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Non-linear long-term autocorrelations present in empirical and synthetic high-frequency financial time-series. Possibility of risk classification

Ryszard Kutner ,  Marzena Kozłowska ,  Filip Świtała 

Warsaw University, Faculty of Physics, Hoża 69, Warszawa 00-681, Poland


We consider the basis of the non-linear, long-term (power-law) autocorrelations present in empirical and our synthetic high-frequency financial time series. The former was studied by physicists since more than one decade [1,2, and refs. therein] while the synthetic time-series was received by us from the recently developed, hierarchical, one-dimensional Continuous-Time Random Walk model [3-8, and refs. therein].

This combined model is defined by the non-separable Weierstrass walk which can be occasionally intermitted by momentary localizations (WWRIL); the localizations themselves are also described by the Weierstrass process. It should be emphasized that the steps of the walk as well as momentary localizations are uncorrelated. This approach makes it possible to study by hierarchical stochastic simulations the whole spatial-temporal region while analytically it is possible to study only the initial, pre-asymptotic and asymptotic ones but not the very important intermediate region.

The basic continuous-time series obtained from this stochastic simulation is shown as a sequence of vectors in the temporal-spatial plane. These vectors connect the turning points of a single realization random walk trajectory (given time-series) expanding in the positive temporal and spatial directions as we study only the absolute values of stock price variations. This simulation is supported by the waiting-time distribution which is the main quantity of our two-state (walking-localization) model. These states are again characterized by their own waiting-time distributions.

The synthetic, discrete time-series was obtained by discretization of the original (basic) continuous-time series at fixed time horizon. The autocorrelation function was studied versus time just for this discrete time-series. We found that the autocorrelation exhibits a persistent power-low relaxation both for the Gaussian and non-Gaussian basic processes. Our study shows that this relaxation is the result of so-called 'domino effect' occurring within the discrete time-series. We suppose that this effect is responsible for the analogous long-term autocorrelatios commonly occurring in the empirical financial high-frequency time-series.

For example, by applying the Continuous-Time Weierstrass Flights model [9] developed in the framework of the nonseparable Continuous-Time Random Walks formalism, we constructed a series of diffusion phase diagrams of increasing orders [9,6] on the plane defined by the spatial and temporal fractional dimensions of the Weierstrass flights. To define the risk of a given asset the moments of increasing orders should be calculated (by using a moving average) from the time series represented the price dynamics of this asset. Hence, we are able to locate the asset on these phase diagrams and define its global and local risk of arbitrary order as well as the level of its mobility and activity again of arbitrary order [2].


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[2] J.-P. Bouchaud and M. Potter, Theory of Financial Risks. From Statistical Physics to Risk Management. Cambridge Univ. Press, Cambridge 2001.
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[4] Anomalous Diffusion. From Basics to Applications. LNP 519. Eds. R. Kutner, A. Pękalski, K. Sznajd-Weron. Springer-Verlag, Berlin 1999.
[5] W. Paul, J. Baschnagel, Stochastic Processes. From Physics to Finance. Springer-Verlag, Berlin 1999.
[6] R. Kutner, F. Świtała, Study of the non-linear autocorrelations within the Gaussian regime. Eur. Phys. J. B 33 (2003) 495-503; R. K. and F. Ś., Possible origin of the non-linear long-term autocorrelations within the Gaussian regime, Physica A 330 (2003) 177-188; R. K. and F. Ś., Stochastic simulations of time series within Weierstrass-Mandelbrot walks, Quant. Finance 3 (2003) 201-211; R. K., Extreme events as foundation of Levy walks with varying velocity, Chem. Phys. 284 (2002) 481-505.
[7] W. Schoutenes, Levy Processes in Finance. Pricing Financial Derivatives. Wiley Series in Probability and Statistics, J. Wiley & Sons Ltd., Chichester 2003.
[8] K.I.Ilinski, Physics of finance. Gauge modeling in non-equilibrium pricing. J. Wiley & Sons Ltd., New York 2001.
[9] R. Kutner, Hierarchical spatio-temporal coupling in fractional wanderings. I. Continuous-time Weierstrass flights, Physica A 264 (1999) 84-106.


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Presentation: oral at Symposium on Econo- and Sociophysics 2004, by Ryszard Kutner
See On-line Journal of Symposium on Econo- and Sociophysics 2004

Submitted: 2004-09-19 17:54
Revised:   2009-06-08 12:55