Multivariate Rough Volatility

TL;DR

This paper develops a multivariate rough-volatility model by letting log-volatilities follow a multivariate fractional Ornstein-Uhlenbeck process driven by a mfBm, allowing component-wise Hurst exponents and cross-component interactions. It introduces a two-step GMM estimator to jointly identify all parameters, proves asymptotic normality, and validates performance through extensive Monte Carlo experiments. Applying the methodology to twenty years of Oxford-Man realized-volatility data (and intraday spot-volatility proxies) reveals strong cross-asset correlations, asymmetric cross-covariances, and significant spillover effects, with the model effectively capturing both cross-sectional dependence and slow mean reversion. The results support rough volatility as a robust framework for modeling multivariate realized volatility and offer a tractable approach to spillover analysis and multivariate forecasting in financial risk management.

Abstract

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.

Figures (16)

• Figure 1: Kernel estimates of the densities of the elements in $(\hat{\theta}_n-\theta_0)/\widehat{s.e.}(\hat{\theta}_n)$, where $\widehat{s.e.}(\hat{\theta}_n)$ is the MC standard error of the GMM estimator $\hat{\theta}_n$. Parameter settings are in Table \ref{['tab:mc1']}. Simulation parameters: $N = 2,\ M = 10^4,\ \Delta = 1/252,\ T = 20,\ \mu_{i}=0,\ i=1,2$.
• Figure 2: Bias and standard error of the GMM estimator as a function of the dimensionality of the mfOU process. Parameters are identical across components, $\alpha_i=1,\nu_i=1,H_i=0.1,\rho_{i,j}=0.5,\eta_{i,j}=0,\ i,j=1,\dots,N,\ M=10^3,\ \Delta = 1/252,\ T=20.$
• Figure 3: Kernel estimates of the densities of the elements in $(\hat{\theta}_n-\theta_0)/\widehat{s.e.}(\hat{\theta}_n)$, where $\hat{\theta}_n$ denotes the GMM estimator that uses asymptotic cross-covariance conditions, and $\widehat{s.e.}(\hat{\theta}_n)$ is the MC standard error of $\hat{\theta}_n$. Parameter settings are in Table \ref{['tab:mc2']}. Simulation parameters: $N = 2,\ M = 10^4,\ \Delta = 1/252,\ T = 20,\ \mu_{i}=0,\ i=1,2$.
• Figure 4: Undirected graph representation of the estimates of $\rho_{i,j},\ i,j=1,\dots,N,\ i\ne j$. The nodes correspond to indices and the edges among pairs of indices have length inversely proportional to the GMM estimates of $\rho_{i,j}$ reported in Table \ref{['tab:rho']}.
• Figure 5: Empirical cross-covariances of log-realized volatilities as blue bars, alongside the theoretical cross-covariances from our model, indicated by the red curves and based on the estimated parameters from Tables \ref{['tab:univ']}, \ref{['tab:rho']}, and \ref{['tab:eta']}. The upper panel shows the pair FTSE-SPX, while the lower panel shows the pair FTSE-SSEC.

Theorems & Definitions (8)

• Proposition 1
• Proposition 2
• Proposition 3
• Remark 1
• Remark 2
• Proposition 4
• Remark 3
• Remark 4