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

TL;DR

This paper compares multi-scale ARCH models to market models for volatility forecasting, showing that long memory ARCH processes best replicate realized volatility and implied volatility dynamics across multiple time horizons.

Contribution

It introduces a multi-components ARCH approach to volatility forecasting and analyzes its relation to market models, highlighting the effectiveness of long memory ARCH processes.

Findings

Long memory LM-ARCH accurately replicates realized volatility dynamics.

Two-scale I-GARCH(2) improves over single-scale I-GARCH(1).

Forecast equations share structure but differ in coefficients from market models.

Abstract

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.