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.

Counterparty Credit Risk (CCR) represents one of the major sources of uncertainty in many financial contracts. The role of credit value adjustment (CVA) is, in fact, that of rewarding the parties for the exposure to such risk. A key driver of CVA is the recovery risk, generated by the variability of recovery rates. In this paper, we develop a framework to assess the CCR accounting for the recovery risk that arises from the introduction of stochastic recovery rates. Adopting the structural model for the time to default that exploits a time-changed Lévy process for the risk driver of the equity value, we provide a complete picture to monitor the CCR and gauge the effects of the stochastic recovery rates. The model extracts information on the creditworthiness of the parties in the OTC contract combining Fourier Cosine Expansion and Monte Carlo simulations methods to price CDS spreads, the related underlying, and to retrieve the default barrier. We apply the model proposed to a business case analyzing the CCR of two parties involved in the OTC contract with underlying energy commodities. Low average recovery rates reveal to be associated with high implied volatility and depart from the fixed value of 40%, especially during periods of market distress.