Modeling Tail Dependence in Multivariate Stochastic Volatility

# Multivariate Stochastic Volatility with Jumps and strong Tail Correlation


# Multivariate Stochastic Volatility with Jumps and strong Tail Correlation












How to model two correlated stocks where the innovations are asymmetric and heavy-tailed? The model is estimated with MCMC on historical returns (not option data) and later used for VaR simulation.

Consider the univariate SV model:

$$ \begin{aligned} r_t &= \mu + e^{h_t}\epsilon_t \qquad \epsilon_t \sim N(0,1)\\ h_t &= \omega + \phi(h_{t-1}-\omega) + \bigl(J_t-\mathbb E[J_t]\bigr),\\ J_t &= \log \eta_t,\qquad \eta_t\sim \mathrm{InvGamma}(\nu/2,\nu/2),\\ \end{aligned} $$

For two assets, I want dependence in both returns and volatility shocks. Plain Gaussian correlation for $\epsilon_t$ is easy, but it does not capture asymmetric tail dependence in the volatility innovations. A full copula for $(J_t^a,J_t^b)$ would be more appropriate, but may be too slow for MCMC.

One possible approximation is to couple the inverse-gamma shocks through quantile transforms:

$$ \begin{aligned} r_t^{a} &= \mu^{a} + e^{h_t^{a}} \epsilon_t^{a}, \\ r_t^{b} &= \mu^{b} + e^{h_t^{b}} \epsilon_t^{b}, \\ h_t^{a} &= \omega^{a} + \phi^{a}(h_{t-1}^{a}-\omega^{a}) + (J_t^{a} - \mathbb E[J_t^{a}]), \\ h_t^{b} &= \omega^{b} + \phi^{b}(h_{t-1}^{b}-\omega^{b}) + (J_t^{b} - \mathbb E[J_t^{b}]), \\ J_t^{a} &= \log\!\big(k_1^{a}\eta_t^{a} + k_2^{a}Q^a(\eta_t^{b})\big), \\ J_t^{b} &= \log\!\big(k_1^{b}Q^b(\eta_t^{a}) + k_2^{b}\eta_t^{b}\big), \end{aligned} $$

where

$$ \eta_t^{a} \sim \mathrm{InvGamma}(\nu^{a}/2,\nu^{a}/2),\qquad \eta_t^{b} \sim \mathrm{InvGamma}(\nu^{b}/2,\nu^{b}/2), $$

and

$$ Q^a(x)=F_a^{-1}(F_b(x)),\qquad Q^b(x)=F_b^{-1}(F_a(x)). $$

Here $Q^a,Q^b$ are fast polynomial approximations to the exact quantile transforms.

Another possibility is to introduce a shared heavy-tailed shock, but it introduces one more heavy tailed source of randomness for each $t$, I'm afraid MCMC would be too slow.

$$ \begin{aligned} h_t^{a} &= \omega^{a} + \phi^{a}(h_{t-1}^{a}-\omega^{a}) + (J_t^{a}-\mathbb E[J_t^{a}]), \\ h_t^{b} &= \omega^{b} + \phi^{b}(h_{t-1}^{b}-\omega^{b}) + (J_t^{b}-\mathbb E[J_t^{b}]), \\ J_t^{a} &= \log\!\big(k_1^{a}\xi_t + k_2^{a}\eta_t^{a}\big), \\ J_t^{b} &= \log\!\big(k_1^{b}Q^b(\xi_t) + k_2^{b}\eta_t^{b}\big), \end{aligned} $$

with

$$ \xi_t \sim \mathrm{InvGamma}(\nu^{a}/2,\nu^{a}/2),\qquad \eta_t^{a} \sim \mathrm{InvGamma}(\nu^{a}/2,\nu^{a}/2),\qquad \eta_t^{b} \sim \mathrm{InvGamma}(\nu^{b}/2,\nu^{b}/2). $$

Does this kind of approximation make sense, or is there a better way to introduce tail dependence in the volatility shocks without making MCMC slow?

My goal is to estimate the model using about 5,000 observations per asset (roughly 20 years of daily returns) and apply the same framework to both very strongly related pairs, such as gold and gold miners, and more weakly related pairs, such as oil and uranium.

## Answer by pandashark (score 1)

https://quant.stackexchange.com/a/85571

Both proposed couplings run into a distributional issue: the sum of inverse-gamma random variables is not inverse-gamma. So the marginal distributions of $\exp(J_t^a)$ and $\exp(J_t^b)$ are no longer InvGamma under either scheme, which destroys the distributional properties that motivated your univariate model and breaks MCMC conjugacy.

### Issues with both proposals

Option A (quantile-transform coupling): The quantile transform $Q^a(\eta_t^b) = F_a^{-1}(F_b(\eta_t^b))$ is marginally InvGamma$({\nu^a}/{2},{\nu^a}/{2})$ by the probability integral transform, which is fine. But $J_t^a = \log(k_1^a \eta_t^a + k_2^a Q^a(\eta_t^b))$ is the log of a sum of InvGamma variables. This sum has no closed-form density, so:

- The marginal distribution of $\exp(J_t^a)$ is not InvGamma, regardless of how you normalise $(k_1, k_2)$.

- The conditional posterior of $\eta_t^a$ given $(r_t^a, r_t^b, h_t^a, h_t^b, \eta_t^b)$ involves the Jacobian of the quantile-transform mapping, which has density ratios $f_b(\eta_t^b)/f_a(Q^a(\eta_t^b))$. Conjugacy is lost — you need Metropolis-Hastings for every $(\eta_t^a, \eta_t^b)$ pair at every $t$, so $2T$ additional MH steps per MCMC iteration with likely poor acceptance rates.

- The parameters $(k_1, k_2, \omega, \nu)$ interact in complex ways, creating identifiability problems.

Option B (shared shock): Structurally closer to a factor model, but $J_t^a = \log(k_1^a \xi_t + k_2^a \eta_t^a)$ has the same sum-of-InvGamma problem. Plus you add $T$ extra latent variables ($\xi_t$), each requiring its own MH step.

### What the literature does instead

Note that the InvGamma multiplicative shock is the scale-mixture representation of the Student-$t$ distribution, applied in the volatility equation. Recall: if $\eta \sim \text{InvGamma}(\nu/2, \nu/2)$, then $\sqrt{\eta}\cdot z$ with $z \sim N(0,1)$ is $t_\nu$-distributed. So you are putting $t$-type tails on the volatility innovations.

For short-horizon VaR (1-day, 10-day), heavy tails in the return innovations are more directly relevant — they fatten the conditional return distribution where VaR quantiles live. Heavy-tailed vol innovations also matter (they generate more extreme vol regimes), but their effect is primarily on longer horizons. Either way, the multivariate coupling is much easier to handle in the return innovations. The standard approach is:

#### Bivariate SV with multivariate- $t$ returns

$$ \begin{aligned} r_t^i &= \mu^i + e^{h_t^i/2}\,\epsilon_t^i, \qquad (\epsilon_t^a,\,\epsilon_t^b)' \sim t_\nu(0, R)\\ h_t^i &= \omega^i + \phi^i(h_{t-1}^i - \omega^i) + \sigma_\eta^i u_t^i, \qquad (u_t^a,\,u_t^b)' \sim N(0, \Sigma_u) \end{aligned} $$

You will also want leverage (correlation between return and vol innovations, $\text{Corr}(\epsilon_t^i, u_t^i) = \rho^i < 0$ for equities). This is empirically important for VaR because it generates the asymmetry: bad returns drive vol up, which fattens the left tail. See Jacquier, Polson, and Rossi (2004) below.

This gives you:

- Tail dependence in returns — the multivariate $t$ has non-zero tail dependence (unlike the multivariate normal), which matters for VaR.

- Correlated volatility dynamics — $\Sigma_u$ captures vol co-movement (when one asset's vol rises, the other tends to follow).

- Leverage — the return-vol correlation $\rho^i$ captures the asymmetric "bad news increases vol" effect.

- Standard MCMC — represent the $t$ as a scale mixture: $\epsilon_t^i = z_t^i / \sqrt{\lambda_t}$, $(z_t^a, z_t^b)' \sim N(0, R)$, $\lambda_t \sim \text{Gamma}(\nu/2, \nu/2)$. This restores conjugacy — the $\lambda_t$ are Gibbs-sampled, and the rest follows the Kim-Shephard-Chib (1998) mixture approximation.

For only 2 assets you do not need factor structure — just estimate the $2 \times 2$ correlation/covariance directly. Factor models (below) become necessary at $\sim$10+ assets.

#### If you want heavy tails in vol and returns

Use the multiplicative (log-additive) factor structure instead of your additive-in-levels formulation:

$$h_t^a = \omega^a + \phi^a(h_{t-1}^a - \omega^a) + \lambda_a f_t + \sigma_\eta^a u_t^a$$

where $f_t$ is a shared log-vol factor. This is the standard factor SV model of Chib, Nardari, and Shephard (2006), with the loading $\lambda_a$ controlling how much asset $a$'s vol responds to the common factor. Everything stays Gaussian in log-vol space, so standard MCMC applies.

### Practical path

For your setting (2 assets, 5000 obs, MCMC, VaR):



- If you need multivariate-$t$ returns specifically: Implement in Stan or custom MCMC. The scale-mixture representation makes this a modest extension — you are adding one $\lambda_t$ per time step with a conjugate Gamma full conditional.



For VaR simulation in all cases: draw $h$ paths from the posterior predictive, draw correlated returns from the fitted dependence structure, and compute portfolio losses.

### Key references

- Jacquier, Polson, and Rossi (2004), "Bayesian Analysis of Stochastic Volatility Models with Fat-Tails and Correlated Errors," Journal of Econometrics 122, 185–212. — Heavy tails and leverage in univariate SV via MCMC.

- Chib, Nardari, and Shephard (2006), "Analysis of High Dimensional Multivariate Stochastic Volatility Models," Journal of Econometrics 134(2), 341–371. — Factor SV with efficient MCMC.

- Kastner, Fruhwirth-Schnatter, and Lopes (2017), "Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models," JCGS 26(4), 905–917. — The `factorstochvol` methodology.

- Kim, Shephard, and Chib (1998), "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," Review of Economic Studies 65(3), 361–393. — The mixture approximation for SV estimation.

- Asai, McAleer, and Yu (2006), "Multivariate Stochastic Volatility: A Review," Econometric Reviews 25(2–3), 145–175. — Comprehensive survey of model classes.