Most volatility models live in a one-dimensional world. They chart the ups and downs of a single market’s risk, ignoring the complex web of connections across global exchanges. But in practice, volatility is a frequent flyer — shocks in New York can ripple into London, Frankfurt, and beyond within hours. The paper Multivariate Rough Volatility takes a decisive step toward modelling this interconnected reality.

From Rough Paths to a Rough Network

The authors extend the rough fractional stochastic volatility model of Gatheral et al. (2018) into a multivariate fractional Ornstein–Uhlenbeck (mfOU) framework. This move adds two vital capabilities:

  • Different roughness per market — each index gets its own Hurst exponent $H_i$, capturing how jagged and short-memory its volatility path is.
  • Structured cross-market dependence — contemporaneous correlation ($\rho_{ij}$) and a time-asymmetry parameter ($\eta_{ij}$) let the model account for both synchronised moves and lead–lag effects.

These enhancements let the model reproduce empirical quirks often seen in real data: volatility in one market persistently nudging another, asymmetric cross-covariance decay, and near–non-stationary mean reversion.

Estimating the Global Web

To fit this high-dimensional beast, the authors deploy a two-step Generalised Method of Moments (GMM), matching model-implied auto- and cross-covariances at selected lags to their empirical counterparts. The parameters are estimated jointly, avoiding the “one series at a time” blind spot of many previous approaches.

Monte Carlo tests show that the method recovers most parameters accurately, with a known challenge: when mean reversion speeds $\alpha$ are very low, bias creeps in — a familiar issue in persistent volatility processes.

A 22-Market Reality Check

Using 22 global equity volatility series from the Oxford–Man Realized Library (2000–2022), the fitted model reveals:

  • Roughness is universal — all $H_i < 0.5$, consistent with the rough volatility literature.
  • High transatlantic synchronicity — $\rho$ between SPX and DJI hits 0.98; European blue chips also cluster tightly.
  • Asia–Pacific’s distinct tempo — correlations with Europe/US are moderate, and asymmetry $|\eta_{ij}|$ is strongest for pairs involving Korea, China, and Pakistan.

Quantifying Volatility Spillovers

The model’s time-asymmetry feature feeds naturally into a Diebold–Yilmaz-style spillover analysis. Results:

  • Total spillovers consume ~86% of forecast error variance — risk is highly interconnected.
  • Net transmitters: US (SPX, DJI), UK (FTSE), and parts of Europe send more volatility than they receive.
  • Net receivers: Several Asian indices, notably Shanghai’s SSEC and Pakistan’s KSE.
  • FTSE stands out as the only index with positive net pairwise spillovers to every other market.

This fine-grained map of volatility flows can inform global hedging, portfolio diversification, and stress-testing strategies.

Why It Matters for Practitioners

For risk managers and quants, the takeaways are clear:

  • Forecasting gains — joint modelling captures lead–lag structure missed by univariate models.
  • Stress propagation insight — identifying net transmitters helps in scenario design.
  • Customised roughness — tailoring $H_i$ per asset improves fit and realism.

By treating volatility as a networked phenomenon, the mfOU approach offers a richer toolkit for understanding and managing global market risk.

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