Measuring the self-similarity exponent in Lévy stable processes of financial time series

Geometric method-based procedures, which will be called GM algorithms herein, were introduced in [M.A. Sanchez Granero, J.E. Trinidad Segovia, J. Garcia Perez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543-5551], to efficiently calculate the self-similarity exponent of a time series. In that paper, the authors showed empirically that these algorithms, based on a geometrical approach, are more accurate than the classical algorithms, especially with short length time series. The authors checked that GM algorithms are good when working with (fractional) Brownian motions. Moreover, in [J.E. Trinidad Segovia, M. Fernandez-Martinez, M.A. Sanchez-Granero, A note on geometric method-based procedures to calculate the Hurst exponent, Phys. A 391 (2012) 2209-2214], a mathematical background for the validity of such procedures to estimate the self-similarity index of any random process with stationary and self-affine increments was provided. In particular, they proved theoretically that GM algorithms are also valid to explore long-memory in (fractional) Levy stable motions.

[1]  Krzysztof Burnecki,et al.  The Lamperti transformation for self-similar processes , 1997 .

[2]  A. Weron,et al.  Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes , 1993 .

[3]  Svetlozar T. Rachev,et al.  Financial market models with Lévy processes and time-varying volatility. , 2008 .

[4]  M. Taqqu,et al.  Estimating long-range dependence in the presence of periodicity: An empirical study , 1999 .

[5]  H. Stanley,et al.  Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series , 2002, physics/0202070.

[6]  J. R. Wallis,et al.  Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence , 1969 .

[7]  R. Weron.  Correction to: "On the Chambers–Mallows–Stuck Method for Simulating Skewed Stable Random Variables" , 1996 .

[8]  R. Adler,et al.  A practical guide to heavy tails: statistical techniques and applications , 1998 .

[9]  J. E. T. Segovia,et al.  Some comments on Hurst exponent and the long memory processes on capital markets , 2008 .

[10]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[11]  O. Barndorff-Nielsen.  Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling , 1997 .

[12]  B. Mandelbrot.  Fractals and Scaling In Finance: Discontinuity, Concentration, Risk , 2010 .

[13]  Harry V. Roberts,et al.  STOCK‐MARKET “PATTERNS” AND FINANCIAL ANALYSIS: METHODOLOGICAL SUGGESTIONS , 1959 .

[14]  C. Peng,et al.  Mosaic organization of DNA nucleotides. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  E. Panas.  Estimating fractal dimension using stable distributions and exploring long memory through ARFIMA models in Athens Stock Exchange , 2001 .

[16]  William B. Riley,et al.  Asset Allocation and Individual Risk Aversion , 1992 .

[17]  R. Weron.  Estimating long range dependence: finite sample properties and confidence intervals , 2001, cond-mat/0103510.

[18]  M. Osborne.  Brownian Motion in the Stock Market , 1959 .

[19]  James B. McDonald,et al.  14 Probability distributions for financial models , 1996 .

[20]  A. Lo.  Long-Term Memory in Stock Market Prices , 1989 .

[21]  R. Weron.  Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime , 2001, cond-mat/0103256.

[22]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[23]  Tomaso Aste,et al.  Non-stationary multifractality in stock returns , 2013 .

[24]  F. Lillo,et al.  The Long Memory of the Efficient Market , 2003, cond-mat/0311053.

[25]  Tomasz J. Kozubowski,et al.  Geometric stable laws: Estimation and applications , 1999 .

[26]  K. Podgórski,et al.  Asymmetric laplace laws and modeling financial data , 2001 .

[27]  Marcel Ausloos,et al.  Statistical physics in foreign exchange currency and stock markets , 2000 .

[28]  P. Lee,et al.  14. Simulation and Chaotic Behaviour of α‐Stable Stochastic Processes , 1995 .

[29]  H. Markowitz,et al.  The Random Character of Stock Market Prices. , 1965 .

[30]  Koponen,et al.  Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  Vicsek,et al.  Multifractality of self-affine fractals. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[32]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[33]  Robert C. Blattberg,et al.  A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices: Reply , 1974 .

[34]  Patrick Flandrin,et al.  On the spectrum of fractional Brownian motions , 1989, IEEE Trans. Inf. Theory.

[35]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[36]  B. Mandelbrot.  The Variation of Certain Speculative Prices , 1963 .

[37]  T. Aste,et al.  Understanding the source of multifractality in financial markets , 2012, 1201.1535.

[38]  Boris Podobnik,et al.  Asymmetric Lévy flight in financial ratios , 2011, Proceedings of the National Academy of Sciences.

[39]  M. Fernández-Martínez,et al.  A note on geometric method-based procedures to calculate the Hurst exponent , 2012 .

[40]  C. D. Vries,et al.  The Limiting Distribution of Extremal Exchange Rate Returns , 1991 .

[41]  George Michailidis,et al.  Simulating sample paths of linear fractional stable motion , 2004, IEEE Transactions on Information Theory.

[42]  M. Yor,et al.  Mathematical Methods for Financial Markets , 2009 .

[43]  B. Mandelbrot.  When Can Price Be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models , 1971 .

[44]  Uwe Hassler,et al.  Long Memory in Inflation Rates: International Evidence , 1995 .

[45]  Edgar E. Peters.  Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility , 1996 .

[46]  S. James Press,et al.  A Compound Events Model for Security Prices , 1967 .

[47]  Svetlozar T. Rachev,et al.  The Modifled Tempered Stable Distribution, GARCH Models and Option Pricing , 2008 .

[48]  Tomaso Aste,et al.  Scaling behaviors in differently developed markets , 2003 .

[49]  Edgar E. Peters.  R/S Analysis Using Logarithmic Returns , 1992 .

[50]  J. Lamperti.  Semi-stable stochastic processes , 1962 .

[51]  Karina Weron,et al.  Enigma of Self-Similarity of Fractional Levy Stable Motions , 2003 .

[52]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[53]  E. Seneta,et al.  Chebyshev Polynomial Approximations and Characteristic Function Estimation , 1987 .

[54]  E. Eberlein,et al.  The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures , 2002 .

[55]  T. D. Matteo,et al.  Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development , 2004, cond-mat/0403681.

[56]  T. Kozubowski,et al.  Multivariate geometric stable distributions in financial applications , 1999 .

[57]  Ladislav Kristoufek,et al.  On Hurst exponent estimation under heavy-tailed distributions , 2010, 1201.4786.

[58]  P. Cizeau,et al.  CORRELATIONS IN ECONOMIC TIME SERIES , 1997, cond-mat/9706021.

[59]  Svetlana Boyarchenko,et al.  OPTION PRICING FOR TRUNCATED LÉVY PROCESSES , 2000 .

[60]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[61]  Gary S. Shea,et al.  Uncertainty and implied variance bounds in long-memory models of the interest rate term structure , 1991 .

[62]  T. D. Matteo,et al.  Multi-scaling in finance , 2007 .

[63]  Murad S. Taqqu,et al.  A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy-Tailed Distributions , 1998 .

[64]  Walter Willinger,et al.  Stock market prices and long-range dependence , 1999, Finance Stochastics.

[65]  M. T. Greene,et al.  Long-term dependence in common stock returns , 1977 .