Persistent and Rough Volatility

This paper contributes to an ongoing debate on volatility dynamics. We introduce a discrete-time fractional stochastic volatility (FSV) model based on the fractional Gaussian noise. The new model has the same limit as the fractional integrated stochastic volatility (FISV) model under the in-fill asymptotic scheme. We study the theoretical properties of both models and introduce a memory signature plot for a model-free initial assessment. A simulated maximum likelihood (SML) method, which maximizes the time-domain log-likelihoods obtained by the importance sampling technique, is employed to estimate the model parameters. Simulation studies suggest that the SML method can accurately estimate both models. Our empirical analysis of several financial assets reveals that volatilities are both persistent and rough. It is persistent in the sense that the estimated autoregressive coefficients of the log volatilities are very close to unity, which explains the observed long-range dependent feature of volatilities. It is rough as the estimated Hurst (fractional) parameters of the FSV (FISV) model are significantly less than half (zero), which is consistent with the findings of the recent literature on ‘rough volatility’.