Scaling properties of long-range correlated noisy signals: application to financial markets

Long-range correlation properties of financial stochastic time series y(i) have been, investigated with the main aim to demonstrate the ability of a recently proposed method to extract the scaling parameters of a stochastic series. According to this technique, the Hurst coefficient H is calculated by means of the following function: DMA = root1/N(max)-n(max) Sigma(i=nmax)(Nmax) [y(i) - (y) over tilde (n)(i)](2) where (y) over tilde (n)(i) is the moving average of y(i), defined as 1/n Sigma(k=0)(n-1)y(i-k), n the moving average window and N(max) is the dimension of the stochastic series. The method is called Detrending Moving Average Analysis (DMA) on account of the several analogies with the well-known Detrended Fluctuation Analysis (DFA). The DMA technique has been widely tested on stochastic series with assigned H generated by suitable algorithms. It has been demonstrated that the ability of the proposed technique relies on very general grounds: the function C(n)(i) = y(i) - (y) over tilde (n)(i) generates indeed a sequence of cluster's with power-law distribution of amplitudes and lifetimes. In particular the exponent of the distribution of cluster lifetime varies as the fractal dimension 2 - H of the series, as expected on the basis of the box-counting method. In the present paper we will report on the scaling coefficients of real data series (the BOBL and DAX German future) calculated by the DMA technique.