Characterizing Multi-Scale Self-Similar Behavior and Non-Statistical Properties of Financial Time Series

TL;DR

This paper uses wavelet transforms to analyze multi-scale self-similar and non-statistical properties of stock market returns, revealing fractal characteristics, deviations, and power-law behaviors across different scales.

Contribution

It introduces a wavelet-based method to isolate local trends and analyze scale-dependent statistical properties of financial time series, highlighting multi-fractal and non-Gaussian behaviors.

Findings

Stock returns exhibit multi-fractal characteristics.

Fat-tailed, non-Gaussian fluctuations are observed at finer scales.

Power-law behavior ($k^{-3}$) emerges at larger scales.

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 \cite{hes5} 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.