Modeling Logarithmic Volatility Under the Rough Fractional Langevin Equation: Approximation Equation and Parameter Estimation

The rough fractional stochastic volatility (RFSV) model is inherently susceptible to arbitrage due to the non-semimartingale and non-Markov properties of its fractional Brownian motion (fBm)-based volatility process. To mitigate this issue, we propose a semimartingale approximation for the volatility process, substituting the fBm with its semimartingale equivalent. We examine the convergence of the proposed volatility process toward the volatility process of the RFSV model and subsequently obtain a unique solution for the approximated process. To facilitate parameter estimation for the proposed volatility process, we establish a link between this process and the VIX index. A discretization scheme for the semimartingale approximation of fBm is developed and analyzed. Our findings suggest that the proposed volatility process effectively captures the dynamics of the VIX index and reproduces the skewness of the volatility distribution function.