Fractional Stochastic Volatility Model

This paper introduces a discrete-time fractional stochastic volatility model (FSV) based on fractional Gaussian noise. The new model includes the standard stochastic volatility model as a special case and has the same limit as the fractional integrated stochastic volatility (FISV) model. A simulated maximum likelihood method, which maximizes the time-domain log-likelihood function calculated by the importance sampling technique, and a frequency-domain quasi maximum likelihood method (or Whittle) are employed to estimate the model parameters. Simulation studies suggest that, while both estimation methods can accurately estimate the model, the simulated maximum likelihood method outperforms the Whittle method. As an illustration, we fit the FSV and FISV models with the proposed estimation techniques to the S\&P 500 composite index over a sample period spanning 45 years. Our results reveal that the volatilities of the data series are persistent and rough.