Fractal properties, information theory, and market efficiency

TL;DR

This paper explores the relationship between entropy-based market information and the Hurst exponent, providing theoretical insights and a multiscale method to analyze market efficiency across different assets and data frequencies.

Contribution

It offers a theoretical expression linking market information and the Hurst exponent for fractional Brownian motion models and introduces a multiscale approach for deeper entropy analysis.

Findings

High informativeness linked to Hurst exponent near 1/2 in stationary models

Multiscale method reveals different entropy characteristics at various scales

Applications demonstrate the method's effectiveness on diverse financial data

Abstract

Considering that both the entropy-based market information and the Hurst exponent are useful tools for determining whether the efficient market hypothesis holds for a given asset, we study the link between the two approaches. We thus provide a theoretical expression for the market information when log-prices follow either a fractional Brownian motion or its stationary extension using the Lamperti transform. In the latter model, we show that a Hurst exponent close to 1/2 can lead to a very high informativeness of the time series, because of the stationarity mechanism. In addition, we introduce a multiscale method to get a deeper interpretation of the entropy and of the market information, depending on the size of the information set. Applications to Bitcoin, CAC 40 index, Nikkei 225 index, and EUR/USD FX rate, using daily or intraday data, illustrate the methodological content.