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

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Written by Forex Automaton   
Friday, 14 January 2011 12:44

I have been looking at the temporal distributions of daily highs and lows in forex for quite 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.

This report uses arithmetic random walk without drift (Wiener process) simulation algorithm already used and described before. In brief, the simulation does not impose any variation on the hourly volatility during the day. The simulated price evolution contains about 50,000 hourly candles of continuous data.

1.1 1.2 1.3

Fig.1. Statistical analysis of random walk simulations. 1.1: Averaged hourly logarithmic return vs hour of the day. 1.2: Hourly logarithmic return variance vs hour of the day. 1.3: 1-hour lag autocorrelation in hourly logarithmic returns vs hour of the day. Vertical bars are RMS estimates of the precision of the mean.

Fig.1 presents hourly "seasonal" averages derived from the hourly logarithmic return in the simulated random walk market. Fig.1.1 shows flat behavior as expected. Fig.1.2 and 1.3 present phoney patterns which look credible if the Gaussian estimates of the precision of the mean are taken at a face value. However, Gaussian confidence levels can not be assigned to the intervals indicated by the bars, and in case of Fig.1.2 and 1.3, the confidence levels are much smaller. This exercise serves to qualitatively indicate the level of scepticism with which to treat results of similar analyses performed on the real data.

Temporal distribution of daily high and lows in the simulated random walk time series

Fig.2. Distribution of time moments (accurate up to an hour) when daily high and low in the simulated random walk time series are achieved.

Daily variations in volatility can be studied by observing probabilities of establishing daily extremes of price (low and high) during particular hours of the day. Fig.2 demonstrates the shape of the probability distribution for the random walk in the absence of seasonal variations in volatility, a baseline for such studies.

For the arithmetic random walk, the distribution indicates the increased probability for the time series to reach the extremes during the first and last hours of the trading day. This may be counterintuitive given the apparent "lack of order" in the shape of any daily piece of the time series, and inherent "lack of information" to account for the parameters of any stable shape. However, this is not our first "counterintuitive" finding regarding the random walk. These effects can be understood intuitively: it is not the price time series but the time series of its regular increments that's truly random and devoid of memory effects. In the time series of prices, there is a memory effect due to the fact that every sucessive price change adds to the sum of its predecessors, no matter how random these prices changes are in themselves.

Needless to say, the location of maximum at the first hour of the day is independent of which hour is chosen to serve as such (which time zone is chosen), and this has been numerically tested.

This observation invalidates any naive interpretation of the similar daily low and high probability peaks found to lie in the first and last hours of the day in the historical forex data.

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Last Updated ( Wednesday, 23 February 2011 16:18 )