Fourth order cumulant in EUR/USD falsifies random walk hypothesis

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Written by Forex Automaton   
Wednesday, 15 September 2010 16:14

Ability to predict the up-coming changes in daily and hourly high and low of price (of course, in a statistical sense, as measured by correlation coefficients between prediction and reality) by using adaptive black-box models has been well documented on this site. Observation of statistical dependence of the extreme levels of price (high and low) within a time bin on the immediate past of the time series, reported for the random walk model, explains and, particularly in the context of searching for market inefficiencies, even trivializes this achievement. Indeed, market inefficiencies are not required for the diffusion equation (cf. Black-Scholes theory) to work. Are we merely creating black-box equivalents of the popular tools of financial engineering? Enter higher-order cumulants. Shown here are measurements of the fourth order cumulants among the 24-hour high and low and the respective forecasts in real-life EUR/USD data; these can now be compared with the values they had in the random walk data. The difference revealed is dramatic.

As usual, the quantities we are going to look at are not the actual low and high. Since these quantities themselves are always positive, they are trivially correlated; this feature is absent in the correlations of the so-called logarithmic returns which are the ratios of low and high to the values they had during the previous 24-hour interval (day bar).

We generate predictions for daily changes in price's high, such that the correlation between the real and predicted change for high are positive. Same for low. If we take a trading position having yesterday's low as a stop-loss and yesterday's high as a profit target, we want to make sure that not only there is a tendency for low not to be hit when the prediction says so (shown by the positive correlation between the daily change in low and its forecast), and not only there is a tendency for the high to be hit when the prediction says so (shown by the positive correlation between the daily change in high and its forecast), but that these two things tend to coincide within the same trade. This is the essence of the difference between the "genuine" fourth order correlation and a mere superposition of two second order ones.

Fourth order cumulant, defined by Eq.1, is a measure of the genuine fourth order correlation. The fourth order cumulants used in this article and in the above-mentioned article on random walk are normalized:

c4 = E[1,2,3,4]
- E[1,2,3]E[4] - E[1]E[2,3,4] - E[1,3,4]E[2] - E[1,2,4]E[3]
- E[1,2]E[3,4] - E[1,3]E[2,4] - E[1,4]E[2,3]
+ 2(E[1,2]E[3]E[4] + E[1,3]E[2]E[4] + E[1,4]E[2]E[3] + E[2,3]E[1]E[4] + E[2,4]E[1]E[3] + E[3,4]E[1]E[2])
- 6E[1]E[2]E[3]E[4].
(1)

where E[] (expectation) is the averaging operator. The normalized cumulant is

C4 = c4/(Var[1]Var[2]Var[3]Var[4])1/2 (2)

where Var[] is variance.

The normalization, Eq.2, guarantees that the overall scale of price and its variation is taken out of consideration when comparing time series of different origin.

4th order cumulant for 24-hour high, low, and their forecasts, as a function of the optimization parameter 1.1 Pearson correlation coefficient for 24-hour high and its forecast, as a function of the optimization parameter 1.2 Pearson correlation coefficient for 24-hour low and its forecast, as a function of the optimization parameter 1.3

Fig.1. Correlation measures of 4th and 2nd order as a function of the optimization parameter responsible for the forecasting. 1.1: Normalized 4th order cumulant for the 24-hour high, low, and their forecasts. 1.2: Pearson correlation coefficient between 24-hour high and its forecast. 1.3: Pearson correlation coefficient between 24-hour low and its forecast. Different symbols mark different temporal binnings, 1 denotes a binning with intervals ending at 1am, 3 -- at 3am, 13 -- at 1pm Eastern time and so on. The quantity plotted along the horizontal axis, nicknamed Fred, determines how the recent past interacts with longer-term accumulated knowledge to perform the projection into the future.

In the 24-hour market like forex, there are many ways to form day bars out of the tick data. In this study, we compare 12 such ways: there is a bar starting at 1am and closing at 1am next day, starting at 3am and closing at 3am next day, and so on. In Fig.1, the different symbols denote different such sub-samples of daily data created from the same master sample of tick data, beginning on August 20, 2002 and ending on August 1, 2010. Each sub-sample is labeled according to its closing time.

Fig.1.1 shows that the forth order cumulant for logarithmic returns in 24-hour high, low, and their respective forecasts for EUR/USD is significantly positive. In contrast, same quantity reported for the random walk data was found to be slightly negative, lying between 0 and -0.1 for all values of Fred. Behavior of the cumulant with Fred resembles behavior of the Pearson correlation coefficients.

The Pearson correlation coefficients are quite close in magnitude and in shape of the Fred dependence to what was seen in the random walk data.

Variation of the cumulant with the closing hour of the 24-hour bar is very interesting. If indeed the positive cumulant signifies the best environment for trading on the basis of the daily high and low forecasts, as the intuitive and qualitative arguments above suggest, then the best hours to place the trades are those corresponding to the highest cumulant. The daily pattern of activity in EUR/USD has been studied; there is no simple correspondence: for example, 7am ET corresponds to a local minimum of activity, yet this is one of the best hours according to Fig.1.1; 11am ET is the local maximum of activity, yet it is also good according to Fig.1.1. The timing rule according to the cumulant appears to be simpler than the daily activity chart -- there is essentially a broad maximum of cumulant between 5 and 11am ET.

When optimizing Demi, a model portfolio trading system based on the predictions of 24-hour extremes, decisions which 24-hour periods to trade were based on the back-tested performance bottom line. The intervals chosen were 3am, 5am, 9am and 11am -- a good overlap with what the overall trend in the cumulant data suggests, although for the Fred value chosen (10) the situation may have local features.

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Last Updated ( Saturday, 10 March 2012 15:44 )