A bivariate rough volatility model

Dugo Ranieri, University of Rome Tor Vergata
Giorgio Giacomo , Department of Mathematics, University of Rome Tor Vergata

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 bivariate fractional Ornstein-Uhlenbeck (2fOU) process. This is a mean reverting Gaussian process with fractal features living in R^2. It is the solution of the Langevin equation with the multivariate fractional Brownian motion, in the sense of Amblard et al. (2012), as driving term. This model is a multivariate version of the rough fractional stochastic volatility (RFSV) model introduced by Gatheral, Jaisson, and Rosenbaum (2018). We discuss the main features of the process and propose different estimation procedures to identify its parameters. One is a two-step method that takes as given the parameters governing the univariate marginals, whereas the other tries to identify all the parameters at once. Regarding the first one, the estimation of the univariate process is well documented in the literature, but it often presents a bias in the mean-reversion parameter, an issue that we also try to tackle. We derive the asymptotic properties of the estimators and adopt a finite-sample bootstrap implementation for inference. Finally, an empirical investigation is carried out on 8 realized volatility time series overlapping in time and available on a long sampling period. Our results show how realized volatility time series are strongly correlated and present different degrees of asymmetry in their cross-covariance structure, often referred to as spillover effects. These features can all be well captured by our model. Moreover, in accordance with the existing literature, we observe behaviors close to nonstationarity and roughness in the trajectories. A forecasting exercise provides further evidence of the consistency of our model with observed data.

Area: IS17 - Fractional volatility modeling in mathematical finance: from estimation to approximation methods (Stefano De Marco)

Keywords: rough volatility, realized volatility, mathematical finance, financial econometrics

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