Parametric Estimation in Fractional Stochastic Differential Equation

Fractional Stochastic Differential Equations are becoming more popular in the literature as they can model phenomena in financial data that typical Stochastic Differential Equations models cannot. In the formulation considered here, the Hurst parameter, H, controls the Fraction of Differentiation, which needs to be estimated from the data. Fortunately, the covariance structure among observations in time is easily expressed in terms of the Hurst parameter which means that a likelihood is easily defined. This work derives the Maximum Likelihood Estimator for H, which shows that it is biased and is not a consistent estimator. Simulation data used to understand the bias of the estimator is used to create an empirical bias correction function and a bias-corrected estimator is proposed and studied. Via simulation, the bias-corrected estimator is shown to be minimally biased and its simulation-based standard error is created, which is then used to create a 95% confidence interval for H. A simulation study shows that the 95% confidence intervals have decent coverage probabilities for large n. This method is then applied to the S&P500 and VIX data before and after the 2008 financial crisis.