We consider the basis of the nonlinear, longterm (powerlaw) autocorrelations present in empirical and our synthetic highfrequency financial time series. The former was studied by physicists since more than one decade [1,2, and refs. therein] while the synthetic timeseries was received by us from the recently developed, hierarchical, onedimensional ContinuousTime Random Walk model [38, and refs. therein].
This combined model is defined by the nonseparable 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 spatialtemporal region while analytically it is possible to study only the initial, preasymptotic and asymptotic ones but not the very important intermediate region.
The basic continuoustime series obtained from this stochastic simulation is shown as a sequence of vectors in the temporalspatial plane. These vectors connect the turning points of a single realization random walk trajectory (given timeseries) 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 waitingtime distribution which is the main quantity of our twostate (walkinglocalization) model. These states are again characterized by their own waitingtime distributions.
The synthetic, discrete timeseries was obtained by discretization of the original (basic) continuoustime series at fixed time horizon. The autocorrelation function was studied versus time just for this discrete timeseries. We found that the autocorrelation exhibits a persistent powerlow relaxation both for the Gaussian and nonGaussian basic processes. Our study shows that this relaxation is the result of socalled 'domino effect' occurring within the discrete timeseries. We suppose that this effect is responsible for the analogous longterm autocorrelatios commonly occurring in the empirical financial highfrequency timeseries.
For example, by applying the ContinuousTime Weierstrass Flights model [9] developed in the framework of the nonseparable ContinuousTime 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].
Bibliography
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