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The fine-structure of volatility feedback I: multi-scale self-reflexivity

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  • R\'emy Chicheportiche
  • Jean-Philippe Bouchaud
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    Abstract

    We attempt to unveil the fine structure of volatility feedback effects in the context of general quadratic autoregressive (QARCH) models, which assume that today's volatility can be expressed as a general quadratic form of the past daily returns. The standard ARCH or GARCH framework is recovered when the quadratic kernel is diagonal. The calibration of these models on US stock returns reveals several unexpected features. The off-diagonal (non ARCH) coefficients of the quadratic kernel are found to be highly significant both In-Sample and Out-of-Sample, but all these coefficients turn out to be one order of magnitude smaller than the diagonal elements. This confirms that daily returns play a special role in the volatility feedback mechanism, as postulated by ARCH models. The feedback kernel exhibits a surprisingly complex structure, incompatible with models proposed so far in the literature. Its spectral properties suggest the existence of volatility-neutral patterns of past returns. The diagonal part of the quadratic kernel is found to decay as a power-law of the lag, in line with the long-memory of volatility. Finally, QARCH models suggest some violations of Time Reversal Symmetry in financial time series, which are indeed observed empirically, although of much smaller amplitude than predicted. We speculate that a faithful volatility model should include both ARCH feedback effects and a stochastic component.

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    File URL: http://arxiv.org/pdf/1206.2153
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    Paper provided by arXiv.org in its series Papers with number 1206.2153.

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    Date of creation: Jun 2012
    Date of revision: Sep 2013
    Publication status: Published in Physica A 410 (2014) 174-195
    Handle: RePEc:arx:papers:1206.2153

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    Web page: http://arxiv.org/

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    8. Bekaert, Geert & Wu, Guojun, 2000. "Asymmetric Volatility and Risk in Equity Markets," Review of Financial Studies, Society for Financial Studies, vol. 13(1), pages 1-42.
    9. Jean-Philippe Bouchaud & Julien Kockelkoren & Marc Potters, 2004. "Random walks, liquidity molasses and critical response in financial markets," Science & Finance (CFM) working paper archive 500063, Science & Finance, Capital Fund Management.
    10. David H. Cutler & James M. Poterba & Lawrence H. Summers, 1988. "What Moves Stock Prices?," Working papers 487, Massachusetts Institute of Technology (MIT), Department of Economics.
    11. Jean-Philippe Bouchaud & Yuval Gefen & Marc Potters & Matthieu Wyart, 2003. "Fluctuations and response in financial markets: the subtle nature of `random' price changes," Papers cond-mat/0307332, arXiv.org, revised Aug 2003.
    12. Gilles Zumbach, 2010. "Volatility conditional on price trends," Quantitative Finance, Taylor & Francis Journals, vol. 10(4), pages 431-442.
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