Detrending Moving Average Algorithm: Quantifying Heterogeneity in Financial Data

The Detrending Moving Average (DMA) algorithm can be implemented to estimate the Shannon entropy of a long-range correlated sequence which will be shown to be of particular relevance for its significance in finance. The entropy is written as the sum of two terms corresponding respectively to power-law (ordered) and exponentially (disordered) distributed blocks (clusters). Interestingly, the behaviour of the ordered clusters is found, on the average, comparable to the one of the whole analysed sequence, while that of the disordered clusters contribute excess fluctuations. These results mean that the power-law correlated clusters carry the same information of the whole sequence, whereas the disordered clusters are related to the deviations from the stationary behaviour of the series. The approach will be illustrated on historical financial data sets. The time series, investigated in this study, are tick-by-tick data of DAX and FIB30 over six years (from 1998 to 2004). This work might add clues to the microscopic dynamics underlying the technical trading and help understanding the issue of profitability.