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

PDF
MPRA_paper_2126.pdf Download (719kB)  Preview 
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 nonGaussian, scale in time, and have powerlaw (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 EuroDollar exchange rate between 19992004. 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 fattailed 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 
Original Title:  Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets 
Language:  English 
Keywords:  Nonstationary increments; autocorrelations; scaling; Hurst exponents; Markov process 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C16  Specific Distributions G  Financial Economics > G0  General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General 
Item ID:  2126 
Depositing User:  Joseph L. McCauley 
Date Deposited:  09. Mar 2007 
Last Modified:  19. Feb 2013 05:45 
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 NonGaussian 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, Timedependent 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 SelfSimilar 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. AlejandroQuinones, 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 FokkerPlanck 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 Interday 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). 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/2126 