Volatility processes and volatility forecast with long memory

We introduce a new family of processes that include the long memory (LM) (power law) in the volatility correlation. This is achieved by measuring the historical volatilities on a set of increasing time horizons and by computing the resulting effective volatility by a sum with power law weights. The processes have two parameters (linear processes) or four parameters (affine processes). In the limit where only one component is included, the processes are equivalent to GARCH(1, 1) and I-GARCH(1). Volatility forecast is discussed in the context of processes with quadratic equations, in particular as a means to estimate process parameters. Using hourly data, the empirical properties of the new processes are compared to existing processes (GARCH, I-GARCH, FIGARCH,…), in particular log-likelihood estimates and volatility forecast errors. This study covers time horizons ranging from 1 h to 1 month. We also study the variation of the estimated parameters with respect to changing sample by introducing a natural coordinate invariant distance. The LM processes show a small but systematic quantitative improvement with respect to the standard GARCH(1, 1) process. Yet, the main advantage of the new LM processes is that they give a good description of the empirical data from 1 h to 1 month, with the same parameters. Their other advantages are that they are efficient to evaluate numerically, that they behave well with respect to the cut-off (i.e. the largest time horizon included in the process) and that they can be extended along several directions.