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?