Even in the random walk, locations of daily high and low are non-uniformly distributed in time

I have been looking at the temporal distributions of daily highs and lows in forex since for some time now (see the EUR/USD report, the first one on the subject). Later I began complementing the data with a similar distribution of variance and 1-hour lag autocorrelation (here is the distribution for EUR/USD), trying to infer whether there is a systematic change in correlation regime from trend following to mean reversion during the day. The inference would be based on combining second order statistics and daily extreme distribution data. It had been tacitly assumed that random walk would have a featureless temporal distribution of daily extremes. In this post I document the "base-line" shape of this distribution: due to peculiar memory effects found even in the random walk (continuity of price), the shape of such distributions has certain characteristic features.

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