Characterizing multi-scale self-similar behavior and non-statistical properties of fluctuations in financial time series

Abstract

We make use of wavelet transform to study the multi-scale, self similar behavior and deviations thereof, in the stock prices of large companies, belonging to different economic sectors. The stock market returns exhibit multi-fractal characteristics, with some of the companies showing deviations at small and large scales. The fact that, the wavelets belonging to the Daubechies' (Db) basis enables one to isolate local polynomial trends of different degrees, plays the key role in isolating fluctuations at different scales. One of the primary motivations of this work is to study the emergence of the k −3 behavior [1] of the fluctuations starting with high frequency fluctuations. We make use of Db4 and Db6 basis sets to respectively isolate local linear and quadratic trends at different scales in order to study the statistical characteristics of these financial time series. The fluctuations reveal fat tail non-Gaussian behavior, unstable periodic modulations, at finer scales, from which the characteristic k −3 power law behavior emerges at sufficiently large scales. We further identify stable periodic behavior through the continuous Morlet wavelet.

Network dynamics: quantitative analysis of complex behavior in metabolism, organelles, and cells, from experiments to models and back

“…Like Fourier, wavelet analysis attempts to express a signal as a sum of component waveforms. Whereas in Fourier analysis waveforms are sines and cosines with fixed frequencies, wavelets are nonperiodic but localized in different frequency bands, thus enabling more flexible signal decomposition while gaining time localization . Another advantage of wavelet analysis is the flexibility for choosing the mother wavelet according to the characteristics of the data under investigation.…”

Intelligent Ensemble Forecasting System of Stock Market Fluctuations Based on Symetric and Asymetric Wavelet Functions

“…In the first component, the continuous wavelet transform (CWT) is applied to stock market returns for analysis purpose. Although the DWT is popular in signal processing and financial prediction, the CWT is more appropriate to analyze data or time series to discover patterns and hidden information; particularly to analyze periodic modulations present in the time series at different scales [17], and analyze localized intermittent oscillations in time series [18]. In addition, the CWT is more suitable in time series analysis than the DWT for features extraction purpose [18].…”

Variable diffusion in stock market fluctuations

“…which can be evaluated [17] by using Itō's calculus in the scaling form of (5) and variable diffusion SDE (6), and taking the limit of τ ≪ t. This quantity is useful in empirical analysis in that it provides an alternative way to validate the scaling (in contrast to multi-scaling [38][39][40][41]) of empirical time series, as discussed below.…”