Bayesian analysis of static and dynamic Hurst parameters under stochastic volatility

Despite the fact that the use of Hurst exponent is well-established, few attempts have been made to embed this parameter in a wider framework to account for stylized facts in stock returns. For example, fat tails and volatility clustering are well known to exist in financial time series, yet a comprehensive model involving the Hurst exponent is unavailable. In this paper, we generalize standard fractional Brownian motion to account for time-varying conditional heteroskedasticity when the Hurst parameter is static or, alternatively, when it also evolves over time, in which case we use a multi-fractional Brownian motion. The generalized model delivers persistence parameters for both time-varying volatility as well as time-varying Hurst exponents. In an application to Eurostoxx-50 (daily data for 1990–2018) we find that once we allow for time variation in the Hurst parameter, the persistence of conditional variances is substantially smaller. Model comparison shows that there are aspects introduced by the Hurst exponent that are not captured by stochastic volatility or other models with time-varying conditional variances. Log-predictive densities are used to compare different models out-of-sample.