A Markov model of financial returns

We address the general problem of how to quantify the kinematics of time series with stationary first moments but having non stationary multifractal long-range correlated second moments. We show that a Markov process is sufficient to model important aspects of the multifractality observed in financial time series and propose a kinematic model of price fluctuations. We test the proposed model by analyzing index closing prices of the New York Stock Exchange and the DEM/USD tick-by-tick exchange rates obtained from Reuters EFX. We show that the model captures the characteristic features observed in actual financial time series, including volatility clustering, time scaling and fat tails in the probability density functions, power-law behavior of volatility correlations and, most importantly, the observed nonuniversal multifractal singularity spectrum. Motivated by our finding of strong agreement between the model and the data, we argue that at least two independent stochastic Gaussian variables are required to adequately model price fluctuations.

[1]  T. Lux.  Turbulence in financial markets: the surprising explanatory power of simple cascade models , 2001 .

[2]  H. Stanley,et al.  Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions , 2001 .

[3]  R. Mantegna,et al.  Scaling behaviour in the dynamics of an economic index , 1995, Nature.

[4]  Sergio Da Silva,et al.  Stockmarket comovements revisited , 2005 .

[5]  Stanley,et al.  Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. , 1994, Physical review letters.

[6]  J. Bouchaud,et al.  Leverage Effect in Financial Markets , 2001 .

[7]  A comparison between several correlated stochastic volatility models , 2003, cond-mat/0312121.

[8]  Spot foreign exchange market and time series , 2003, cond-mat/0303071.

[9]  M. Serva,et al.  Multiscale behaviour of volatility autocorrelations in a financial market , 1998, cond-mat/9810232.

[10]  B. Mandelbrot.  A Multifractal Walk down Wall Street , 1999 .

[11]  Nuno Crato,et al.  Long-range dependence in the conditional variance of stock returns , 1994 .

[12]  J. Masoliver,et al.  Random diffusion and leverage effect in financial markets. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[14]  H. Stanley,et al.  Multifractal properties of price fluctuations of stocks and commodities , 2003, cond-mat/0308012.

[15]  V. Plerou,et al.  Scaling of the distribution of fluctuations of financial market indices. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  R. Gomory,et al.  Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E , 1997 .

[17]  E. Bacry,et al.  Multifractal random walk. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  H. Stanley,et al.  Effect of trends on detrended fluctuation analysis. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  P. Ivanov,et al.  Variance fluctuations in nonstationary time series: a comparative study of music genres , 2003, cond-mat/0312380.

[20]  Marcel Ausloos,et al.  Coherent and random sequences in financial fluctuations , 1997 .

[21]  C. Granger,et al.  A long memory property of stock market returns and a new model , 1993 .

[22]  B. Mandelbrot.  Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier , 1974, Journal of Fluid Mechanics.

[23]  E. Bacry,et al.  Wavelets and multifractal formalism for singular signals: Application to turbulence data. , 1991, Physical review letters.

[24]  J. Gani,et al.  Statistical Models and Turbulence. , 1972 .

[25]  Maurizio Serva,et al.  The origin of fat-tailed distributions in financial time series , 2001 .

[26]  Andrew J. Patton,et al.  What good is a volatility model? , 2001 .

[27]  Richard T. Baillie,et al.  Long memory processes and fractional integration in econometrics , 1996 .

[28]  A. Vulpiani,et al.  Anomalous scaling laws in multifractal objects , 1987 .

[29]  Guido Caldarelli,et al.  Scaling in currency exchange , 1997 .

[30]  Vulpiani,et al.  Anomalous scaling and generalized Lyapunov exponents of the one-dimensional Anderson model. , 1987, Physical review. B, Condensed matter.

[31]  Alessandro Chessa,et al.  Systems with correlations in the variance: Generating power law tails in probability distributions , 1999, cond-mat/9910433.

[32]  Gilles Teyssière,et al.  Microeconomic Models for Long Memory in the Volatility of Financial Time Series , 2001 .

[33]  E. Fama.  EFFICIENT CAPITAL MARKETS: A REVIEW OF THEORY AND EMPIRICAL WORK* , 1970 .

[34]  Maurizio Serva,et al.  Clustering of Volatility as a Multiscale Phenomenon , 1999 .

[35]  M. Levy,et al.  POWER LAWS ARE LOGARITHMIC BOLTZMANN LAWS , 1996, adap-org/9607001.

[36]  T. Lux.  DETECTING MULTIFRACTAL PROPERTIES IN ASSET RETURNS: THE FAILURE OF THE "SCALING ESTIMATOR" , 2004 .

[37]  Didier Sornette,et al.  The Black-Scholes option pricing problem in mathematical finance : generalization and extensions for a large class of stochastic processes , 1994 .

[38]  J. Bouchaud,et al.  Leverage effect in financial markets: the retarded volatility model. , 2001, Physical review letters.

[39]  L. Bachelier,et al.  Théorie de la spéculation , 1900 .

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

[41]  Gilles O. Zumbach,et al.  Volatility processes and volatility forecast with long memory , 2002 .

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

[43]  P. F. Panter.  Modulation, noise, and spectral analysis , 1965 .

[44]  G. Viswanathan,et al.  A STOCHASTIC MODEL FOR MULTIFRACTAL BEHAVIOR OF STOCK PRICES , 2004 .

[45]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[46]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

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

[48]  Rosario N. Mantegna,et al.  Turbulence and financial markets , 1996, Nature.

[49]  P. Cizeau,et al.  Statistical properties of the volatility of price fluctuations. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[50]  Rosario N. Mantegna,et al.  Book Review: An Introduction to Econophysics, Correlations, and Complexity in Finance, N. Rosario, H. Mantegna, and H. E. Stanley, Cambridge University Press, Cambridge, 2000. , 2000 .