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

Bassler, Kevin E. and McCauley, Joseph L. and Gunaratne, Gemunu H. (2006): Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets.

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.

• Item Type: MPRA Paper
• Institution: University of Houston
• Language: English
• Keywords: Nonstationary increments; autocorrelations; scaling; Hurst exponents; Markov process
• Subjects: G - Financial Economics > G0 - General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General

References

[1] B. B. Mandelbrot, The Variation of Certain Speculative Prices, J. Bus. 36, 394 (1963). [2] J. L. McCauley and G. H. Gunaratne, An Empirical Model of Volatility Returns and Options Pricing, Physica A 329, 170 (2003). [3] R. N. Mantegna and H. E. Stanley, Scaling Behavior in the Dynamics of an Economic Index, Nature 376, 46 (1995); Turbulence in Financial Markets, Nature 383, 587 (1996). [4] R. Friedrich, J. Peinke, and Ch. Renner, How to Quantify Deterministic and Random Influences on the Statistics of the Foreign Exchange Market, Phys. Rev. Lett. 84, 5224 (2000). [5] L. Borland, A Theory of Non-Gaussian Option Pricing, Quan. Finance 2, 415 (2002). [6] S. Galluccio, G. Caldarelli, M. Marsili, and Y. C. Zhang, Scaling in Currency Exchange, Physica A 245, 423 (1997). [7] A. Carbone, G. Castelli, and H. E. Stanley, Time-dependent Hurst Exponents in Financial Time Series, Physica A 344, 267 (2004). [8] B. Mandlebrot and J. W. van Ness, Fractional Brownian Motion, Fractional Noise and Applications, SIAM Rev. 10, 422 (1968). [9] G. H. Gunaratne, J. L. McCauley, M. Nicole, and A. T¨or¨ok. Variable Step Random Walks and Self-Similar Distributions, J. Stat. Phys. 121, 887 (2005). [10] S. Chandrasekhar, Stochastic Problems in Physics and Astronomy, Rev. Mod. Phys., 15, 1 (1943). [11] A. A. Alejandro-Quinones, K. E. Bassler, M. Field, J. L. McCauley, M. Nicol, I. Timofeyev, A. T¨or¨ok, and G. H. Gunaratne, A Theory of Fluctuations in Stock Prices, Physica A 363, 383 (2006). [12] K. E. Bassler, G. H. Gunaratne, and J. L. McCauley, Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with Applications to Finance, Physica A, 369, 343 (2006). [13] L. Borland, Microscopic Dynamics of the Nonlinear Fokker-Planck Equation: A Phenomeno- 7 logical Model, Phys. Rev. E 57, 6634 (1998). [14] S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, and Y. Dodge, Turbulent Cascades in Foreign Exchange Markets, Nature 381, 767 (1996). [15] H. C. Fogedby, T. Bohr, and H. J. Jensen, Fluctuations in a L´evy Flight Gas, J. Stat. Phys. 66, 583 (1992). [16] U. A. M¨uller, M. M. Dacorogna, R. B. Olsen, O. V. Pictet, M. Schwarz, and C. Morgenegg, Statistical Study of Foreign Exchange Rates, Empirical Evidence of a Price Change Scaling Law, and Inter-day Analysis, J. Bank. Fin. 14, 1189 (1990). [17] R. Cont, M. Potters, and J.-P. Bouchard, Scaling in Stock Market Data: Stable Laws and Beyond, in ”Scale Invariance and Beyond”, eds. B. Dubrulle, F. Graner, and D. Sornette, Springer, Berlin, 1997. [18] C. C. Heyde and Y. Yang, On Defining Long Range Dependence, J. Appl. Prob. 34, 939 (1997). [19] C. C. Heyde and N. N. Leonenko, Student Processes, Adv. Appl. Prob. 37, 342 (2005). [20] M. Couillard and M. Davison, A Comment on Measuring the Hurst exponent of Financial Time Series, Physica A 348, 404 (2005).