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

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  • 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.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 2126.

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Date of creation: 30 Sep 2006
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Handle: RePEc:pra:mprapa:2126

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

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References

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  1. Lisa Borland, 2002. "A theory of non-Gaussian option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 415-431.
  2. 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.
  3. 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.
  4. 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.
  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. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  7. 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.
  8. 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.
  9. 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.
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Cited by:
  1. 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.
  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. 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.
  4. 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.
  5. 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.
  6. Kerry W. Fendick, 2013. "Pricing and Hedging Derivative Securities with Unknown Local Volatilities," Papers 1309.6164, arXiv.org, revised Oct 2013.

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