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CONCLUDING REMARKS

The record of stock market ticker transactions displays four nonrandom properties: (1) There is a general tendency for price reversal between trades. (2) Reversals are relatively more concentrated at integers where stable slow-moving participants offer to buy and sell. There is a concentration of particular types of reversals just above and below these barriers. (3) Quick moving com ­petitors cognizant of these barriers can take positions at nearby prices, thus "getting the trade" and hoping to make a profit. (4) After two changes in the same direction, the chances of continuation in that direction are greater than after changes in opposite directions.

It would be interesting to see if these properties of stock market prices hold in other markets. We remarked that the tendency to reversal holds in wheat and coin markets. As far as we know, no one has provided information concern ­ing properties (2)-(4) in other markets.

Although the specific properties reported in this study have a significance from a statistical point of view, the reader may well ask whether or not they arc helpful in a practical sense. Certain trading rules emerge as a result of our analysis. One is that limit and stop orders should be placed at odd eighths, preferably at 7/8 for sell orders and at 1/8 for buy orders. Another is to buy when a stock advances through a barrier, and to sell when it sinks through a barrier. Professional traders will recognize these rules or their equivalent as quite familiar.6 Since the tendency of traders to prefer integers seems to be a fundamental and stable principle of stock market psychology, we may have confidence that the transactions of those who follow the proposed rules will not destroy the effect [3, 20].

Godfrey and his co-workers, have looked for periodicities and other regu ­larities in the record of ticker transactions of 2 NYSE issues. Their conclusions are opposite to ours in a great many respects. The interested reader is invited to form his own conclusion by perusal of the references [refs. 5, and 8-14]. We shall be content here to record our impression that spectral analysis, the tech ­nique they utilized, seems unsuited to the analysis of stock market prices.

At a more fundamental level, the present writers believe that the discoveries of regularities in price movements of consecutive transactions reported herein provide a stepping stone for further and more exhaustive studies. The first step in this direction would be to derive the probability density function for daily stock price changes by letting the second order Markov process we have described run for the actual number of transactions that occur in different stocks during the day. Will the distribution of daily price changes approach normality? Will it he dependent on previous daily price changes and volume?

What is the best way to incorporate any existing dependence between price and volume movements into this process? Somehow one must incorporate both a "transaction number time scale," and a "calendar time scale" into the process, since there is evidence that both are significant [see, e.g., ref. 12, Fig. 9].

One fruitful approach might be to apply central limit theorems for dependent variables to the sum of price changes differenced over a constant number of transactions. The distribution of this sum, for n>30 is probably very close to normal. But daily price changes may be the sum of widely differing numbers of transactions. Perhaps daily price changes can be envisioned as a mixture of normal processes with weights proportional to observed classes of transaction numbers.

It is our hope that this paper will suggest questions and tests of this kind, and also help to solve them. Certainly the findings of structure, regularities, and dependence effects, which have been the subject of this study, ought to be valuable guides in the formulation of more sophisticated models of stock price movement.