Volatility Spillover Index

• Volatility Spillover Index is a measure that quantifies how volatility shocks in one asset transmit to others using VAR and variance decomposition techniques.
• It employs high-frequency data to compute realized variance and semivariances, carefully distinguishing between downside and upside risks.
• The indices capture directional and asymmetric spillovers, offering actionable insights for portfolio diversification and systemic risk management.

A volatility spillover index quantifies the extent to which volatility shocks in one financial asset or sector are transmitted to others, tracing interdependencies in risk dynamics and information flow. This class of measures is crucial for understanding systemic risk, portfolio diversification, and the sectoral or directional propagation of uncertainty. Modern volatility spillover indices are constructed using high-frequency data, advanced variance decomposition techniques, and often incorporate asymmetries between "bad" (downside) and "good" (upside) volatility connectedness. Sectoral, directional, and time-varying spillover indices enable detailed mapping of how shocks propagate during stable versus crisis regimes.

1. Foundations: Volatility and Spillover Quantification

Volatility is chiefly measured using realized variance (RV) constructed from high-frequency asset returns:

$\text{RV} = \sum_{i=1}^{n}(p_i - p_{i-1})^2,$

where $p_i$ is the $i$th high-frequency log-price within the trading day. The realized variance converges to the quadratic variation of the underlying price process.

To discriminate between downside and upside risk, realized semivariances are introduced (Barunik et al., 2013):

• Downside ("bad") realized semivariance:

$\text{RS}^- = \sum_{i=1}^n r_i^2 I(r_i < 0)$

• Upside ("good") realized semivariance:

$\text{RS}^+ = \sum_{i=1}^n r_i^2 I(r_i > 0)$

yielding the decomposition: $\text{RV} = \text{RS}^- + \text{RS}^+$.

The core of spillover measurement is a vector autoregressive (VAR) modeling approach. For an $N$-variate vector of volatility measures (e.g., $\mathbf{RV}_t$), the VAR($p$) model is specified as:

$$\mathbf{RV}_t = \Phi_1 \mathbf{RV}_{t-1} + \dots + \Phi_p \mathbf{RV}_{t-p} + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0,\Sigma_\varepsilon)$$

The moving average (MA) representation for VAR enables generalized forecast error variance decompositions, partitioning the forecast error variance of one component into proportions attributable to shocks in itself and in other variables. The generalized approach (à la Diebold–Yilmaz) ensures invariance to variable ordering.

2. Computation and Structure of Volatility Spillover Indices

The central computational object is the $H$-step-ahead generalized forecast error variance decomposition:

$$\omega_{ij}^H = \frac{\sigma_{jj}^{-1} \sum_{h=0}^{H-1} (e_i' \Psi_h \Sigma_\varepsilon e_j)^2}{\sum_{h=0}^{H-1} e_i' \Psi_h \Sigma_\varepsilon \Psi_h' e_i}$$

where $e_j$ is a selection vector and $\Psi_h$ are the MA coefficients.

Normalization is necessary due to non-orthogonalized innovations:

$$\widetilde{\omega}_{ij}^H = \omega_{ij}^H/\sum_j \omega_{ij}^H$$

The total spillover index is then defined as:

$$S^H = 100 \times \frac{1}{N} \sum_{i \neq j} \widetilde{\omega}_{ij}^H$$

Directional spillovers, both "from" and "to" a given asset, as well as net spillovers (the difference between transmitted and received) are extractable by appropriately summing normalized entries.

This VAR-based framework is extendable to realized semivariances, yielding separate spillover indices for $\text{RS}^-$ and $\text{RS}^+$ components.

3. Asymmetric Spillover Measurement and Sectoral Disaggregation

Recognizing that volatility spillovers may not be symmetric for positive and negative return innovations, the Spillover Asymmetry Measure (SAM) is introduced:

$$\text{SAM} = 100 \times \frac{S^+ - S^-}{0.5 (S^+ + S^-)}$$

where $S^+$ and $S^-$ are spillover indices calculated on the upside and downside semivariances, respectively (Barunik et al., 2013). At the asset or sector level, analogous directional asymmetry measures are defined—for example, for spillovers received by or transmitted from asset $i$:

$$\text{SAM}_{i \leftarrow \bullet} = 100 \frac{S_{i \leftarrow \bullet}^+ - S_{i \leftarrow \bullet}^-}{0.5 (S_{i \leftarrow \bullet}^+ + S_{i \leftarrow \bullet}^-)}$$

Statistical significance of asymmetry is assessed via a bootstrap procedure within a null symmetric process.

Empirical results using high-frequency U.S. equity data show that, at the aggregate portfolio level, spillover asymmetry vanishes—$\text{SAM}$ is statistically indistinguishable from zero. In contrast, strong asymmetry arises at the sector level and for individual stocks. For example, negative spillovers (bad volatility) may dominate in sectors like Consumer Discretionary during crisis periods, while positive spillovers can be more prominent in Health Care or Information Technology during phases of optimism (Barunik et al., 2013).

4. Dynamic Behavior and Macroeconomic Drivers

Time-variation in the volatility spillover index is prominent. Using rolling-window estimation, the total spillover index surges during episodes of macro-financial uncertainty (e.g., the 2007–2008 financial crisis), reaching or exceeding 90% at the peak of systemic shocks. This is highly correlated with external risk proxies such as the VIX and TED spread, reflecting increased market interconnectedness and the breakdown of traditional diversification during distressed regimes (Barunik et al., 2013).

Disaggregation by sector or stock reveals episodic realignments; for example, in the Financials sector, risk transmission patterns shifted dramatically during the subprime crisis, with some banks switching from net contributors to net receivers of volatility.

5. Implications for Risk Valuation and Portfolio Strategies

Volatility spillover indices offer actionable signals for risk management. During tranquil periods, relatively low spillover suggests sectoral or portfolio diversification is effective. In stress regimes, synchronized (high) spillover signifies elevated systemic risk and compressed diversification. Sectoral asymmetry further implies that the transmission channels of downside and upside risks differ, and managers should test resilience of portfolio choices both to aggregate and sector-specific transmission patterns.

Knowledge of directional asymmetries—derived via $\text{SAM}$ and directional variants—enables more granular assessment of risk sources and recipients. For example, policy shifts or industry-specific shocks are traceable through sectoral spillover indices; asset allocators can recalibrate positions dynamically in anticipation of changing transmission patterns.

6. Recent Advances and Extensions

Recent research advances include more sophisticated modeling of nonlinear (e.g., VAR with Student $t$ errors) and networked (e.g., GARCH–BEKK, Graph Neural Network-based) spillover relationships. Asymmetric and sectoral analyses have been extended to commodities, foreign exchange (Barunik et al., 2016), and cross-asset correlations. The empirical relevance of asymmetry has been documented across asset classes, with bad volatility spillovers found to be more contagious than good volatility in petroleum and FX markets (Barunik et al., 2014, Barunik et al., 2016).

Dynamic connectivity measures, such as time-varying rolling-window spillover graphs, further allow for quantification of how sudden changes in global conditions impact risk transmission—enabling real-time systemic risk monitoring.

7. Summary Table of Key Indices and Their Mathematical Forms

Measure	Mathematical Formula	Interpretation
Realized Variance (RV)	$\sum_{i=1}^n (p_i - p_{i-1})^2$	Total daily volatility
Realized Semivariances ($\text{RS}^-, \text{RS}^+$)	$$\text{RS}^- = \sum r_i^2 I(r_i<0)\quad \text{RS}^+ = \sum r_i^2 I(r_i>0)$$	Downside / upside volatility
Total Spillover Index	$$S^H = 100 \times \frac{1}{N} \sum_{i\neq j} \widetilde{\omega}_{ij}^H$$	Mean cross-asset volatility transmission
Spillover Asymmetry Measure	$$\text{SAM} = 100 \times \frac{S^+ - S^-}{0.5(S^+ + S^-)}$$	Degree of asymmetry: positive (good) vs. negative (bad) spillovers
Directional Asymmetry (from $i$)	$$\text{SAM}_{i \rightarrow \bullet} = 100 \frac{S_{i \rightarrow \bullet}^+ - S_{i \rightarrow \bullet}^-}{0.5 (S_{i \rightarrow \bullet}^+ + S_{i \rightarrow \bullet}^-)}$$	Disaggregated directional asymmetry

8. Significance and Future Research

The volatility spillover index family—augmented by semivariances and asymmetry measures—enables nuanced monitoring of systemic risk and transmission channels. The revelation that aggregate connectedness may be symmetric while sector-level asymmetry persists challenges risk management paradigms relying solely on aggregate metrics.

Ongoing research trends focus on refining these indices with nonlinear dynamics, regime-switching, machine learning (e.g., neural network–inferred covariance filtering), and cross-asset adaptation. The aim is to further improve early-warning capabilities, directional sensitivity, and the detection of transmission patterns under structural changes or unprecedented events.

These innovations support the evolving requirements for systemic risk monitoring, active portfolio management, and cross-sector capital allocation in complex, interconnected, and occasionally turbulent financial markets.