Scaling transformation and probability distributions for financial time series

TL;DR

This paper explores the scaling transformations of financial time series, linking group theory and multifractal analysis to reveal the complex, multifractal nature of asset price increments.

Contribution

It introduces a novel connection between non-linear group theoretical methods and multifractal analysis in financial data.

Findings

Scaling transformation acts as a non-linear group action on moments

Spectral decomposition reveals multifractal behavior

Financial time series exhibit complex scaling properties

Abstract

The price of financial assets are, since Bachelier, considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works. In this letter we investigate the question of scaling transformation of price processes by establishing a new connexion between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.