via Metropolis-Hastings with Kalman-filter-based proposal. (b) Draw $(\{h_t\}, \mu)$ jointly via the Gaussian simulation smoother conditional on $\theta$ and $\{s_t\}$ .

Importance reweighting. Exact inference is recovered by reweighting MCMC draws by $w_k \propto \prod_{t=1}^n f / g$ , where $f$ is the true joint density and $g$ is the mixture approximation. For TOPIX (1998–2002, $n = 1232$ ), the log-weight standard deviation is $\approx 0.34$ , confirming negligible approximation error.

Empirical results (TOPIX 1998–2002). Posterior means: $\hat\phi = 0.951$ , $\hat\sigma = 0.134$ , $\hat\rho = -0.362$ [95% CI: $-0.593, -0.107$ ]. Inefficiency factors: $\phi = 9.3$ , $\sigma = 13.0$ , $\rho = 6.8$ , $b = \exp(\mu/2) = 2.7$ — all small, indicating near-i.i.d. draws. Log Bayes factor (Chib 1995 identity) for ASV vs. SV $\approx 2.24$ : leverage is decisively present. Model ranking: ASV-t $\approx$ ASV $\approx$ superposition $\gg$ SV-t $\gg$ SV.

Jensen-Maheu (2008): Semiparametric SV with Dirichlet Process Mixture

Jensen and Maheu (2008) attack a different axis of misspecification: not leverage, but the parametric form of the return innovation distribution. Standard SV models assume $\varepsilon_t \sim \mathcal{N}(0,1)$ ; they replace this with a nonparametric Dirichlet Process Mixture (DPM) while keeping the parametric AR(1) log-volatility process intact.

SV-DPM model. Returns $y_t = \exp(h_t/2)\varepsilon_t$ where $\varepsilon_t \sim G$ and $G \sim \text{DP}(\alpha, G_0)$ with Normal-IG base distribution $G_0$ . After stick-breaking, $y_t | h_t \sim \sum_{j=1}^\infty V_j \mathcal{N}(\mu_j,\, \sigma_j^{-2}\exp\{h_t\})$ : an infinite Gaussian mixture with volatility-scaled variances. Log-volatility follows the standard AR(1): $h_{t+1} = \mu + \phi(h_t - \mu) + \eta_t$ , $\eta_t \sim \mathcal{N}(0,\sigma_v^2)$ , independent of $\varepsilon_t$ . No leverage effect.

Key finding: misspecification inflates $\sigma_v^2$ . A Gaussian SV model confounds excess kurtosis and skewness in returns with volatility variation — it attributes non-Gaussian tail behavior to spikes in the latent variance rather than to the innovation distribution. The DPM corrects this.

CRSP daily returns (1980–2006, $n=6{,}815$ ): SV-DPM $\hat\phi=0.9877$ , $\hat\sigma_v^2=0.0103$ , avg. $k=7.16$ clusters; SV-N $\hat\phi=0.9795$ , $\hat\sigma_v^2=0.0276$ — 2.7 $\times$ inflation.
Simulation (3-component Gaussian mixture DGP): SV-N $\hat\sigma_v^2 \approx 20\times$ true value; SV-DPM recovers true parameters and correctly assigns mass to 3 clusters.

4-block MCMC. (1) Truncated-Normal/IG for $(\phi,\sigma_v^2)$ ; (2) Fleming-Kirby (2003) random-length block sampler for $\{h_t\}$ — groups consecutive observations into randomly-sized blocks and runs a Kalman smoother within each block, more efficient than single-move for persistent volatility; (3) Chinese Restaurant Process updates for cluster allocations $\{k_t\}$ and cluster parameters $\{(\mu_j,\sigma_j^2)\}$ (West-Müller-Escobar 1994 / MacEachern-Müller 1998); (4) Gamma posterior update for $\alpha$ .

Place in literature. Jensen-Maheu (2008) complements Omori-Chib-Shephard-Nakajima (2007): OCSN (2007) handles leverage ( $\rho\neq0$ ) with a fixed parametric return distribution; Jensen-Maheu handles distributional misspecification with no leverage. A model combining both extensions — bivariate mixture for leverage + DPM for the innovation shape — remains an open problem. See Dirichlet Process Mixture for the statistical framework.

Chib-Nardari-Shephard (2002): SVt and SVJt with Efficient Gibbs Sampling

Chib et al. (2002) extend the basic SV0 model in two directions and develop efficient MCMC algorithms for each.

Model SVt. The observation equation replaces Gaussian errors with Student-t via scale mixing:

$y_t = x_t'\beta + w_t^\phi \exp(h_t/2)\, u_t, \qquad u_t = \lambda_t^{-1/2}\varepsilon_t,\quad \lambda_t \sim \mathrm{Gamma}(\nu/2,\,\nu/2)$

$h_t = \mu + z_t'\gamma + \rho(h_{t-1}-\mu) + \sigma\eta_t$

The level effect $w_t^\phi$ allows variance to scale with the mean level; $z_t'\gamma$ adds covariates to the log-variance equation; $x_t$ includes lagged returns for the mean equation.

Model SVJt. Adds a Bernoulli jump term $k_t q_t$ where $q_t \sim \mathrm{Bernoulli}(\lambda)$ and $\log(1+k_t) \sim \mathcal{N}(-\delta^2/2, \delta^2)$ .

The seven-component Gaussian mixture. After augmenting with $\lambda_t$ , the squared and logged observation $y_t^* = \log(y_t - x_t'\beta)^2 + \log\lambda_t = h_t + z_t + \log w_t^2$ where $z_t = \log\chi^2(1)$ . Kim-Shephard-Chib (1998) approximate the log $\chi^2(1)$ distribution by a 7-component Gaussian mixture with fixed weights $\{q_j\}$ , means $\{m_j\}$ , and variances $\{v_j^2\}$ (Table 1). Conditional on discrete indicators $s_t \in \{1,\ldots,7\}$ , the model is a linear Gaussian state space, enabling the de Jong-Shephard (1995) simulation smoother to draw the entire latent $\{h_t\}$ path in one block.

Algorithm 1 (SVt — 4-block Gibbs):

Draw scale mixtures $\{\lambda_t\}$ from Gamma full conditionals.
Draw $[\psi, \{h_t\}]$ jointly: (a) $\psi = (\mu, \rho, \sigma, \phi, \gamma, \beta)$ by M-H with tailored multivariate- $t$ proposal (Chib-Greenberg 1994): numerically find mode $m$ and $V = (-\nabla^2 \log g(m))^{-1}$ , propose from $t_\nu(m, V)$ ; (b) $\{h_t\}$ by simulation smoother in one block given $\psi$ and $\{s_t\}$ .
Draw mixture indicators $s_t$ from 7-point discrete full conditional: $\Pr(s_t = j) \propto q_j \mathcal{N}(y_t^* \mid m_j + h_t, v_j^2)$ .
Draw $[\nu, \tau_t]$ : $\nu$ by M-H; $\tau_t \sim \mathrm{Gamma}$ .

Algorithm 2 (SVJt — 6-block Gibbs): adds blocks for jump indicators $\{q_t\}$ , jump sizes $\{\xi_t\}$ , and jump intensity $\lambda_\mathrm{jump}$ .

Marginal likelihood. The Chib (1995) identity $\log m(y) = \log f(y \mid \theta^*) + \log \pi(\theta^*) - \log \hat\pi(\theta^* \mid y)$ requires evaluating $f(y \mid \theta^*)$ (done via the auxiliary particle filter, Pitt-Shephard 1999, with $M=20{,}000$ particles) and the posterior ordinate $\hat\pi(\theta^* \mid y)$ (done via Chib-Jeliazkov (2001) for the M-H blocks, which uses the proposal and acceptance probability directly).

Inefficiency factor $= 1 + 2\sum_{k=1}^\infty \rho(k)$ ; its inverse equals Geweke's numerical efficiency. Values of 1–5 obtained with tailored M-H proposals and blocking; much higher without careful implementation.

S&P 500 results (8,849 obs, July 1962 – August 1997):

Bayes factor ( $\log_{10}$ )	vs. SV0
SVt	10.75 — decisive
SVJ	5.18 — decisive
SVJt	9.76 — decisive
SVt vs. SVJ	+5.57 in favor of SVt
SVJt vs. SVt	−0.99 (essentially tied)

Key parameter estimates: $\rho \approx 0.990$ (strong persistence); $\hat\nu \approx 12.5$ for SVt; jump intensity $\hat\kappa \approx 0.0034$ /day in SVJ (1 per $\approx 300$ days) falls to 0.00195 in SVJt (1 per $\approx 500$ days). The decrease in jump frequency when moving from SVJ to SVJt shows that SVt absorbs most "jump-like" observations as ordinary tail realizations. SV0 systematically overestimates volatility because large returns are attributed entirely to variance spikes rather than being filtered as tails or jumps.

Prior and simulation robustness: G=5,000 and G=50,000 sweeps yield identical posteriors; rankings robust to 7 alternative prior specifications including priors strongly favoring Gaussian errors.

Chib-Nardari-Shephard (2006): High-Dimensional Multivariate Factor-SV

Chib, Nardari, and Shephard (2006) extend the univariate SV framework to high-dimensional multivariate settings via a factor structure: each of $p$ asset returns loads on $k$ latent factors, with both asset-specific and factor-specific log-volatilities following independent AR(1) processes.

MSVJt model. The observation equation is:

$y_t = B f_t \cdot q_t + u_t$

where $f_t \sim \mathcal{N}_k(0, D_t)$ are latent factors with diagonal variance $D_t = \mathrm{diag}(e^{h_{p+1,t}}, \ldots, e^{h_{p+k,t}})$ ; $q_t$ is a Bernoulli jump indicator; and $u_{jt} = \lambda_{jt}^{-1/2} e_{jt}$ with $\nu_j / \lambda_{jt} \sim \chi^2(\nu_j)$ (Student-t scale mixing). Each log-volatility follows:

$h_{jt} = m_j + \phi_j(h_{j,t-1} - m_j) + \sigma_j Z_{jt}, \qquad j = 1, \ldots, p+k$

Identification. $B$ is lower-triangular with positive diagonal elements, leaving $pk - (k^2+k)/2$ free parameters. At $p=50$ , $k=8$ : 688 total parameters — a scale previously infeasible for multivariate SV estimation.

Reduced blocking scheme. Naive alternation of $b|f$ then $f|b$ produces inefficiency factors exceeding 1,000, because $b$ and $f$ are nearly collinear. The key innovation: sample $b$ marginalized over $f$ via a Newton-Raphson-tuned multivariate- $t$ Metropolis-Hastings proposal (eq. 7), then draw $f | b, h, \lambda$ analytically ( $\hat f_t = F_t B'(V_t^*)^{-1} \tilde y_t$ ). After this step the model separates into $p+k$ conditionally independent univariate SV state spaces, each sampled via the KSC (1998) 7-component mixture + de Jong-Shephard (1995) simulation smoother. Resulting inefficiency factors: 1–30 (vs. >1,000 without the refinement).

Marginal likelihood. An auxiliary particle filter (Pitt-Shephard 1999, $M = 20{,}000$ particles) estimates $\hat f(y_t | F_{t-1}, c^*)$ at the posterior mode $c^*$ . The simulation-consistent likelihood feeds into the Chib (1995) identity; posterior ordinates for M-H blocks use the Chib-Jeliazkov (2001) framework. Bayes factors select among MSV / MSVt / MSVJ / MSVJt variants.

Application (10 international equity indices, weekly 1973–2003, $T = 1{,}605$ ). Model selection: Bayes factors favor MSVt with 3 factors (fat tails, no jumps). Forecast comparison: MSV mean absolute deviation (MAD) at 1-week horizon = 3.45 vs. BEKK = 3.63, Dynamic Conditional Correlation (DCC) = 3.72 — MSV best or tied across all multivariate GARCH (MGARCH) alternatives. value at risk (VaR): MSV achieves best conditional coverage across 4 portfolios (World, Europe, Pacific Rim, US) at both 1% and 5% levels.

MCMC Maximum Likelihood for SV (Jacquier-Johannes-Polson 2007)

Jacquier, Johannes, and Polson (2007) show how to compute the frequentist MLE of SV parameters via MCMC, without gradient methods. $J$ independent copies of the volatility path $\{V_t\}$ are drawn at each Gibbs step. The augmented joint density $p_J(\theta, \tilde{V}^J) \propto [\prod_j p(Y|\theta,V^j)p(V^j|\theta)]\cdot m(\theta)$ has a $\theta$ -marginal concentrating at the MLE as $J\to\infty$ .

For the basic SV model ( $\log V_t = a + d\cdot\log V_{t-1} + \sigma_v v_t$ ), stacking $J$ volatility copies in a single regression gives:

$p(a, d \mid \sigma_v, \tilde{V}^J, Y) \sim \mathcal{N}(\hat{b}, \sigma_v^2 (X'X)^{-1}), \qquad p(\sigma_v^2 \mid \tilde{V}^J, Y) \sim \mathrm{IG}(J, S)$

where X is the $JT \times 2$ design matrix of stacked copies and S the residual sum. The volatility copies are drawn via M-H (as in JPR 1994). The convergence theorem guarantees $\sqrt{J}(\theta^{(g)} - \hat\theta) \to \mathcal{N}(0, \ell''(\hat\theta)^{-1})$ , providing standard errors as $\mathrm{Var}(\sqrt{J}\text{ draws})$ . Normality of scaled draws (Jarque-Bera) is the diagnostic.

$J=20$ , $G=25{,}000$ suffices for the basic SV model ( $\approx 20$ min on 2006-era hardware); smoothed volatility estimates are essentially identical to the Bayesian smoother, confirming that the MCMC smoothing efficiency is preserved at the MLE. See MCMC Maximum Likelihood for the general framework.

Brandt-Kang (2004): Latent VAR for Mean-Volatility Joint Dynamics

Whereas the models above treat volatility as a univariate latent process, Brandt and Kang (2004) model both the conditional mean $m_t$ and conditional volatility $v_t$ of stock returns as jointly latent states in a bivariate VAR(1):

$y_t = \bar{\mu}e^{m_{t-1}} + \bar{\sigma}e^{v_{t-1}}\varepsilon_t, \qquad s_t = As_{t-1} + \eta_t, \quad \eta_t \sim MVN(0,\Sigma)$

The key parameter is $\rho = \mathrm{Corr}(\eta_t^m, \eta_t^v)$ — the contemporaneous correlation between mean and volatility innovations. Estimated by simulated maximum likelihood with a VAR importance-sampling correction.

Main finding (Center for Research in Security Prices (CRSP) monthly returns, 1946–1998): $\hat{\rho} \approx -0.56$ ( $t = -5.80$ ) — strongly negative. A positive shock to expected returns is simultaneously accompanied by a drop in volatility. The lag risk-return tradeoff coefficient $a_{12}$ (lagged volatility predicting mean) is small and insignificant ( $t = 1.01$ ).

This distinguishes two correlations with opposite signs: the contemporaneous innovation correlation (negative, strong) vs. the unconditional long-run correlation (positive — both mean and volatility peak at recession troughs). See Risk-Return Tradeoff.

Bollerslev-Zhou (2006): Three Return-Volatility Regression Puzzles

Bollerslev and Zhou (2006) show that three apparently contradictory empirical regularities all follow analytically from the Heston (1993) model with just two parameters: $\rho < 0$ (instantaneous leverage) and $\lambda_v < 0$ (negative volatility risk premium).

Model: Heston (1993) objective dynamics and risk-neutral dynamics:

$dp_t = (\mu + \lambda_s V_t)\,dt + \sqrt{V_t}\,dB_t, \qquad dV_t = \kappa(\theta - V_t)\,dt + \sigma\sqrt{V_t}\,dW_t, \qquad \text{corr}(dB_t, dW_t) = \rho$

$dV_t = \kappa^*(\theta^* - V_t)\,dt + \sigma\sqrt{V_t}\,dW_t^*, \quad \kappa^* = \kappa + \lambda_v < \kappa \text{ (since } \lambda_v < 0\text{)}$

The gap between $\kappa$ and $\kappa^*$ is central to all three results.

Proposition 1 — Volatility feedback: Population slope in realized-vol regression is $\beta = \lambda_s + \rho\kappa/\sigma < \lambda_s$ ; can be negative when $0 < \lambda_s < -\rho\kappa/\sigma$ . Implied-vol slope is $\beta^* = \lambda_s(a_\Delta/a^*_\Delta) > 0$ always, where $a_\Delta/a^*_\Delta < 1$ due to $\kappa^* < \kappa$ . S&P500 (Jan 1990–Feb 2002): $\hat\beta = -0.53$ (realized), $\hat\beta^* = +0.24$ (implied).

Proposition 2 — Leverage asymmetry: $\delta^* < \delta < 0$ : implied vol always exhibits stronger asymmetric response to lagged returns than realized vol. Long regressions confirm: realized leverage is statistically insignificant while implied leverage is highly significant ( $\hat\delta^* \approx -0.31$ to $-0.55$ ).

Proposition 3 — Implied-vol forecasting bias: Population slope in $RV = \phi_0 + \phi_1 \cdot IV^* + \varepsilon$ is $\phi_1 = a_\Delta/a^*_\Delta < 1$ . Slower risk-neutral mean-reversion ( $\kappa^* < \kappa$ ) makes implied vol systematically overshoot future realized vol. Not a market inefficiency — it is structural. Empirical $\hat\phi_1 = 0.68$ (std), $0.41$ (variance).

Monte Carlo finding: 5-minute realized vol introduces negligible measurement error for monthly regressions. Daily-squared-return-based RV introduces large, persistent biases that do not shrink with sample size.

Multivariate SV for BVAR: Conjugate Wishart Random Walk (Uhlig 1997)

Standard BVAR estimation fixes the error covariance $\Sigma$ over the full sample. Uhlig (1997) extends the conjugate BVAR framework to allow a time-varying $m \times m$ precision matrix $H_t$ (the inverse covariance), maintaining analytical tractability via a multiplicative random walk driven by multivariate Beta variates.

Law of motion for H_t:

$H_{t+1} = A_{t+1}^{1/2}\, H_t\, A_{t+1}^{1/2}$

where $A_{t+1}$ is drawn from the matrix Beta distribution (Uhlig 1994b: ratio of two independent Wishart variates). The scalar parameter $n > 0$ controls persistence — large $n$ means H_t is nearly constant, small $n$ allows large jumps. In the univariate case, $\lambda_{t+1} = A_t \lambda_t$ with $A_t \sim \mathrm{Beta}(n/2, 1/2)$ : a scalar multiplicative random walk for the precision.

Conjugate updating. The prior on $(B, H_0)$ is matrix-normal inverse-Wishart (MNIW). The key result: despite time-varying $H_t$ , the posterior at each step remains MNIW, updated by closed-form recursions. At each period:

Discount step: reduce the effective degrees of freedom $\nu_t^{\mathrm{prior}} = \frac{n}{n+1}\nu_{t-1}^{\mathrm{post}}$ , equivalent to a discount factor $\delta = n/(n+1)$ in the West-Harrison (1989) sense — past information is "forgotten" at rate $1 - \delta$ .
Data update: standard MNIW conjugate update with the new observation $Y_t$ .

No MCMC is needed for the per-step update; the recursion is a nonlinear Kalman filter.

Importance sampling for the marginal posterior of B. The full joint posterior $p(B, H_1,\ldots,H_T | Y_{1:T})$ is not closed-form. Uhlig evaluates it by: (a) drawing $H_t$ paths from their conditional posteriors, (b) weighting by importance ratios. Figure 2 of the paper shows log-weight vs. log-posterior scatter near-diagonal for 4,000 draws on a 4-variable system — confirming the sampler does not collapse.

Contrast with other approaches:

GARCH: $H_t$ is a deterministic function of past residuals, not a latent process. Uhlig's model captures genuine randomness in volatility.
Standard univariate SV (Taylor 1982; JPR 1994): log-normal state equation destroys conjugacy, requiring MCMC. The multiplicative Wishart law of motion restores conjugacy by working with the precision directly.
TVP-VAR with SV (Cogley-Sargent 2005; Primiceri 2005): later literature extended Uhlig's insight to allow simultaneous time variation in both B and Σ, using MCMC for the joint posterior.

Empirical finding (Uhlig 1996 draft, 4-variable U.S. macro VAR): "Modest evidence of a genuine change" in volatility. Results are robust to discount factor choices $\delta \in \{3/5, 2/3\}$ .

Eraker (2001): MCMC for Continuous-Time CEV and SV Interest-Rate Models

Eraker (2001) solves the continuous-time estimation problem by data augmentation: $m-1$ latent data points between each discrete observation set $\Delta t = 1/m$ , making the Euler Gaussian transition density arbitrarily accurate. The augmented posterior is sampled by a two-block Gibbs cycle.

CEV one-factor model. $dr_t = (\theta_r + \kappa_r r_t)dt + \sigma r_t^\beta dW_t$ . Given augmented data $\widehat{Y}$ , the drift parameters $(\theta_r, \kappa_r)$ have a Normal posterior via linear regression; $\sigma^2$ has an IG posterior; $\beta$ (log-concave) uses a second-order Taylor–expanded Metropolis step. Key empirical result on 2,288 weekly US T-bill yields (1954–1997): $\hat\beta \approx 0.75$ vs. GMM $\approx 1.5$ (Chan et al. 1992) — a large discrepancy attributable to discretization bias in non-Bayesian estimates. CEV fails diagnostically: heavy-tailed residuals (Q-Q) and ARCH-like autocorrelations in squared residuals that the model cannot explain.

CEV+SV two-factor model. $dr_t = (\theta_r + \kappa_r r_t)dt + \sigma_r\exp\!\left(\tfrac{1}{2}Z_t\right)r_t^\beta dW_{1,t}, \quad dZ_t = (\theta_z + \kappa_z Z_t)dt + \sigma_z dW_{2,t}$ $Z_t$ (log-volatility) is latent, augmented alongside the missing path. Key results: $\hat\kappa_z \Rightarrow$ first-order autocorrelation (AC) $\approx 0.976$ (consistent with GARCH/SV literature); $\hat\sigma_z \approx 0.27$ (vs. Andersen-Lund 1997a EMM: 0.176 — Eraker attributes the difference to higher short-run volatility variation); $\kappa_r$ entirely negative in SV model (strong mean reversion), whereas CEV finds only $\approx 8\%$ posterior mass below 0.

Reparameterization fix for $\kappa_z$ bias (Appendix D). High serial dependence in the volatility chain's running mean $\bar{Z}^{(g)}$ biases $\kappa_z$ downward. Fix: at each step subtract $k = \bar{Z}^{(g)}$ and adjust $Z_t^* \leftarrow Z_t^{(g)} - k$ , $\sigma_r^* \leftarrow \sigma_r^{(g)}\exp(\tfrac{1}{2}k)$ . This recenters the log-volatility chain without affecting the stationary distribution.

Discretization-convergence trade-off. As $\Delta t \to 0$ Gibbs conditionals collapse toward Dirac measures, slowing convergence. At $\Delta t$ large, discretization bias dominates. Practical recommendation: start with m=1 and increase until posterior means stabilize (convergence at $m\approx2$ – $4$ for CEV parameters). See Diffusion Process for the Brownian bridge AR-MH proposal and theoretical justification.

Cappuccio-Lubian-Raggi (2006): Skew-GED Return Distribution

Cappuccio, Lubian, and Raggi (2006) extend the standard SV model by replacing Gaussian return shocks with a Skew-GED distribution built via the Azzalini (1985) device applied to the generalized error distribution (GED) base. Two parameters: $\kappa$ (skewness; $\kappa<0$ = left-skew) and $\nu$ (tail thickness; $\nu=2$ is Gaussian). The model nests Normal ( $\nu=2,\kappa=0$ ), Skew-Normal ( $\nu=2,\kappa\neq0$ ), and GED ( $\kappa=0$ ). No leverage effect: return and volatility shocks are independent.

MCMC: Log-volatilities via Delayed-Rejection MH (Tierney-Mira 1999); $\kappa$ and $\nu$ via Adaptive-Rejection Metropolis Sampling. Specification testing via Savage-Dickey density ratios: $\text{SD}_{j|1} = p(\theta_j^0|y)/p(\theta_j^0)$ , computed by kernel smoothing MCMC output at the restriction.

Empirical results (DJ30, S&P500, Nasdaq; daily and weekly from Datastream): $\hat\phi \approx 0.991$ – $0.993$ (near-IGARCH); $\hat\nu \approx 1.49$ – $1.94$ (heavy tails throughout). Daily: $\hat\kappa \approx 0$ for DJ30/S&P500 (GED sufficient), $\hat\kappa \approx -0.54$ for Nasdaq. Weekly: decisive left-skew across all indexes ( $\hat\kappa \approx -0.82$ to $-1.06$ ). Gaussianity rejected universally. Frequency dependence: asymmetry is a weekly phenomenon, stronger than at daily frequency — consistent with the broader stylized-fact literature.

Why It Matters

SV models provide a theoretically motivated alternative to GARCH for financial econometrics, especially when the volatility process itself has economic meaning (e.g., uncertainty about fundamentals).
In macroeconometrics, SV shocks on VAR error covariances (Cogley-Sargent 2005; Primiceri 2005) model changing macroeconomic volatility — e.g., the Great Moderation — within time-varying parameter VARs.
Continuous-time SV with OU volatility (Stein-Stein 1991) generates a volatility smile and power-law fat tails; the mixture-of-lognormals representation makes the relationship between volatility dynamics and option prices transparent.
Continuous-time SV (Heston 1993, which adds correlation between $dz_1$ and $dz_2$ ) is the standard framework for equity option pricing when returns exhibit both a volatility smile and a volatility skew (put leverage premium).

Open Questions

Distinguishing GARCH from SV is difficult in practice: both can fit autocorrelations of squared returns well, and formal non-nested tests have low power in typical samples.
Particle-filter-based inference is computationally intensive for large systems; approximations like INLA or variational Bayes are active research areas.