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Random cascade model in the limit of infinite integral scale as the exponential of a non-stationary $1/f$ noise. Application to volatility fluctuations in stock markets

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  • J. F. Muzy
  • R. Baile
  • E. Bacry

Abstract

In this paper we propose a new model for volatility fluctuations in financial time series. This model relies on a non-stationary gaussian process that exhibits aging behavior. It turns out that its properties, over any finite time interval, are very close to continuous cascade models. These latter models are indeed well known to reproduce faithfully the main stylized facts of financial time series. However, it involve a large scale parameter (the so-called "integral scale" where the cascade is initiated) that is hard to interpret in finance. Moreover the empirical value of the integral scale is in general deeply correlated to the overall length of the sample. This feature is precisely predicted by our model that turns out, as illustrated on various examples from daily stock index data, to quantitatively reproduce the empirical observations.

Suggested Citation

  • J. F. Muzy & R. Baile & E. Bacry, 2013. "Random cascade model in the limit of infinite integral scale as the exponential of a non-stationary $1/f$ noise. Application to volatility fluctuations in stock markets," Papers 1301.4160, arXiv.org.
    • Handle: RePEc:arx:papers:1301.4160
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    Cited by:

    1. Wu, Peng & Muzy, Jean-François & Bacry, Emmanuel, 2022. "From rough to multifractal volatility: The log S-fBM model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    2. Grahovac, Danijel, 2020. "Multifractal processes: Definition, properties and new examples," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    3. Grahovac, Danijel, 2022. "Intermittency in the small-time behavior of Lévy processes," Statistics & Probability Letters, Elsevier, vol. 187(C).
    4. Forde, Martin & Fukasawa, Masaaki & Gerhold, Stefan & Smith, Benjamin, 2022. "The Riemann–Liouville field and its GMC as H→0, and skew flattening for the rough Bergomi model," Statistics & Probability Letters, Elsevier, vol. 181(C).
    5. Forde, Martin & Smith, Benjamin, 2020. "The conditional law of the Bacry–Muzy and Riemann–Liouville log correlated Gaussian fields and their GMC, via Gaussian Hilbert and fractional Sobolev spaces," Statistics & Probability Letters, Elsevier, vol. 161(C).
    6. Dai, Meifeng & Hou, Jie & Ye, Dandan, 2016. "Multifractal detrended fluctuation analysis based on fractal fitting: The long-range correlation detection method for highway volume data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 722-731.
    7. Cristina Sattarhoff & Marc Gronwald, 2018. "How to Measure Financial Market Efficiency? A Multifractality-Based Quantitative Approach with an Application to the European Carbon Market," CESifo Working Paper Series 7102, CESifo.

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