Volatility tail risk under fractionality

We study the probabilistic properties of the fractional Ornstein–Uhlenbeck process, which is a relevant framework for volatility modeling in continuous time. First, we compute an expression for its variance for any value of the Hurst parameter, H ∈ (0, 1). Second, we derive the density of the process and we calculate the probability of its supremum to be above a given threshold. We provide a number of illustrations based on fractional stochastic volatility models, such as those of Comte and Renault (1998), Bayer et al. (2016) and Gatheral et al. (2018). Finally, the empirical analysis, based on the realized variance series of S&P500, shows the usefulness of these theoretical results for risk management purposes, especially when a characterization of the volatility tail risk is needed.