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Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets

Author

Listed:
  • Bassler, Kevin E.
  • McCauley, Joseph L.
  • Gunaratne, Gemunu H.

Abstract

Arguably the most important problem in quantitative finance is to understand the nature of stochastic processes that underlie market dynamics. One aspect of the solution to this problem involves determining characteristics of the distribution of fluctuations in returns. Empirical studies conducted over the last decade have reported that they are non-Gaussian, scale in time, and have power-law (or fat) tails [1–5]. However, because they use sliding interval methods of analysis, these studies implicitly assume that the underlying process has stationary increments. We explicitly show that this assumption is not valid for the Euro-Dollar exchange rate between 1999-2004. In addition, we find that fluctuations in returns of the exchange rate are uncorrelated and scale as power laws for certain time intervals during each day. This behavior is consistent with a diffusive process with a diffusion coefficient that depends both on the time and the price change. Within scaling regions, we find that sliding interval methods can generate fat-tailed distributions as an artifact, and that the type of scaling reported in many previous studies does not exist.

Suggested Citation

  • Bassler, Kevin E. & McCauley, Joseph L. & Gunaratne, Gemunu H., 2006. "Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets," MPRA Paper 2126, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:2126
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    File URL: https://mpra.ub.uni-muenchen.de/2126/1/MPRA_paper_2126.pdf
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    References listed on IDEAS

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    1. J.L. McCauley & G.h. Gunaratne, 2002. "An empirical model of volatility of returns and option pricing," Computing in Economics and Finance 2002 186, Society for Computational Economics.
    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. Rama Cont & Marc Potters & Jean-Philippe Bouchaud, 1997. "Scaling in stock market data: stable laws and beyond," Science & Finance (CFM) working paper archive 9705087, Science & Finance, Capital Fund Management.
    5. Carbone, A. & Castelli, G. & Stanley, H.E., 2004. "Time-dependent Hurst exponent in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 267-271.
    6. 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.
    7. 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.
    8. 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..
    9. Couillard, Michel & Davison, Matt, 2005. "A comment on measuring the Hurst exponent of financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 348(C), pages 404-418.
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    Citations

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    Cited by:

    1. 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.
    2. McCauley, Joseph L. & Bassler, Kevin E. & Gunaratne, Gemunu H., 2008. "Martingales, nonstationary increments, and the efficient market hypothesis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3916-3920.
    3. Kerry W. Fendick, 2013. "Pricing and Hedging Derivative Securities with Unknown Local Volatilities," Papers 1309.6164, arXiv.org, revised Oct 2013.
    4. 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.
    5. McCauley, Joseph L., 2008. "Nonstationarity of efficient finance markets: FX market evolution from stability to instability," International Review of Financial Analysis, Elsevier, vol. 17(5), pages 820-837, December.
    6. McCauley, Joseph L., 2009. "ARCH and GARCH models vs. martingale volatility of finance market returns," International Review of Financial Analysis, Elsevier, vol. 18(4), pages 151-153, September.
    7. Jovanovic, Franck & Schinckus, Christophe, 2017. "Econophysics and Financial Economics: An Emerging Dialogue," OUP Catalogue, Oxford University Press, number 9780190205034.

    More about this item

    Keywords

    Nonstationary increments; autocorrelations; scaling; Hurst exponents; Markov process;

    JEL classification:

    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • G0 - Financial Economics - - General
    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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