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On the short term stability of financial ARCH price processes

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  • Gilles Zumbach

Abstract

For many financial applications, it is important to have reliable and tractable models for the behavior of assets and indexes, for example in risk evaluation. A successful approach is based on ARCH processes, which strike the right balance between statistical properties and ease of computation. This study focuses on quadratic ARCH processes and the theoretical conditions to have a stable long-term behavior. In particular, the weights for the variance estimators should sum to 1, and the variance of the innovations should be 1. Using historical data, the realized empirical innovations can be computed, and their statistical properties assessed. Using samples of 3 to 5 decades, the variance of the empirical innovations are always significantly above 1, for a sample of stock indexes, commodity indexes and FX rates. This departure points to a short term instability, or to a fast adaptability due to changing conditions. Another theoretical condition on the innovations is to have a zero mean. This condition is also investigated empirically, with some time series showing significant departure from zero.

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  • Gilles Zumbach, 2021. "On the short term stability of financial ARCH price processes," Papers 2107.06758, arXiv.org.
    • Handle: RePEc:arx:papers:2107.06758
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    1. Asger Lunde & Peter R. Hansen, 2005. "A forecast comparison of volatility models: does anything beat a GARCH(1,1)?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 20(7), pages 873-889.
    2. Gilles Zumbach, 2009. "Time reversal invariance in finance," Quantitative Finance, Taylor & Francis Journals, vol. 9(5), pages 505-515.
    3. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2000. "Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian," Multinational Finance Journal, Multinational Finance Journal, vol. 4(3-4), pages 159-179, September.
    4. Gilles Zumbach, 2021. "Tile test for back-testing risk evaluation," Quantitative Finance, Taylor & Francis Journals, vol. 21(10), pages 1605-1619, October.
    5. Fulvio Corsi, 2009. "A Simple Approximate Long-Memory Model of Realized Volatility," Journal of Financial Econometrics, Oxford University Press, vol. 7(2), pages 174-196, Spring.
    6. Andersen, Torben G & Bollerslev, Tim, 1998. "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 885-905, November.
    7. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
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    Cited by:

    1. Alexander Musaev & Andrey Makshanov & Dmitry Grigoriev, 2022. "Statistical Analysis of Current Financial Instrument Quotes in the Conditions of Market Chaos," Mathematics, MDPI, vol. 10(4), pages 1-16, February.

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