Time-dependent Hurst exponent in financial time series

We calculate the Hurst exponent H(t) of several time series by dynamical implementation of a recently proposed scaling technique: the detrending moving average (DMA). In order to assess the accuracy of the technique, we calculate the exponent H(t) for artificial series, simulating monofractal Brownian paths, with assigned Hurst exponents H. We next calculate the exponent H(t) for the return of high-frequency (tick-by-tick sampled every minute) series of the German market. We find a much more pronounced time-variability in the local scaling exponent of financial series compared to the artificial ones. The DMA algorithm allows the calculation of the exponent H(t), without any a priori assumption on the stochastic process and on the probability distribution function of the random variables, as happens, for example, in the case of the Kitagawa grid and the extended Kalmann filtering methods. The present technique examines the local scaling exponent H(t) around a given instant of time. This is a significant advance with respect to the standard wavelet transform or to the higher-order power spectrum technique, which instead operate on the global properties of the series by Legendre or Fourier transform of qth-order moments.