Long memory and periodicity in intraday volatilities of stock index futures

This paper investigates the intraday volatility pattern of the E-mini SP500, quoted at the Chicago Mercantile Exchange, one of the most traded American Stock Index futures. The data set consists of round-the-clock hourly returns. The squared (and absolute) returns are characterized by long memory and periodicity. In order to jointly model the long memory and the periodic components in the returns volatility we introduce two new parameterizations. The Fractionally Integrated Periodic EGARCH (FI-PEGARCH) and the Seasonal Fractional Integrated Periodic EGARCH (SFI-PEGARCH). For both models we compute the population kurtosis and the autocorrelation function of power transformations of absolute returns. We find that during the Asian and European trading time the volatility is lower than during the American trading time when we observe a sharp increase. The results seem to confirm the fact that hourly returns sampled over the 24 hours across different markets are characterized by a strong seasonal pattern with a statistically significant persistence. Finally we present the in-sample and out-of-sample forecasts results of unrestricted and restricted long memory periodic volatility models.