IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v419y2015icp221-233.html
   My bibliography  Save this article

Variable diffusion in stock market fluctuations

Author

Listed:
  • Hua, Jia-Chen
  • Chen, Lijian
  • Falcon, Liberty
  • McCauley, Joseph L.
  • Gunaratne, Gemunu H.

Abstract

We analyze intraday fluctuations in several stock indices to investigate the underlying stochastic processes using techniques appropriate for processes with nonstationary increments. The five most actively traded stocks each contains two time intervals during the day where the variance of increments can be fit by power law scaling in time. The fluctuations in return within these intervals follow asymptotic bi-exponential distributions. The autocorrelation function for increments vanishes rapidly, but decays slowly for absolute and squared increments. Based on these results, we propose an intraday stochastic model with linear variable diffusion coefficient as a lowest order approximation to the real dynamics of financial markets, and to test the effects of time averaging techniques typically used for financial time series analysis. We find that our model replicates major stylized facts associated with empirical financial time series. We also find that ensemble averaging techniques can be used to identify the underlying dynamics correctly, whereas time averages fail in this task. Our work indicates that ensemble average approaches will yield new insight into the study of financial markets’ dynamics. Our proposed model also provides new insight into the modeling of financial markets dynamics in microscopic time scales.

Suggested Citation

  • Hua, Jia-Chen & Chen, Lijian & Falcon, Liberty & McCauley, Joseph L. & Gunaratne, Gemunu H., 2015. "Variable diffusion in stock market fluctuations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 221-233.
  • Handle: RePEc:eee:phsmap:v:419:y:2015:i:c:p:221-233
    DOI: 10.1016/j.physa.2014.10.024
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437114008620
    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

    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. 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.
    2. Lisa Borland, 2002. "A theory of non-Gaussian option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 415-431.
    3. Galluccio, S. & Caldarelli, G. & Marsili, M. & Zhang, Y.-C., 1997. "Scaling in currency exchange," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 245(3), pages 423-436.
    4. Joseph L. McCauley, 2008. "Time vs. Ensemble Averages for Nonstationary Time Series," Papers 0804.0902, arXiv.org.
    5. Anirban Chakraborti & Ioane Muni Toke & Marco Patriarca & Frederic Abergel, 2011. "Econophysics review: II. Agent-based models," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 1013-1041.
    6. Silva, A. Christian & Yakovenko, Victor M., 2007. "Stochastic volatility of financial markets as the fluctuating rate of trading: An empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 382(1), pages 278-285.
    7. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    8. Anirban Chakraborti & Ioane Muni Toke & Marco Patriarca & Frederic Abergel, 2011. "Econophysics review: I. Empirical facts," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 991-1012.
    9. Bassler, Kevin E. & Gunaratne, Gemunu H. & McCauley, Joseph L., 2008. "Empirically based modeling in financial economics and beyond, and spurious stylized facts," International Review of Financial Analysis, Elsevier, vol. 17(5), pages 767-783, December.
    10. Stanley, H.E. & Buldyrev, S.V. & Franzese, G. & Havlin, S. & Mallamace, F. & Kumar, P. & Plerou, V. & Preis, T., 2010. "Correlated randomness and switching phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(15), pages 2880-2893.
    11. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    12. Bassler, Kevin E. & Gunaratne, Gemunu H. & McCauley, Joseph L., 2006. "Markov processes, Hurst exponents, and nonlinear diffusion equations: With application to finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 369(2), pages 343-353.
    13. Gunaratne, Gemunu H. & Nicol, Matthew & Seemann, Lars & Török, Andrei, 2009. "Clustering of volatility in variable diffusion processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(20), pages 4424-4430.
    14. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    15. Zhou, Bin, 1996. "High-Frequency Data and Volatility in Foreign-Exchange Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 45-52, January.
    16. Ghosh, Sayantan & Manimaran, P. & Panigrahi, Prasanta K., 2011. "Characterizing multi-scale self-similar behavior and non-statistical properties of fluctuations in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4304-4316.
    17. McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2007. "Hurst exponents, Markov processes, and fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 1-9.
    18. Muller, Ulrich A. & Dacorogna, Michel M. & Olsen, Richard B. & Pictet, Olivier V. & Schwarz, Matthias & Morgenegg, Claude, 1990. "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis," Journal of Banking & Finance, Elsevier, vol. 14(6), pages 1189-1208, December.
    19. A. Christian Silva, 2005. "Applications of physics to finance and economics: returns, trading activity and income," Papers physics/0507022, arXiv.org.
    20. Seemann, Lars & Hua, Jia-Chen & McCauley, Joseph L. & Gunaratne, Gemunu H., 2012. "Ensemble vs. time averages in financial time series analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(23), pages 6024-6032.
    21. Anand Banerjee & Victor M. Yakovenko, 2009. "Universal patterns of inequality," Papers 0912.4898, arXiv.org, revised Apr 2010.
    22. Alejandro-Quiñones, Ángel L. & Bassler, Kevin E. & Field, Michael & McCauley, Joseph L. & Nicol, Matthew & Timofeyev, Ilya & Török, Andrew & Gunaratne, Gemunu H., 2006. "A theory of fluctuations in stock prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 363(2), pages 383-392.
    23. Fiaschi, Davide & Marsili, Matteo, 2012. "Distribution of wealth and incomplete markets: Theory and empirical evidence," Journal of Economic Behavior & Organization, Elsevier, vol. 81(1), pages 243-267.
    24. McCauley, Joseph L. & Bassler, Kevin E. & Gunaratne, Gemunu H., 2008. "Martingales, detrending data, and the efficient market hypothesis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(1), pages 202-216.
    25. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    26. Jim Griffin & Roel Oomen, 2008. "Sampling Returns for Realized Variance Calculations: Tick Time or Transaction Time?," Econometric Reviews, Taylor & Francis Journals, vol. 27(1-3), pages 230-253.
    27. Gopikrishnan, P & Plerou, V & Liu, Y & Amaral, L.A.N & Gabaix, X & Stanley, H.E, 2000. "Scaling and correlation in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 362-373.
    28. Gençay, Ramazan & Dacorogna, Michel & Muller, Ulrich A. & Pictet, Olivier & Olsen, Richard, 2001. "An Introduction to High-Frequency Finance," Elsevier Monographs, Elsevier, edition 1, number 9780122796715, August.
    29. Kevin E. Bassler & Gemunu H. Gunaratne & Joseph L. McCauley, 2006. "Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance," Papers cond-mat/0602316, arXiv.org.
    30. Oomen, Roel C.A., 2006. "Properties of Realized Variance Under Alternative Sampling Schemes," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 219-237, April.
    31. Chan, Kalok & Chung, Y. Peter & Johnson, Herb, 1995. "The Intraday Behavior of Bid-Ask Spreads for NYSE Stocks and CBOE Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 30(03), pages 329-346, September.
    32. Seemann, Lars & McCauley, Joseph L. & Gunaratne, Gemunu H., 2011. "Intraday volatility and scaling in high frequency foreign exchange markets," International Review of Financial Analysis, Elsevier, vol. 20(3), pages 121-126, June.
    33. McCauley, Joseph L., 2008. "Time vs. ensemble averages for nonstationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(22), pages 5518-5522.
    34. Dacorogna, Michael M. & Muller, Ulrich A. & Nagler, Robert J. & Olsen, Richard B. & Pictet, Olivier V., 1993. "A geographical model for the daily and weekly seasonal volatility in the foreign exchange market," Journal of International Money and Finance, Elsevier, vol. 12(4), pages 413-438, August.
    35. Frédéric Abergel & Aymen Jedidi, 2013. "A Mathematical Approach to Order Book Modelling," Post-Print hal-00621253, HAL.
    36. T. Di Matteo, 2007. "Multi-scaling in finance," Quantitative Finance, Taylor & Francis Journals, vol. 7(1), pages 21-36.
    37. Victor M. Yakovenko & J. Barkley Rosser, 2009. "Colloquium: Statistical mechanics of money, wealth, and income," Papers 0905.1518, arXiv.org, revised Dec 2009.
    38. Jain, Prem C. & Joh, Gun-Ho, 1988. "The Dependence between Hourly Prices and Trading Volume," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(03), pages 269-283, September.
    39. Frederic Abergel & Aymen Jedidi, 2010. "A Mathematical Approach to Order Book Modeling," Papers 1010.5136, arXiv.org, revised Mar 2013.
    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. Pirvu Daniela & Barbuceanu Mircea, 2016. "Recent Contributions Of The Statistical Physics In The Research Of Banking, Stock Exchange And Foreign Exchange Markets," Annals - Economy Series, Constantin Brancusi University, Faculty of Economics, vol. 2, pages 85-92, April.
    2. Hua, Jia-Chen & Roy, Sukesh & McCauley, Joseph L. & Gunaratne, Gemunu H., 2016. "Using dynamic mode decomposition to extract cyclic behavior in the stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 172-180.

    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:419:y:2015:i:c:p:221-233. See general information about how to correct material in RePEc.

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

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.