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AN OBSERVATIONAL TEST OP PROPERTIES INDUCED BY MARKET MAKING

In this section we test a second sample of data for the properties suggested by the preceding discussion of market making. The data consist of the comВ­plete record of ticker transactions during a randomly chosen day in January of each of seven consecutive years. To reduce the magnitude of the computaВ­tions to a manageable level, and to maintain a relatively homogeneous sample, we eliminated from consideration all transactions in which there was no change from the previous price. Quantitatively this reduced the number of transactions by almost 60%, b\it qualitatively the loss of information was small. We took this set of data as the sample and concentrated upon all price movements which followed two consecutive changes of +1/8. The sequences were classified in the following manner:

Event RR: A rise of ( + 1/8) followed by a rise of ( + 1/8). Event PR: A fall of (-1/8) followed by a rise of (+1/8). Event FF: A fall of (-1/8) followed by a fall of (-1/8). Event RF: A rise of ( + 1 /ft) followed by СЏ full of (-1/8).

Examples of these events may be found in Figure 1. It may be observed that on the fourth transaction, Event FR (fall-rise) occurred at the fractional price 1/8, and a decline followed. On the fifth transaction, Event RF (rise-fall) ocВ­curred at the fractional price 0/8, and an advance followed.

This data sample is intended primarily to examine price structure at the different eighth positions. It is still reassuring that the total number of currences of the four events confirms the conclusions of our previous analysis. Recall that events RF and FR arc reversals; events RR and FF are continuaВ­tions. In the sample of 12,777 occurrences of these events, the ratio of events FR and RF to events RR and FF is 8912/3025 = 2.34/1, a ratio quite comВ­patible to the 3/1 ratio shown in Equations (3) and (4).

Additional confirmation of our previous conclusions can also be found in Table V and Figures 2 and 3. Table V contains the number of falls and rises {not restricted to В±1/8) which follow each of the four events for each of the eight fractional levels. The total number of rises and falls following all events terminating at a specific fractional price are given in the last column to the right. The total at the bottom of each column represents the total number of rises or falls for all fractional prices after a specific event. Figuro 2 contains the derived probability of a rise (1/8 or more) after patterns, or Events RR and FR. Figure 3 is a graph of the probability of a fall (1/8 or more) after Events FF and RF. The solid line in both figures indicates the probability of a continuation after two moves in the same direction. The right hand ordinate scale gives the probability of reversal.

The dotted line gives the probability of a continuation in the direction of the last move after two moves in an opposite direction (FR, R for Figure 2; RF, F for Figure 3). The last pair of lines to the right of the Figures gives the average probabilities for all the fractional price movements after the event. These were derived from the column marginals in Table V. For example, in the bottom row of Table V under Event FF there were 602 falls and 1148 rises. The probability of continuation was 602/1750 = 0.35. The last solid line at the righl of I'iguro 3 rises to this mark on the scale.

It is apparent that the solid line is taller than the dotted line at all positions in Figures 2 and 3- This indicates that continuations are more likely after two changes of the same sign than after two changes in opposite directions. But note that even after two changes in the same direction, reversals are still more probable than continuations at all eighths. Although these results come from a separate sample, they arc in accord with the information displayed in section 2. A tendency to reversal is a property of market making, and the consistent difference between full and dotted lines of Figures 2 and 3 imply that continua­tions are slightly more probable after a previous continuation, than after a previous reversal (i.e., ДУ(_« does influence ДК,).

It is noteworthy that the differences between the size of the solid and dotted lines at each fractional price is systematically greater for Figure 2 than for Figure 3. This shows that for this sample, rises at 1 2 have a more pronounced effect (in the probability sense) on Р” Yt than do falls. Somehow this runs counter to our intuitive picture of the market in its stochastic structure as up and down symmetric. The result may arise from a slight preponderance of advances beВ­tween consecutive transactions in the sample.

Before turning to the specific predictions, we repeat that a continuation or reversal of a move after events RR, RF, FR, FF is determined by agreement or disagreement with the sign of the terminating move of the event. Thus a continuation after event FR occurs when a fall of 1/8 followed by a rise of 1/8 is followed by a rise (of 1/8 or more); a reversal occurs if the third move is a fall. The notation Pe (7/8) refers to the probability ot continuation after one of the four events ending at fractional price 7/8.