ARCH Processes

The ARCH processes are very important as they can capture correctly the empirical stylized facts regarding the heteroscedasticity. The GARCH(1,1) process is introduced first. Two equivalent formulations are presented: the original formulation of Engle and another form that is easier to understand intuitively and that allows for natural multiscale generalizations. This process is investigated in detail, in particular volatility forecasts and lagged correlations as they are the step stones for more complex analytical computations. Some variations around GARCH(1,1) are introduced, like I-GARCH(1), I-GARCH(2), and EGARCH(1,1). This set of simple processes allows one to understand the generic differences between linear and affine processes, and the implications for mean reversion, integrated processes, and the asymptotic properties. These simple processes pave the way for the rich family of multicomponent ARCH processes. Several variations of the multicomponent processes are studied, with a trade-off between simplicity and analytical computations versus more accurate stylized facts. In particular, these processes can reproduce the long memory observed in the empirical data. The mug shots are used to display the long-memory heteroscedasticity and the non-time reversal invariance. For a subclass of the multicomponent ARCH processes, a volatility forecast can be derived analytically. Then, the FIGARCH process is presented, and the difficulties related to the fractional difference operator with a finite cut-off are discussed. Advanced sections are devoted to the trend effect, the sensitivity with respect to the estimated parameters, the long-term dynamics of the mean volatility, and the empirical finding that the ARCH processes are always close to an instability limit. The induced dynamic for the volatility is derived from the ARCH equations, showing that it is a new type of process, clearly different from the stochastic volatility equation. The chapter concludes with a suggestion for a good and robust overall ARCH volatility process, with fixed parameters, that can be used in most practical situations.