  For a given time horizon DT, this article explores the relationship between
the realized volatility (the volatility that will occur between t and t+DT),
the implied volatility (corresponding to at-the-money option with expiry at
t+DT), and several forecasts for the volatility build from multi-scales linear
ARCH processes. The forecasts are derived from the process equations, and the
parameters set a priori. An empirical analysis across multiple time horizons DT
shows that a forecast provided by an I-GARCH(1) process (1 time scale) does not
capture correctly the dynamic of the realized volatility. An I-GARCH(2) process
(2 time scales, similar to GARCH(1,1)) is better, while a long memory LM-ARCH
process (multiple time scales) replicates correctly the dynamic of the realized
volatility and delivers consistently good forecast for the implied volatility.
The relationship between market models for the forward variance and the
volatility forecasts provided by ARCH processes is investigated. The structure
of the forecast equations is identical, but with different coefficients. Yet
the process equations for the variance are very different (postulated for a
market model, induced by the process equations for an ARCH model), and not of
any usual diffusive type when derived from ARCH.
