Multifractal Diffusion Entropy Analysis: Applications to Financial Time Series

Scaling behavior of time series is the key aspect that enables one to reveal the relevant temporal dynamical scales. In this respect, a multifractal theory represents a powerful approach in complex dynamical systems that is designed for the study of scaling behavior. In this framework the estimation of multifractal spectrum is one of popular methods allowing one to detect and quantify the underlying complexity present in the system. Among many different approaches, the Multifractal Diffusion Entropy Analysis, based on estimation of Renyi entropy, provides an innovative approach for evaluation of multifractal exponents. In the recent article [1], we have shown that the proper estimation of probability histograms is crucial for correct evaluation of Renyi entropy and ensuing multifractal exponents. In this paper we summarize our recent results and apply them to various real financial time series, recorded both on minute and daily basis. Our aim is to illustrate the potency of the method proposed in the field of financial time series. In particular we show that the method works well for daily data, but for minute data, which are usually distributed with power-laws, we observe discontinuities in the estimated spectra.