# Estimating the Hurst exponent in short terms in developed markets

In the Proceedings of the Estonian Academy of Sciences, Physics and Mathematics (2003), I saw the following sentence:

Surprisingly, in the case of developed markets, short-term $H$ results showed almost no persistance in memory.

If I understand the meaning of the Hurst exponent well enough, this means that developed markets are close to being efficient in micro-scales.

I've done some calculations on a week of data of EUR/USD prices from February 2012, and the Hurst exponent I've found (using the algorithm by sagemath) was around 0.5 (actually floating from 0.495 to 0.505). The prices were mid prices, sampled every second.

Do you think it is a "safe" to assume that in such small timescales (~1 sec.), in (highly-)developed markets, the Hurst exponent is ~0.5? That is, is it "safe" to assume that in these markets/timescales the pattern of the prices may be modelled by geometric Brownian motion, in which the standard deviation may change, but also may be assumed to be constant over short (~1 min.) time periods?

## 1 Answer

I don't believe it means they are efficient. It could only imply that your sample has no persistence (notice I'm not using the term auto-correlation) in the returns, if you did use returns instead of prices.

Evidence of EMH via the Hurst exponent is an extrapolation that you cannot make. It just says you cannot reject the null regarding this test. There are several ways in which markets could have inefficiencies while displaying H=0.5. It does not imply allocation efficiency, operational efficiency nor information efficiency.

A good counter-example is finding heteroskedasticity in the variance, which is not consistent with the EMH and consistent with finding no auto-correlation in the returns.