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

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  • Gilles Zumbach

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.

Suggested Citation

  • Gilles Zumbach, 2009. "Volatility forecasts and the at-the-money implied volatility: a multi-components ARCH approach and its relation with market models," Papers 0901.2275, arXiv.org.
    • Handle: RePEc:arx:papers:0901.2275
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    References listed on IDEAS

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    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    3. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
    4. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    5. Zumbach, Gilles & Lynch, Paul, 2001. "Heterogeneous volatility cascade in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 298(3), pages 521-529.
    6. Paul Lynch & Gilles Zumbach, 2003. "Market heterogeneities and the causal structure of volatility," Quantitative Finance, Taylor & Francis Journals, vol. 3(4), pages 320-331.
    7. Gilles Zumbach & Paul Lynch, 2001. "Heterogeneous volatility cascade in financial markets," Papers cond-mat/0105162, arXiv.org.
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