The volatility of financial returns is characterized by rapid and large increments. We propose an extension of the Heterogeneous Autoregressive model to incorporate jumps into the dynamics of the ex-post volatility measures. Using the realized-range measures of 36 NYSE stocks, we show that there is a positive probability of jumps in volatility. A common factor in the volatility jumps is shown to be related to a set of financial covariates (such as variance risk premium, S&P500 volume, credit-default swap, and federal fund rates). The credit-default swap on US banks and variance risk premium have predictive power on expected jump moves, thus confirming the common interpretation that sudden and large increases in equity volatility can be anticipated by credit deterioration of the US bank sector as well as changes in the market expectations of future risks. Finally, the model is extended to incorporate the credit-default swap and the variance risk premium in the dynamics of the jump size and intensity.

Intraday return volatilities are characterized by the contemporaneous presence of periodicity and long memory. This paper proposes two new parameterizations of the intraday volatility: the Fractionally Integrated Periodic EGARCH and the Seasonal Fractional Integrated Periodic EGARCH, which provide the required flexibility to account for both features. The periodic kurtosis and periodic autocorrelations of power transformations of the absolute returns are computed for both models. The empirical application shows that volatility of the hourly Emini S&P 500 futures returns are characterized by a periodic leverage effect coupled with a statistically significant long-range dependence. An out-of-sample forecasting comparison with alternative models shows that a constrained version of the FI-PEGARCH provides superior forecasts. A simulation experiment is carried out to investigate the effects that sample frequency has on the fractional differencing parameter estimate.

The realized volatility of financial returns is characterized by persistence and occurrence of unpredictable large increments. To capture those features, we introduce the Multiplicative Error Model with jumps (MEM-J). When a jump component is included in the multiplicative specification, the conditional density of the realized measure is shown to be a countably infinite mixture of Gamma and K distributions. Strict stationarity conditions are derived. A Monte Carlo simulation experiment shows that maximum likelihood estimates of the model parameters are reliable even when jumps are rare events. We estimate alternative specifications of the model using a set of daily bipower measures for 7 stock indexes and 16 individual NYSE stocks. The estimates of the jump component confirm that the probability of jumps dramatically increases during the financial crises. Compared to other realized volatility models, the introduction of the jump component provides a sensible improvement in the fit, as well as for in-sample and out-of-sample volatility tail forecasts.

Multivariate GARCH models are in principle able to accommodate the features of the dynamic conditional correlations processes, although with the drawback, when the number of financial returns series considered increases, that the parameterizations entail too many parameters. In general, the interaction between model parametrization of the second conditional moment and the conditional density of asset returns adopted in the estimation determines the fitting of such models to the observed dynamics of the data. This paper aims to evaluate the interactions between conditional second moment specifications and probability distributions adopted in the likelihood computation, in forecasting volatilities and covolatilities. We measure the relative performances of alternative conditional second moment and probability distributions specifications by means of Monte Carlo simulations, using both statistical and financial forecasting loss functions.

The no-arbitrage relation between futures and spot prices implies an analogous relation between futures and spot daily ranges. The long-memory features of the range-based volatility estimators are analyzed, and fractional cointegration is tested in a semi-parametric framework. In particular, the no-arbitrage condition is used to derive a long-run relationship between volatility measures and to justify the use of a fractional vector error correction model (FVECM) to study their dynamic relationship. The out-of-sample forecasting superiority of FVECM, with respect to alternative models, is documented. The results highlight the importance of incorporating the long-run equilibrium in volatilities to obtain better forecasts, given the information content in the volatility of futures prices.