This paper revisits several existing volatility models by the light of extremal dependence, that is, serial dependence in extreme returns. First, we investigate the extremal properties of different high-frequency-based volatility processes and show that only a subset of them can generate dependence in the extremes. Second, we corroborate the empirical evidence on extremal dependence in financial returns, showing that extreme returns present strong and persistent correlation and that extreme negative returns are much more correlated than positive ones. Finally, a large empirical analysis suggests that only models exhibiting extremal dependence and endowed with a leverage component can appropriately explain extreme events.

One of the key components of financial risk management is risk measurement. This typically requires modeling, estimating and forecasting tail-related quantities of the asset returns’ conditional distribution. Recent advances in the financial econometrics literature have developed several models based on Extreme Value Theory (EVT) to carry out these tasks. The purpose of this paper is to review these methods.

The Tukey's gh distribution is widely used in situations where skewness and elongation are important features of the data. As the distribution is defined through a quantile transformation of the normal, the likelihood function cannot be written in closed form and exact maximum likelihood estimation is unfeasible. In this paper we exploit a novel approach based on a frequentist reinterpretation of Approximate Bayesian Computation for approximating the maximum likelihood estimates of the gh distribution. This method is appealing because it only requires the ability to sample the distribution. We discuss the choice of the input parameters by means of simulation experiments and provide evidence of superior performance in terms of Root-Mean-Square-Error with respect to the standard quantile estimator. Finally, we give an application to operational risk measurement.