  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.
