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Why a Hurst Exponent Near 0.5 Does Not Prove Market Efficiency

Article Quant Q&A · Author: Bach
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Summary

The document asks whether a Hurst exponent near 0.5 in one-second EUR/USD prices supports modeling developed markets as geometric Brownian motion, with constant volatility over short intervals. The author reports estimates around 0.5 from a week of February 2012 mid-price data and relates them to a published observation about weak short-term memory in developed markets.

The response cautions that such an estimate, particularly if computed on prices rather than returns, does not establish market efficiency. A result near 0.5 indicates only that this test did not reject a no-persistence null; it does not rule out other forms of inefficiency. For example, returns may have little autocorrelation while their variance remains heteroskedastic. The discussion therefore does not justify assuming constant volatility or geometric Brownian motion from the Hurst estimate alone. Its evidence is conceptual rather than a broader empirical study, and it does not evaluate the reported calculation or establish how the result generalizes across markets, samples, or time horizons.

Key ideas

  • A Hurst estimate near 0.5 does not by itself prove that a market is efficient.
  • The interpretation depends on whether the statistic was estimated from prices or returns.
  • Failure to reject persistence in one test does not rule out other market inefficiencies.
  • Returns can show little autocorrelation while still having changing variance.
  • The reported EUR/USD estimate from one week of data does not establish a general modeling rule.

Tags

ForexStatisticsVolatilityMarket microstructure
Full text
# Estimating the Hurst exponent in short terms in developed markets


# 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?

## Answer by not.so.quanty (score 5, accepted)

https://quant.stackexchange.com/a/12616

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.