IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v343y2004icp603-622.html
   My bibliography  Save this article

Multifractal model of asset returns with leverage effect

Author

Listed:
  • Eisler, Z.
  • Kertész, J.

Abstract

Multifractal processes are a relatively new tool of stock market analysis. Their power lies in the ability to take multiple orders of autocorrelations into account explicitly. In the first part of the paper we discuss the framework of the Lux model and refine the underlying phenomenological picture. We also give a procedure of fitting all parameters to empirical data. We present a new approach to account for the effective length of power-law memory in volatility. The second part of the paper deals with the consequences of asymmetry in returns. We incorporate two related stylized facts, skewness and leverage autocorrelations into the model. Then from Monte Carlo measurements we show, that this asymmetry significantly increases the mean squared error of volatility forecasts. Based on a filtering method we give evidence on similar behavior in empirical data.

Suggested Citation

  • Eisler, Z. & Kertész, J., 2004. "Multifractal model of asset returns with leverage effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 343(C), pages 603-622.
    • Handle: RePEc:eee:phsmap:v:343:y:2004:i:c:p:603-622
    DOI: 10.1016/j.physa.2004.05.061
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437104007307
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2004.05.061?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bacry, E. & Delour, J. & Muzy, J.F., 2001. "Modelling financial time series using multifractal random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 84-92.
    2. Thomas Lux, 2003. "The Multi-Fractal Model of Asset Returns:Its Estimation via GMM and Its Use for Volatility Forecasting," Computing in Economics and Finance 2003 14, Society for Computational Economics.
    3. H. E. Stanley & V. Plerou, 2001. "Scaling and universality in economics: empirical results and theoretical interpretation," Quantitative Finance, Taylor & Francis Journals, vol. 1(6), pages 563-567.
    4. Fabrizio Lillo & Rosario N. Mantegna, 2000. "Symmetry alteration of ensemble return distribution in crash and rally days of financial markets," Papers cond-mat/0002438, arXiv.org.
    5. 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.
    6. Yanhui Liu & Parameswaran Gopikrishnan & Pierre Cizeau & Martin Meyer & Chung-Kang Peng & H. Eugene Stanley, 1999. "The statistical properties of the volatility of price fluctuations," Papers cond-mat/9903369, arXiv.org, revised Mar 1999.
    7. Ghysels, E. & Jasiak, J., 1994. "Stochastic Volatility and time Deformation: An Application of trading Volume and Leverage Effects," Cahiers de recherche 9403, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    8. Perelló, Josep & Masoliver, Jaume & Anento, Napoleón, 2004. "A comparison between several correlated stochastic volatility models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 134-137.
    9. Gopikrishnan, P. & Plerou, V. & Gabaix, X. & Amaral, L.A.N. & Stanley, H.E., 2001. "Price fluctuations and market activity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 137-143.
    10. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    11. Parameswaran Gopikrishnan & Vasiliki Plerou & Luis A. Nunes Amaral & Martin Meyer & H. Eugene Stanley, 1999. "Scaling of the distribution of fluctuations of financial market indices," Papers cond-mat/9905305, arXiv.org.
    12. Calvet, Laurent & Fisher, Adlai, 2001. "Forecasting multifractal volatility," Journal of Econometrics, Elsevier, vol. 105(1), pages 27-58, November.
    13. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
    14. Laurent Calvet & Adlai Fisher & Benoit Mandelbrot, 1997. "Large Deviations and the Distribution of Price Changes," Cowles Foundation Discussion Papers 1165, Cowles Foundation for Research in Economics, Yale University.
    15. Bonanno, Giovanni & Lillo, Fabrizio & Mantegna, Rosario N, 2000. "Dynamics of the number of trades of financial securities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 280(1), pages 136-141.
    16. Kantelhardt, Jan W. & Zschiegner, Stephan A. & Koscielny-Bunde, Eva & Havlin, Shlomo & Bunde, Armin & Stanley, H.Eugene, 2002. "Multifractal detrended fluctuation analysis of nonstationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 87-114.
    17. Eric Ghysels & Christian Gouriéroux & Joann Jasiak, 1995. "Trading Patterns, Time Deformation and Stochastic Volatility in Foreign Exchange Markets," CIRANO Working Papers 95s-42, CIRANO.
    18. Brockwell, P. J. & Dahlhaus, R., 2004. "Generalized Levinson-Durbin and Burg algorithms," Journal of Econometrics, Elsevier, vol. 118(1-2), pages 129-149.
    19. Marc Potters & Jean-Philippe Bouchaud, 2001. "More stylized facts of financial markets: leverage effect and downside correlations," Science & Finance (CFM) working paper archive 29960, Science & Finance, Capital Fund Management.
    20. V. Plerou & P. Gopikrishnan & X. Gabaix & L. A. N. Amaral & H. E. Stanley, 2001. "Price fluctuations, market activity and trading volume," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 262-269.
    21. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    22. Laurent Calvet & Adlai Fisher, 2003. "Regime-Switching and the Estimation of Multifractal Processes," NBER Working Papers 9839, National Bureau of Economic Research, Inc.
    23. V. Plerou & P. Gopikrishnan & L. A. N. Amaral & M. Meyer & H. E. Stanley, 1999. "Scaling of the distribution of price fluctuations of individual companies," Papers cond-mat/9907161, arXiv.org.
    24. Kaushik Matia & Yosef Ashkenazy & H. Eugene Stanley, 2003. "Multifractal Properties of Price Fluctuations of Stocks and Commodities," Papers cond-mat/0308012, arXiv.org.
    25. Bouchaud, Jean-Philippe & Potters, Marc, 2001. "More stylized facts of financial markets: leverage effect and downside correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 60-70.
    26. Jean-Philippe Bouchaud & Andrew Matacz & Marc Potters, 2001. "The leverage effect in financial markets: retarded volatility and market panic," Science & Finance (CFM) working paper archive 0101120, Science & Finance, Capital Fund Management.
    27. Ausloos, M., 2000. "Statistical physics in foreign exchange currency and stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 285(1), pages 48-65.
    28. Kullmann, L & Töyli, J & Kertesz, J & Kanto, A & Kaski, K, 1999. "Characteristic times in stock market indices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(1), pages 98-110.
    29. J.-P. Bouchaud & M. Potters & M. Meyer, 2000. "Apparent multifractality in financial time series," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 13(3), pages 595-599, February.
    30. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    31. Benoit Mandelbrot & Adlai Fisher & Laurent Calvet, 1997. "A Multifractal Model of Asset Returns," Cowles Foundation Discussion Papers 1164, Cowles Foundation for Research in Economics, Yale University.
    32. Jaume Masoliver & Josep Perelló, 2002. "A Correlated Stochastic Volatility Model Measuring Leverage And Other Stylized Facts," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(05), pages 541-562.
    33. A. G. Zawadowski & R. Karadi & J. Kertesz, 2002. "Price Drops, Fluctuations, and Correlation in a Multi-Agent Model of Stock Markets," Papers cond-mat/0204574, arXiv.org.
    34. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    35. Zawadowski, A.G & Karádi, R & Kertész, J, 2002. "Price drops, fluctuations, and correlation in a multi-agent model of stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 403-412.
    36. Stanley, H.E & Amaral, L.A.N & Canning, D & Gopikrishnan, P & Lee, Y & Liu, Y, 1999. "Econophysics: Can physicists contribute to the science of economics?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(1), pages 156-169.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dutta, Srimonti & Ghosh, Dipak & Chatterjee, Sucharita, 2016. "Multifractal detrended Cross Correlation Analysis of Foreign Exchange and SENSEX fluctuation in Indian perspective," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 463(C), pages 188-201.
    2. 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).
    3. Wei, Yu & Wang, Yudong & Huang, Dengshi, 2011. "A copula–multifractal volatility hedging model for CSI 300 index futures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4260-4272.
    4. Pan, Zhiyuan & Liu, Li, 2018. "Forecasting stock return volatility: A comparison between the roles of short-term and long-term leverage effects," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 168-180.
    5. Batten, Jonathan A. & Kinateder, Harald & Wagner, Niklas, 2014. "Multifractality and value-at-risk forecasting of exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 401(C), pages 71-81.
    6. Akash P. POOJARI & Siva Kiran GUPTHA & G Raghavender RAJU, 2022. "Multifractal analysis of equities. Evidence from the emerging and frontier banking sectors," Theoretical and Applied Economics, Asociatia Generala a Economistilor din Romania - AGER, vol. 0(3(632), A), pages 61-80, Autumn.
    7. 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).
    8. Wei, Yu & Chen, Wang & Lin, Yu, 2013. "Measuring daily Value-at-Risk of SSEC index: A new approach based on multifractal analysis and extreme value theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(9), pages 2163-2174.
    9. Pablo Su'arez-Garc'ia & David G'omez-Ullate, 2013. "Multifractality and long memory of a financial index," Papers 1306.0490, arXiv.org.
    10. Deniz Erer & Elif Erer & Selim Güngör, 2023. "The aggregate and sectoral time-varying market efficiency during crisis periods in Turkey: a comparative analysis with COVID-19 outbreak and the global financial crisis," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 9(1), pages 1-25, December.
    11. Onali, Enrico & Goddard, John, 2009. "Unifractality and multifractality in the Italian stock market," International Review of Financial Analysis, Elsevier, vol. 18(4), pages 154-163, September.
    12. Suárez-García, Pablo & Gómez-Ullate, David, 2014. "Multifractality and long memory of a financial index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 394(C), pages 226-234.
    13. Morales Martínez, Jorge Luis & Segovia-Domínguez, Ignacio & Rodríguez, Israel Quiros & Horta-Rangel, Francisco Antonio & Sosa-Gómez, Guillermo, 2021. "A modified Multifractal Detrended Fluctuation Analysis (MFDFA) approach for multifractal analysis of precipitation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    14. Zunino, Luciano & Figliola, Alejandra & Tabak, Benjamin M. & Pérez, Darío G. & Garavaglia, Mario & Rosso, Osvaldo A., 2009. "Multifractal structure in Latin-American market indices," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2331-2340.
    15. Zunino, L. & Tabak, B.M. & Figliola, A. & Pérez, D.G. & Garavaglia, M. & Rosso, O.A., 2008. "A multifractal approach for stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(26), pages 6558-6566.
    16. Chen, Wang & Wei, Yu & Lang, Qiaoqi & Lin, Yu & Liu, Maojuan, 2014. "Financial market volatility and contagion effect: A copula–multifractal volatility approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 398(C), pages 289-300.
    17. Challet, Damien & Peirano, Pier Paolo, 2008. "The ups and downs of the renormalization group applied to financial time series," MPRA Paper 9770, University Library of Munich, Germany.
    18. R. P. Datta, 2023. "Analysis of Indian foreign exchange markets: A Multifractal Detrended Fluctuation Analysis (MFDFA) approach," Papers 2306.16162, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zoltan Eisler & Janos Kertesz, 2004. "Multifractal model of asset returns with leverage effect," Papers cond-mat/0403767, arXiv.org, revised May 2004.
    2. Gu, Gao-Feng & Zhou, Wei-Xing, 2007. "Statistical properties of daily ensemble variables in the Chinese stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 383(2), pages 497-506.
    3. 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).
    4. Wang, Yudong & Wu, Chongfeng & Yang, Li, 2016. "Forecasting crude oil market volatility: A Markov switching multifractal volatility approach," International Journal of Forecasting, Elsevier, vol. 32(1), pages 1-9.
    5. Wei, Yu & Wang, Yudong & Huang, Dengshi, 2011. "A copula–multifractal volatility hedging model for CSI 300 index futures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4260-4272.
    6. Stanley, H.Eugene, 2003. "Statistical physics and economic fluctuations: do outliers exist?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 318(1), pages 279-292.
    7. Wei, Yu & Chen, Wang & Lin, Yu, 2013. "Measuring daily Value-at-Risk of SSEC index: A new approach based on multifractal analysis and extreme value theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(9), pages 2163-2174.
    8. Thomas Lux, 2009. "Applications of Statistical Physics in Finance and Economics," Chapters, in: J. Barkley Rosser Jr. (ed.), Handbook of Research on Complexity, chapter 9, Edward Elgar Publishing.
    9. Jovanovic, Franck & Schinckus, Christophe, 2017. "Econophysics and Financial Economics: An Emerging Dialogue," OUP Catalogue, Oxford University Press, number 9780190205034, Decembrie.
    10. Troy Tassier, 2013. "Handbook of Research on Complexity, by J. Barkley Rosser, Jr. and Edward Elgar," Eastern Economic Journal, Palgrave Macmillan;Eastern Economic Association, vol. 39(1), pages 132-133.
    11. Antonio Doria, Francisco, 2011. "J.B. Rosser Jr. , Handbook of Research on Complexity, Edward Elgar, Cheltenham, UK--Northampton, MA, USA (2009) 436 + viii pp., index, ISBN 978 1 84542 089 5 (cased)," Journal of Economic Behavior & Organization, Elsevier, vol. 78(1-2), pages 196-204, April.
    12. Johann Lussange & Ivan Lazarevich & Sacha Bourgeois-Gironde & Stefano Palminteri & Boris Gutkin, 2021. "Modelling Stock Markets by Multi-agent Reinforcement Learning," Computational Economics, Springer;Society for Computational Economics, vol. 57(1), pages 113-147, January.
    13. Jaume Masoliver & Josep Perello, 2006. "Multiple time scales and the exponential Ornstein-Uhlenbeck stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 423-433.
    14. Wang, Lei & Liu, Lutao, 2020. "Long-range correlation and predictability of Chinese stock prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    15. Leopoldo S'anchez-Cant'u & Carlos Arturo Soto-Campos & Andriy Kryvko, 2016. "Evidence of Self-Organization in Time Series of Capital Markets," Papers 1604.03996, arXiv.org, revised Mar 2017.
    16. Selçuk, Faruk & Gençay, Ramazan, 2006. "Intraday dynamics of stock market returns and volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 375-387.
    17. Chen, Shu-Peng & He, Ling-Yun, 2010. "Multifractal spectrum analysis of nonlinear dynamical mechanisms in China’s agricultural futures markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(7), pages 1434-1444.
    18. Chen, Wang & Wei, Yu & Lang, Qiaoqi & Lin, Yu & Liu, Maojuan, 2014. "Financial market volatility and contagion effect: A copula–multifractal volatility approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 398(C), pages 289-300.
    19. Lux, Thomas & Alfarano, Simone, 2016. "Financial power laws: Empirical evidence, models, and mechanisms," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 3-18.
    20. Liu, Chang & Chang, Chuo, 2021. "Combination of transition probability distribution and stable Lorentz distribution in stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:343:y:2004:i:c:p:603-622. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.