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Multifractal analysis of foreign exchange data

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  • François Schmitt
  • Daniel Schertzer
  • Shaun Lovejoy

Abstract

In this paper we perform multifractal analyses of five daily Foreign Exchange (FX) rates. These techniques are currently used in turbulence to characterize scaling and intermittency. We show the multifractal nature of FX returns, and estimate the three parameters in the universal multifactal framework, which characterize all small and medium intensity fluctuations, at all scales. For large fluctuations, we address the question of hyperbolic (fat) tails of the distributions which are characterized by a fourth parameter, the tail index. We studied both the prices fluctuations and the returns, finding no systematic difference in the scaling exponents in the two cases. We discuss and compare our results with several recent studies, and show how the additive models are not compatible with data: Brownian, fractional Brownian, Lévy, Truncated Lévy and fractional Lévy models. We analyse in this framework the ARCH(1), GARCH(1,1) and HARCH (7) models, and show that their structure functions scaling exponents are undistinguishable from that of Brownian motion, which means that these models do not adequately describe the scaling properties of the statistics of the data. Our results indicate that there might exist a multiplicative ‘flux of financial information’, which conditions small‐scale statistics to large‐scale values, as an analogy with the energy flux in turbulence. Copyright © 1999 John Wiley & Sons, Ltd.

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  • François Schmitt & Daniel Schertzer & Shaun Lovejoy, 1999. "Multifractal analysis of foreign exchange data," Applied Stochastic Models and Data Analysis, John Wiley & Sons, vol. 15(1), pages 29-53, March.
    • Handle: RePEc:wly:apsmda:v:15:y:1999:i:1:p:29-53
    DOI: 10.1002/(SICI)1099-0747(199903)15:13.0.CO;2-Z
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    1. Christian Walter, 2020. "Sustainable Financial Risk Modelling Fitting the SDGs: Some Reflections," Sustainability, MDPI, vol. 12(18), pages 1-28, September.
    2. Barunik, Jozef & Aste, Tomaso & Di Matteo, T. & Liu, Ruipeng, 2012. "Understanding the source of multifractality in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(17), pages 4234-4251.
    3. Lux, Thomas & Morales-Arias, Leonardo, 2010. "Forecasting volatility under fractality, regime-switching, long memory and student-t innovations," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2676-2692, November.
    4. Lux, Thomas & Morales-Arias, Leonardo, 2010. "Relative forecasting performance of volatility models: Monte Carlo evidence," Kiel Working Papers 1582, Kiel Institute for the World Economy (IfW Kiel).
    5. Segnon, Mawuli & Lux, Thomas, 2013. "Multifractal models in finance: Their origin, properties, and applications," Kiel Working Papers 1860, Kiel Institute for the World Economy (IfW Kiel).
    6. Jun-ichi Maskawa & Koji Kuroda, 2020. "Model of continuous random cascade processes in financial markets," Papers 2010.12270, arXiv.org.
    7. Wu, Yue & Shang, Pengjian & Chen, Shijian, 2019. "Modified multifractal large deviation spectrum based on CID for financial market system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1331-1342.
    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. J. Doyne Farmer, 2000. "Physicists Attempt To Scale The Ivory Towers Of Finance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(03), pages 311-333.
    10. Jun-ichi Maskawa & Koji Kuroda & Joshin Murai, 2018. "Multiplicative random cascades with additional stochastic process in financial markets," Evolutionary and Institutional Economics Review, Springer, vol. 15(2), pages 515-529, December.
    11. Fernández-Martínez, M. & Sánchez-Granero, M.A. & Casado Belmonte, M.P. & Trinidad Segovia, J.E., 2020. "A note on power-law cross-correlated processes," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    12. Luo, Min & Kontosakos, Vasileios E. & Pantelous, Athanasios A. & Zhou, Jian, 2019. "Cryptocurrencies: Dust in the wind?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1063-1079.
    13. Mishra, Ankit & Bandyopadhyay, Jayendra N. & Jalan, Sarika, 2021. "Multifractal analysis of eigenvectors of small-world networks," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    14. Hallam, Mark & Olmo, Jose, 2014. "Forecasting daily return densities from intraday data: A multifractal approach," International Journal of Forecasting, Elsevier, vol. 30(4), pages 863-881.
    15. Sornette, Didier & Zhou, Wei-Xing, 2006. "Importance of positive feedbacks and overconfidence in a self-fulfilling Ising model of financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(2), pages 704-726.
    16. Richards, Gordon R., 2000. "The fractal structure of exchange rates: measurement and forecasting," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 10(2), pages 163-180, June.
    17. V. Gontis, 2002. "Multiplicative Stochastic Model of the Time Interval between Trades in Financial Markets," Papers cond-mat/0211317, arXiv.org.
    18. Grahovac, Danijel & Leonenko, Nikolai N., 2014. "Detecting multifractal stochastic processes under heavy-tailed effects," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 78-89.
    19. Gordon R. Richards, 2004. "A fractal forecasting model for financial time series," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 23(8), pages 586-601.
    20. 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.
    21. Lux, Thomas & Morales-Arias, Leonardo, 2009. "Forecasting volatility under fractality, regime-switching, long memory and student-t innovations," Kiel Working Papers 1532, Kiel Institute for the World Economy (IfW Kiel).
    22. Lux, Thomas & Morales-Arias, Leonardo & Sattarhoff, Cristina, 2011. "A Markov-switching multifractal approach to forecasting realized volatility," Kiel Working Papers 1737, Kiel Institute for the World Economy (IfW Kiel).
    23. Leonenko, Nikolai & Petherick, Stuart & Taufer, Emanuele, 2013. "Multifractal models via products of geometric OU-processes: Review and applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(1), pages 7-16.
    24. L. Borland & J. -Ph. Bouchaud, 2005. "On a multi-timescale statistical feedback model for volatility fluctuations," Papers physics/0507073, arXiv.org.
    25. Lisa Borland & Jean-Philippe Bouchaud, 2005. "On a multi-timescale statistical feedback model for volatility fluctuations," Science & Finance (CFM) working paper archive 500059, Science & Finance, Capital Fund Management.

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