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For a given time horizon DT, this article explores the relationship betweenthe realized volatility (the volatility that will occur between t and t+DT),the implied volatility (corresponding to at-the-money option with expiry att+DT), and several forecasts for the volatility build from multi-scales linearARCH processes. The forecasts are derived from the process equations, and theparameters set a priori. An empirical analysis across multiple time horizons DTshows that a forecast provided by an I-GARCH(1) process (1 time scale) does notcapture 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-ARCHprocess (multiple time scales) replicates correctly the dynamic of the realizedvolatility and delivers consistently good forecast for the implied volatility.The relationship between market models for the forward variance and thevolatility forecasts provided by ARCH processes is investigated. The structureof the forecast equations is identical, but with different coefficients. Yetthe process equations for the variance are very different (postulated for amarket model, induced by the process equations for an ARCH model), and not ofany usual diffusive type when derived from ARCH.

Volatility forecasts and the at-the-money implied volatility: a multi-components ARCH approach and its relation with market models