Scalas, Enrico and Kim, Kyungsik (2006): The art of fitting financial time series with Levy stable distributions.
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This paper illustrates a procedure for fitting financial data with alpha-stable distributions. After using all the available methods to evaluate the distribution parameters, one can qualitatively select the best estimate and run some goodness-of-fit tests on this estimate, in order to quantitatively assess its quality. It turns out that, for the two investigated data sets (MIB30 and DJIA from 2000 to present), an alpha-stable fit of log-returns is reasonably good.
|Item Type:||MPRA Paper|
|Original Title:||The art of fitting financial time series with Levy stable distributions|
|Keywords:||finance; statistical methods; stable distributions|
|Subjects:||C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General
C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C16 - Specific Distributions
G - Financial Economics > G0 - General > G00 - General
|Depositing User:||Enrico Scalas|
|Date Deposited:||09. Oct 2006|
|Last Modified:||12. Feb 2013 17:45|
 W. Bertram, Modelling asset dynamics via an empirical investigation of Australian Stock Exchange data, Ph.D Thesis, School of Mathematics and Statistics, University of Sydney, 2005.  L. Bachelier, Th´eorie de la sp´eculation, Gauthier-Villar, Paris, 1900. (Reprinted in 1995, Editions Jacques Gabay, Paris).  P.H. Cootner (Ed.), The Random Character of Stock Market Prices, MIT Press, Cambridge, MA, 1964.  B. Mandelbrot, The Variation of Certain Speculative Prices, The Journal of Business 36, 394–419 (1963).  B. Mandelbrot, H.M. Taylor, On the distribution of stock price differences, Oper. Res. 15, 1057-1062 (1967).  P.K. Clark, A subordinated stochastic process model with finite variance for speculative prices, Econometrica 41, 135-156 (1973).  R.C. Merton, Continuous Time Finance, Blackwell, Cambridge, MA, 1990.  R.N. Mantegna, L´evy walks and enhanced diffusion in Milan stock exchange, Physica A 179, 232–242 (1991).  R.N. Mantegna and H.E. Stanley, Stochastic Process With Ultra-Slow Convergence to a Gaussian: The Truncated L´evy Flight, Phys. Rev. Lett. 73, 2946–2949 (1994).  R.N. Mantegna and H.E. Stanley, Scaling behaviour in the dynamics of an economic index, Nature 376, 46–49 (1995).  I. Koponen, Analytic approach to the problem of convergence of truncated L´evy flights towards the Gaussian stochastic process, Phys. Rev. E 52, 1197–1199 (1195).  W. Schoutens, L´evy Processes in Finance: Pricing Financial Derivatives, Wiley, New York, NY, 2003.  R. Weron, L´evy-stable distributions revisited: tail index > 2 does not exclude the L´evy stable regime, Int. J. Mod. Phys. C 12, 209–223 (2001).  J.P. Nolan, Fitting data and assessing goodness-of-fit with stable distributions, in J. P. Nolan and A. Swami (Eds.), Proceedings of the ASA- IMS Conference on Heavy Tailed Distributions, 1999. J.P. Nolan, Maximum likelihood estimation of stable parameters, in O.E. Barndorff-Nielsen, T. Mikosch, and S. I. Resnick (Eds.), L´evy Processes: Theory and Applications, Birkh¨auser, Boston, 2001. The program stable.exe can be downloaded from http://academic2.american.edu/∼jpnolan/stable/stable.html.  J.M. Chambers, C.L. Mallows, B.W. Stuck, A Method for Simulating Stable Random Variables, J. Amer. Stat. Assoc. 71, 340–344 (1976).  R. Weron, On the Chambers-Mallows-Stuck Method for Simulating Skewed Stable Random Variables, Statist. Probab. Lett. 28, 165–171 (1996). R. Weron, Correction to: On the Chambers-Mallows-Stuck Method for Simulating Skewed Stable Random Variables, Research report, Wroclaw University of Technology, 1996.  Various implementations of the Chambers-Mallows-Stuck algorithm are available from the web page of J. Huston McCulloch: http://www.econ.ohio-state.edu/jhm/jhm.html.  P. L´evy, Th´eorie de l’addition de variables al´eatoires, Editions Jacques Gabay, Paris, 1954.  J.P. Nolan, Parameterizations and modes of stable distributions, Statist. Probab. Lett. 38, 187–195 (1998).  G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, NY, 1994.  A. Zolotarev, One-Dimensional Stable Distributions, American Mathematical Society, Providence, RI, 1986.  The adjusted close is the close index value adjusted for all splits and dividends. See http://help.yahoo.com/help/us/fin/quote/quote-12.html for further information.  W.H. DuMouchel, Estimating the Stable Index alpha in Order to Measure Tail Thickness: A Critique, Ann. Statist. 11, 1019–1031 (1983).  J.H. McCulloch. Simple Consistent Estimates of Stable Distribution Parameters, Commun. Statist. - Simula. 15, 1109-1136 (1986).  I.A. Koutrouvelis, Regression-Type Estimation of the Parameters of Stable Laws, J. Amer. Statist. Assoc. 75, 918–928 (1980). I.A. Koutrouvelis, An Iterative Procedure for the Estimation of the Parameters of the Stable Law, Commun. Statist. - Simula. 10 17–28 (1981).  S.M. Kogon, D.B. Williams, Characteristic function based estimation of stable parameters, in R. Adler, R. Feldman, and M. Taqqu, eds., A Practical Guide to Heavy Tails, Birkhaeuser, Boston, MA, 1998.  S. Rachev and S. Mittnik, Stable Paretian Models in Finance, Wiley, New York, NY, 2000.  M.A. Stephens, EDF Statistics for Goodness of Fit and Some Comparisons, J. Amer. Stat. Assoc. 69, 730–737 (1974).