Multivariate Rough Volatility

Motivated by empirical evidence from the joint behavior of realized volatility time series, we propose to model the joint dynamics of log-volatilities using a multivariate fractional Ornstein-Uhlenbeck process. This model is a multivariate version of the Rough Fractional Stochastic Volatility model introduced in [Gatheral, Jaisson, and Rosenbaum, Quant. Finance, 2018]. It allows for different Hurst exponents in the different marginal components and non trivial interdependencies. We discuss the main features of the model and propose a Generalized Method of Moments estimator that jointly identifies its parameters. We derive the asymptotic theory of the estimator and perform a simulation study that confirms the asymptotic theory in finite sample. We conduct an extensive empirical investigation of all realized-volatility time series covering the entire span of about two decades in the Oxford-Man realized library, and of a small spot-volatility system. Our analysis shows that these time series are strongly correlated and can exhibit asymmetries in their empirical cross-covariance function, accurately captured by our model. These asymmetries lead to spillover effects, which we derive analytically within our model and compute based on empirical estimates of model parameters. Moreover, in accordance with the existing literature, we observe behaviors close to non-stationarity and rough trajectories.

💡 Research Summary

The paper introduces a multivariate extension of the Rough Fractional Stochastic Volatility (RFSV) framework by modeling the joint dynamics of log‑realized volatilities with a multivariate fractional Ornstein‑Uhlenbeck (mfOU) process. The driving noise is a multivariate fractional Brownian motion (mfBm) characterized by three sets of parameters: a vector of Hurst exponents (H=(H_1,\dots,H_N)) allowing each component to have its own roughness, a contemporaneous correlation matrix (\rho), and an antisymmetric matrix (\eta) that governs time‑reversibility (asymmetry) of the cross‑covariance function. When (\eta=0) the cross‑covariances are symmetric; otherwise they exhibit the empirically observed lag‑dependent asymmetry, which the authors link to volatility spill‑over effects.

Each component (Y_i(t)) of the mfOU satisfies
(dY_i(t)=\alpha_i(\mu_i-Y_i(t))dt+\nu_i dW^{H_i}t),
with mean‑reversion speed (\alpha_i>0), long‑run mean (\mu_i), and diffusion coefficient (\nu_i>0). The stationary solution can be written as an integral over the past mfBm, inheriting both long‑memory (through the Hurst exponents) and mean‑reverting behavior (through (\alpha_i)). The authors prove two propositions: (1) as (\alpha_i\to0) the mfOU converges in (L^2) to the underlying mfBm, establishing a bridge to the univariate rough volatility literature; (2) for small (\alpha_i,\alpha_j) the cross‑covariance admits a simple expansion (\gamma{ij}(k)=\gamma_{ij}(0)-\rho_{ij}+\eta_{ij}^2\nu_i\nu_j k^{H_i+H_j}+o(1)), highlighting the role of (\eta) and (\alpha) in generating asymmetry.

Because the mfOU is non‑Markovian, maximum‑likelihood estimation is computationally infeasible. The authors therefore develop a two‑step Generalized Method of Moments (GMM) estimator. Moment conditions are built by matching model‑implied cross‑covariances (derived analytically from the mfOU) with empirical sample covariances over a chosen set of lags (L). The estimator minimizes a quadratic loss ((\hat\gamma-\gamma(\theta))‘W_n(\hat\gamma-\gamma(\theta))). Under standard regularity conditions and assuming all Hurst exponents satisfy (H_i<3/4), they establish consistency and asymptotic normality with (\sqrt{n}) convergence, providing explicit expressions for the asymptotic variance matrix involving the Jacobian of the moment functions.

Monte‑Carlo experiments with sample sizes ranging from 500 to 2000 confirm the theoretical rates: bias and variance shrink as predicted, and the estimation of the interdependence parameters (\rho) and (\eta) improves markedly with larger (n).

The empirical application uses the Oxford‑Man realized‑volatility library, covering 22 major equity indices over roughly 20 years. Estimated Hurst exponents cluster around 0.1–0.2, confirming the “rough” nature of volatility. The contemporaneous correlation matrix (\rho) exhibits strong positive dependence across markets, while the antisymmetric matrix (\eta) contains several significant non‑zero entries, indicating pronounced asymmetry in cross‑covariances. By plugging the estimated parameters into the analytical spill‑over formulas, the authors quantify how shocks to one market’s volatility propagate asymmetrically to others, a phenomenon that aligns with observed cross‑covariance decay patterns. A secondary analysis on a small spot‑volatility system for US indices reproduces the same roughness and asymmetry features, demonstrating the model’s flexibility.

In summary, the paper delivers a comprehensive theoretical and empirical framework for multivariate rough volatility. It provides (i) a mathematically rigorous mfOU specification that captures heterogeneous roughness and time‑reversibility, (ii) a feasible GMM estimation procedure with proven asymptotic properties, (iii) simulation evidence validating the estimator, and (iv) an extensive real‑world study showing that the model accurately reproduces observed cross‑covariance asymmetries and spill‑over effects. These contributions advance the state of the art in financial econometrics, offering a powerful tool for risk management, portfolio allocation, and option pricing in environments where multiple assets exhibit jointly rough and interdependent volatility dynamics.