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Multifractal Fluctuations In Finance

Author

Listed:
  • FRANCOIS SCHMITT

    (Dept. of Fluid Mechanics VUB, 2 Pleinlaan, B-1050 Brussels, Belgium)

  • DANIEL SCHERTZER

    (LMM, University of Paris VI, 4, Place Jussieu, F-75005 Paris, France)

  • SHAUN LOVEJOY

    (McGILL University, Physics Department, 3600 University Street, Montreal H3A2T8, Canada)

Abstract

We consider the structure functionsS(q)(τ), i.e. the moments of orderqof the incrementsX(t + τ)-X(t)of the Foreign Exchange rateX(t)which give clear evidence of scaling(S(q)(τ)∝τζ(q)). We demonstrate that the nonlinearity of the observed scaling exponentζ(q)is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity ofζ(q)corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponentqD> 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.

Suggested Citation

  • Francois Schmitt & Daniel Schertzer & Shaun Lovejoy, 2000. "Multifractal Fluctuations In Finance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(03), pages 361-364.
    • Handle: RePEc:wsi:ijtafx:v:03:y:2000:i:03:n:s0219024900000206
    DOI: 10.1142/S0219024900000206
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    Cited by:

    1. Rodriguez-Romo, Suemi & Sosa-Herrera, Antonio, 2013. "Lacunarity and multifractal analysis of the large DLA mass distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(16), pages 3316-3328.
    2. Pont, Oriol & Turiel, Antonio & Pérez-Vicente, Conrad J., 2009. "Empirical evidences of a common multifractal signature in economic, biological and physical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(10), pages 2025-2035.
    3. Cao, Guangxi & Cao, Jie & Xu, Longbing, 2013. "Asymmetric multifractal scaling behavior in the Chinese stock market: Based on asymmetric MF-DFA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 797-807.
    4. Cao, Guangxi & Xu, Longbing & Cao, Jie, 2012. "Multifractal detrended cross-correlations between the Chinese exchange market and stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4855-4866.
    5. Carlos Pedro Gonc{c}alves, 2018. "Financial Risk and Returns Prediction with Modular Networked Learning," Papers 1806.05876, arXiv.org.
    6. Sergio Da Silva & Annibal Figueiredo & Iram Gleria & Raul Matsushita, 2007. "Hurst exponents, power laws, and efficiency in the Brazilian foreign exchange market," Economics Bulletin, AccessEcon, vol. 7(1), pages 1-11.
    7. Subbotin, Alexandre, 2009. "Volatility Models: from Conditional Heteroscedasticity to Cascades at Multiple Horizons," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 15(3), pages 94-138.
    8. Zhou, Wei-Xing, 2012. "Finite-size effect and the components of multifractality in financial volatility," Chaos, Solitons & Fractals, Elsevier, vol. 45(2), pages 147-155.
    9. La Spada Gabriele & Lillo Fabrizio, 2014. "The effect of round-off error on long memory processes," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(4), pages 445-482, September.
    10. Marina Resta & Davide Sciutti, "undated". "A characterization of self-affine processes in finance through the scaling function," Modeling, Computing, and Mastering Complexity 2003 13, Society for Computational Economics.
    11. Rossitsa Yalamova, 2012. "Fractal Measures in Market Microstructure Research," Multinational Finance Journal, Multinational Finance Journal, vol. 16(1-2), pages 137-154, March - J.
    12. Brouty, Xavier & Garcin, Matthieu, 2024. "Fractal properties, information theory, and market efficiency," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    13. V. Gontis, 2002. "Multiplicative Stochastic Model of the Time Interval between Trades in Financial Markets," Papers cond-mat/0211317, arXiv.org.
    14. Indranil Mukherjee & Amitava Sarkar, 2011. "Complexity, Financial Markets and their Scaling Laws," DEGIT Conference Papers c016_008, DEGIT, Dynamics, Economic Growth, and International Trade.
    15. R. P. Datta, 2023. "Analysis of Indian foreign exchange markets: A Multifractal Detrended Fluctuation Analysis (MFDFA) approach," Papers 2306.16162, arXiv.org.

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