<s> Some Quantitative Tests for Stock Price Generating Models and TradingFolklore
 
M. F. M. Osborne
Journal of the American Statistical Association, Volume 62, Issue 318 (Jun., 1967),321-340.
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http://www.jstor.org/Thu May 16 03:59:42 2002JOURNAL OF THE AMERICANSTATISTICAL ASSOCIATION
Number 818 JUNE, 1967 Volume 62
SOME QUANTITATIVE TESTS FOR STOCK PRICE GENERATINGMODELS AND TRADING FOLKLORE
M. F. M. OsBorneWashington, D.C.
Five stock price sequences are examined quantitatively for structureas predicated by: 1) a random walk model; 2) a continuously dif-ferentiable price process; 3) a dynamic model consisting of transientsof a discrete process. The first and third models also make predictionsin agreement with trading lore. The data are examined by the methodof coincident events. Positive evidence is found for both the randomwalk and discrete transient model, and slightly against the continuousprice process. The theoretical predictions seems better confirmed bydata at price minima than price maxima. The data are in partial dis-agreement with the predictions of both the random walk and discretetransient model that large volume and large second differences of priceshould tend to occur at the same time. Some confirmation is found foritems of trading lore not predicted by theory. The non-random prop-erties of stock prices are primarily found in short interval data (dailyand weekly) and in individual stock prices as opposed to an average.
1, INTRODUCTION
N A pREvious paper [7] we described a model for the dynamics of stock
trading, which was based on a generalization of the types of orders used tobuy and sell stocks and their manner of execution. This model led to a pricesequence as a discrete process, being composed of a sequence of finite sectionsof “starting transients” of a difference equation. These transients were of theform Ae (t= an integer) with ) real, complex or imaginary, and were assumedto be started by concentrated bursts of orders, which in turn implies large vol-ume. The conclusions of this model were found to be in qualitative agreementwith the “folklore” of stock trading.
It is the purpose of this paper to subject the above qualitative conclusionsto a more careful quantitative examination. We shall at the same time test forsome properties of other price models, notably the random walk model, and aprocess which generates prices which are functions of the time, continuous withcontinuous derivative, and not further specified. We shall also test for thevalidity of certain concepts in the folklore of stock trading. We shall use themethod of coincident events in our analysis, exploiting the analogy betweenstock market and Geiger counter data [6].
We can motivate our definitions and methods of analysis by some simpleexamples, which are intended to show that the method of coincident events is
321322 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1987
merely a formalization of a simple and well known learning process. Let usimagine, in the days when the mercury barometer was first invented, that itwas observed once a day, and the height of the mercury or pressure, p=p (t),(cf., price of a stock) recorded. Not many days would be required to convincethe observer that the measurements fluctuated in the neighborhood of 30inches, with occasional larger excursions upward or downward. Nor would itbe long before an observant individual might suspect that these excursionswere associated with unusual weather. Let us formalize this for a very simpleexample, and indicate some generalizations.
Define a barometric “event” B (t) as occuring on any day t on which therecording was less than 28 inches. Suppose there were observed N.1;3(B) =4 ofthese in one year, out of T=365 “trials”. Define a storm event S(t) as a dayon which trees were blown down. Suppose N.43(8) =5 of these occurred in ourT =365 trials.
Now suppose there were two days in the year when a barometric event anda storm occurred on the same day. Let us call this event a simple coincidenceor binary event, and denote it by (B(t), S(d)), the time argument ¢ being thesame for both to indicate a coincidence. So we have Nors(B(t), S()) =2. Arethese two coincidences convincing evidence that there is a connection betweenthe occurrence of a B and an S event? Intuitively one would certainly be sus-picious that such was the case. A calculation will show that the suspicion iscertainly justified.
Assume: 1) B events occur independently of other B events; 2) S eventsoccur independently of other S events; 3) B and S events occur independentlyof each other; 4) B and S events have the same probability from day to day(i.e., time independent). Then the relative frequency, or estimated probabilityof a B event, for one trial, is P(B) =N.o1.(B)/T =4/365. For an S event theestimated probability is P(S) =Nors (S)/T =5/365. The estimated probabilityof a coincidence at one trial is the product of these probabilities, P(B)P(S).The “theoretical” or expected number & of coincidences in T trials is then
Niner = &(N(B(é), S()) = 365-(4-5)/(865)? = 0.055
The significance probability, or probability of actually observing two or morecoincidences under the null hypothesis of the four assumptions above is givenapproximately by the Poisson distribution. Let \=0.055 be the theoretical orexpected number of coincidences in a year.
P(Novs = 2| = 0.055) = D) e*/k! & 2/2 = 0.0015 (1)k=2Roughly one year in seven hundred would contain two or more (B(é), S(d))coincidences, if these two events occurred independently, and we conclude, atthe significance level 0.0015, that barometric and storm events are not inde-pendent.Appendix I contains a derivation of the probability of coincidence when thePoisson formula is not valid. It reduces to the Poisson formula in the case ofsmall numbers of events, relative to the number of trials.QUANTITATIVE TESTS FOR STOCK PRICE MODELS 323
Evidently one can define other types of events and coincidences. A pressurechange event would occur if | Ap(t)| = | p(t) —p(t-)| exceeded some specifiedminimum. A “delayed coincidence” (| Ap(t) | , S(t+1)), would denote a pressurechange event followed in one day by a storm. Colloquially, a negative Apevent might be called a storm predictor (forecasting by one day), whereas thelow barometer event B was a storm indicator (happening ‘now’).
It should be evident from the above discussion that much of folklore orpersonal knowledge is acquired and expressed by observation of coincident ordelayed coincidence events. We are going to define certain events in the se-quence in time of stock market prices and volumes, and examine both marketfolklore and the predictions of some price models, using the method of coinci-dent events. Before doing so we must point out some essential elements in theprocess of formalizing the method, and some limitations.
First, the data must be expressed as a discrete sequence of trials, even though(as in the case of the barometer) it may be originally expressed continuously.Second, the events must be defined so that just one event of a given type un-ambiguously did or did not occur just once in each trial. Third, the mathe-matics is much more simplified, if the events are defined so as to be relativelyinfrequent compared to the number of trials—i.e., so that the estimated prob-ability (number of events observed (single or coincidences) /number of trials)is small. How small is “small enough” is sometimes debatable. A rule of thumbfor the probability would suggest 0.10, or perhaps 2/./number of trials, willusually be satisfactorily small. This rule of thumb is suggested by examiningthe neglected terms and approximations required to obtain equation A6 inthe appendix.
We should point out that the Poisson series can be used to test both for anexcess or a deficiency, relative to independence of events. In the case of a de-ficiency, the expected number A of coincidences must be greater than about 4,or the test for deficiency cannot be very effective. The reason for this is thatthe “low frequency tail” of the Poisson distribution (Eq. (1)) starts at k=0(no observed binary events at all) and P(k=0) =e”, or e-*=0.018. No ob-served coincidences, if 4 are expected, is not quite significant at the 1% level,but is at the 5% level. This problem can be alleviated in practice by combiningobstrvations (see Section 7), using the fact that the sum of Poisson variablesis a Poisson variable, to reach a significant conclusion.
While we have indicated the convenience for the events to be infrequentrelative to the number of trials, they must not be so infrequent that there isdoubt, or insufficient evidence, that probability concepts are applicable. Thusa person may (erroneously) have a lifelong phobia because of one unfortunatecoincidence between two events in childhood, one of which was unpleasant.Adults are sometimes impressed by the coincidence of rare events such as someremarkable astronomical phenomenon, and the occurrence of an historicalevent on earth. But unless in practice both types are unambiguously definable,and recognizable as possibly occurring more than once, there is not much pointin using statistics to argue for a connection in the probability sense betweenthem.
Finally we should point out that although the low barometer event was324 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1967
connected with the storm event in the above example, as a storm indicator itwas wrong more than half the time. One should not look for infallibility evenwhen the statistical evidence of a connection is quite good. In a practical sensethere are many other indicators of a storm available to the observer in additionto the barometer. In our language, he can use triple and higher order coinci-dences, either as a storm indicator, or a storm predictor. One can think of manyanalogies to the above statements in the everday, unconscious use of the methodof coincident events.
2. EVENTS IN STOCK MARKET SEQUENCES
Let us now turn to stock market data, define some events in an unambiguousway, and indicate the kind of coincidences that might be expected to occurbetween them on the basis of folklore, or some price-generating models.
The four basic sequences we shall consider, specifying a basic or single“trial” interval of a day, week, or month, are the sequence of high prices p,(t),low prices p,(é), closing prices p,(t), and the volume V(t). These are the fourbasic sequences plotted on the ordinary chart of stock prices. Note that bydefinition the closing price must fall in the interval between the high and lowprice for the same interval, pi(t) < p.(é) < pn(d). Hence it follows from definitionsthat some events defined for one of these three sequences will not be indepen-dent of other events defined for the other two.
The events below are defined in such a way as to be easily and unambiguouslyrecognized on the charts. They are certainly not the only definable events, andthey may not even be the best definitions for the purposes intended. They haveonly the merit of simplicity and ease of recognition. Fig. 1 illustrates the eventsand the coincidences defined below.
A simple maximum (minimum) occurs in a sequence when a member of thesequence is greater (less) than the nearest preceding and following memberwhich is different. Note that price and volume data appear in multiples of §dollar or one round lot, so that this definition allows several maxima to appearin sequence without an intervening minimum, and conversely.
A ten per cent S (Superior) event is a simple maximum in the sequence ofhighs p,(é) for which there were preceding and following trades in the marketfor the particular stock considered at ten per cent less than the price at S andnone greater than at S of closer proximity. A ten per cent I (Inferior) eventis similarly defined from minima in the sequence of lows p;(i). The ten per centfigure is arbitrary; it is simply picked large enough to make the I and S eventssufficiently infrequent. I and S events obviously do not occur independentlyof each other. They tend to avoid each other, and to alternate in appearancefor a given stock or average.
An alternative definition of particular S and J events, are the “day away”events Sua; Iaa (reference 3, p. 334). This means, for Sa, that there are nearbypreceding and following days of trading for which the high-low range is “awayfrom” a closer day’s trading; i.e., the high of one day is less than the low of apreceding day’s trading nearer to or on the Sa, day, and similarly for some daypreceding the Sa. If two Saa events occur without an intervening Ia, chooseonly the highest of these Sz.. If two or more such Sz, events are highest (ex-325
QUANTITATIVE TESTS FOR STOCK PRICE MODELS
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 326 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1967
actly equal), choose all. Sao, Ia events are even easier to pick out on chartsthan per cent S and I events, but occasionally give rise to ambiguities. Forexample, on Fig. 1 the price maximum in the third week of May never be-comes a “day away” maximum. The maximum on June 1 is not identifiable asa “day away” maximum until about June 17.
It is evident that S and I events of either the percentage or day away typecannot be immediately identified as such at the end of the day on which theyoccurred, but are almost always unambiguously dated for a given span of data.
A volume event is defined as occurring when the volume of any day (week,month) is larger than the two preceding simple maxima in the volume series.From this definition, several volume events can occur in immediate sequence,the last of which will itself be a simple maximum. Generally speaking a volumeevent will be a day of relatively large volume. But if there is one volume eventof quite large volume, we can have quite large volumes on succeeding dayswithout their being defined as volume events. A volume event can be immedi-ately identified at the end of the day (week, month) of its occurrence.
A | Ap| event occurs if the change of the closing price, in absolute value,| Ap-(t) | = | pelt) —pe(t—1) | , is greater in absolute value than some pre-assignedminimum, expressed either in dollars or as a percentage. A | Ap| event is(arbitrarily) dated as of the second day (week, month) of the two closing pricesneeded to generate it; i.e., a | Ap| event is considered to occur on a single day.One can also take account of the sign of Ap and refer to +-Ap or —Ap events.
A |A@p| event occurs when the absolute value of the second difference ofclosing price, | Ap.(t) —Ap.(t— 1)| , exceeds some arbitrary minimum. It is con-sidered to “occur” on the last two days (weeks, months) of the three consecutiveprices needed to generate a single second difference. We use this double datingconvention since we interpret a |A@p| event as an indication of large (sequen-tial) dispersion [6] in the sequence of numbers Ap(é), Ap(¢+1), ete. |A®p| isthe range of a sample of two consecutive Ap’s from the population of sequentialAp’s, and a sample of two is the smallest possible sample which can be used toestimate dispersion.
A range event R occurs when the difference of the two sequences, p,(é) — pi(t),on a single day (week, month) is greater than some pre-assigned minimum.
The above six events are defined so that not more than one event of a givenkind occurs* in one basic interval, for one sequence. For any one basic interval(day, week, month) we can from the data decide unambiguously that an eventof a given type did or did not occur.
It should be noted that of the six events defined above, four of them, V,|Ap|, |A@p| , R can be immediately and unambiguously identified as havingoccurred at the end of any given basic interval. Unfortunately this is not true ofthe I and S events, though they can be unambiguously identified from a com-plete sequence of data. I and S events are obviously good points to buy andsell. Hence, if one can find a statistically significant degree of coincidence be-tween I and S events and the remaining four, these four events may provide
 
* This is not strictly correct for | A@p| events, since if four or more large Ap events occurred in sequence,of alternating sign, there would, by our definition be two | Ap| events on one day. In practice we have not seenthis happen, although theoretically it could occur.QUANTITATIVE TESTS FOR STOCK PRICE MODELS 327
some practical trading signals. In the stock market this is called “technicalanalysis”.
A second point to observe is that the six types of events can be divided intothree groups, and within each group the events are certainly not independentof each other. These three groups are:
1. I and S events. These two events tend to avoid coincidence both withthemselves and each other, simply from the way they are defined, Thus coinci-dences of the Type (I(t), S(t)) and delayed coincidences of the Type ((t—1),I()), occur less often than if these individual events occurred completelyindependently.
2. Volume events V. Volume events do not appear independently of eachother. In fact they tend to cluster, since it is known that volume tends to appearin bursts sometimes lasting longer than a month [6]. Delayed coincidences ofthe type (V(t—1), V(é)) probably appear more frequently than if V eventsoccurred independently of each other.
3. |Ap|, [|A@p| and R events also tend to coincide with each other, simplyfrom the way they are defined. The first two are defined from the closing se-quence p-(t), the last from the difference p,(t) — p(t). The high, low and closingsequences are connected by the relation p(t) <p.(t) <pi(t). Hence, large firstor second differences events in p(t); | Ap| , [A®p| , and the range event FR willall tend to coincide. Binary coincidences of the type (| Ap(t), |A@p(t) | ), RW,| Api) | ), RY, |A®@p(#)|) will all tend to occur more frequently than if theseparate events occurred independently (see upper part of Table 1).
What can we say about binary coincidences where the two members aredrawn from different groups above, say an (R(t), V()) or an (S(t), V())coincidence? Our definitions do not give us any clues here. One might thenreasonably assume independence as a null hypothesis, and then use data to seeif the null hypothesis is rejected. Trading folklore, and different price generatingmodels do make definite inferences about coincidences of events from thesethree different groups, and it is just these inferences we wish to test. In sections3 through 6 we identify the coincidences to be expected from folklore and pricemodels, with brief references to the tables where data on these coincidencesappear. In section 7 we examine this data in detail.
3. COINCIDENCES EXPECTED FROM FOLKLORE
Consider the statement “It takes volume to make prices move”. We cantranslate this into a statement about coincidences between our events asfollows: “Move” could mean large motion up or down in one day (week,month), a | Ap| event. “Move” could also mean a violent change of motion ontwo successive days, a |A®p| event, or violently up and down within a singleday, which would mean an R event. So the statement means there are morecoincidences of the type
VO, | APO), VO, |4%PO|), VO, RO)
than are given by the hypotheses that V events occur independently of | Ap| ,[A®p| or R events (see lower half of Table I).A slightly different statement, “It takes volume to put prices up, but they328 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1967
can fall of their own weight”, can be translated to imply more coincidencesof the type (V(t), +Ap(t)) than (V(),—Ap(é)), the former being more frequentthan if V and +Ap events occurred independently. We have not tested forthis item of folklore.
The statement “important moves tend to end on large volume, or climac-tically” can be translated into statements about coincidences as follows. Theend of an important move is likely to be an J or S events, if the subsequent pricemotion is in a direction opposite to the previous motion. It does not have tobe an J or S event if the “subsequent motion” is in the same direction at areduced rate, or in fact no motion at all, and the price sits still (in technicalanalysis a horizontal “line”). “Large volume” evidently implies a volumeevent. “Climactically” could imply violent motion up and down in one or moresuccessive days, i.e., | Ap| , [A®p|, or R events. Taken as a whole this state-ment suggests looking for more frequent occurrence of coincidences of J and Sevents (Group 1) with the remaining four (Groups 2 and 3) than would begiven by the null hypothesis that I and S events occurred independently of theremaining four events (see Table 2).
In our previous paper [1] we argued that “something happening” in themarket implied V, [Ap| , [A®p| , and R events. If “something happening” alsomeans an important turning point, or change of sign of the expected value ofAp, this means an I or S event. This is the same as the conclusion just reached.
Some of the more sophisticated tenets of stock data interpretation seem toimply that the volume sequence foretells or leads the price sequence [1]. Thisstatement suggests looking for an excess of delayed coincidences of the form(Vii—1), S@®) and (V(t—1), I), and a deficiency in delayed coincidences ofthe form (S(t—1), V(#)), and (I(t—1), V(#)), Table 4. The evidence is small butpositive. It may be that different definitions of events and coincidences wouldmake better tests for these folklore statements,
4, COINCIDENCES EXPECTED FROM THE RANDOM WALK MODEL
Let us now translate some properties of different price models into state-ments about coincidences. Suppose the sequence of transacted prices in a giveninterval, say one day, is a simple coin tossing random walk, with the numberof tosses, or steps proportional to the number of transactions, or volume. Wefurther suppose that the random walk on each day starts where it ended at theclose of the previous day. From this price model it follows that: (1) | Ap.(2)| 7the net distance traversed in the walk from its daily starting point, (the previousclose) ; (2) | Ap. (2), a crude measure of the dispersion of the daily sequenceof Ap,’s; and (3) the daily range R(t), all increase with the number of steps, orvolume. In terms of coincidences, this means an increased frequency of coinci-dences of the type (V(é), R()), (VM, | Ap(t)| , and (V(é), |A®p)]), relativeto their frequency under the null hypotheses that V events occur independentlyof R, | Ap| , and. | A@p(e)| events. In other words, groups 2 and 3 are not inde-pendent, for the random walk model. Note that this prediction of the randomwalk model is qualitatively the same as what we inferred from the folklorestatement that “it takes volume to make prices move”. Whether there is aquantitative difference, we do not know, though we suspect there is (Section 7).QUANTITATIVE TESTS FOR STOCK PRICE MODELS 329
The above discussion refers to the relation of big “jumps” ( |A(p)|, A@pevents) to volume, for the random walk model. In contrast, the random walkmodel, so far as we can see, does not make any predictions as to where maximaand minima (J and S events) occur; relative to the volume or to | Ap| or|A@p|. I and 8S events are defined to be of a magnitude such as 10% of theprice, and hence are relatively infrequent. We surmise using the followingheuristic argument that for this model J and 8 events occur independently of| Ap| , |A@p| , and R events. For a random walk model, the probability of anygiven combination of single steps (including those which give I or S events)is a constant, independent of what has gone before. If the number of stepsbetween observations, or volume per day varies, the probability distributionof Ap.(t) does vary (notably the dispersion) from day to day, but the mean ofthe distribution does not. Hence we suspect, but cannot prove, that the appear-ance by chance of an J or S event in the total sequence is independent of theoccurence of V, |Ap|, |A®p| or R events, if the prices are strictly generatedby a random walk. In other words, we conclude group 1 events are independentof both groups 2 and 8 events, in the random walk model.
5, COINCIDENCES EXPECTED FROM A CONTINUOUS AND CONTINUOUSDERIVATIVE PRICE SEQUENCE
Let us now suppose a second model, specified as follows. There exists anunknown mechanism which generates prices which are, for all times, contin-uously differentiable functions of that time. The observed, executed transactionprices are the prices generated by this mechanism, sampled at times correspond-ing to the executions plus an error term, and rounded off to the nearest 1/8. Thekey words are in italics. The actual mechanism is unimportant.
We can illustrate the force of these remarks by an example. Imagine a weighthung on a spring, and set in vertical oscillation. The height of this weight p(¢)(cf. price) is for all times after the start a continuous with continuous derivativefunction of the time. For such functions the calculus gives as a necessary condi-tion that the derivative dp/dt is zero at maxima and minima. This suggeststhat for observations of p(t) at discrete intervals, with an error, the difference,Ap.(t) =pe(t) — pe(t— 1), and the range R=p,(t) — pi(t) should be relatively smallat maxima and minima, provided the interval of differencing (i.e. between ob-servations), a day, week or month, is small compared to the natural dynamicalperiods (real or complex) of the mechanism. In terms of our | Ap| and Revents, implying large | Ape| and R, these events should avoid eoincidence withI and S events. This conclusion is precisely the opposite of what we concludedfrom the statements of folklore.
Concerning second differences of the “observed” prices, we can make nosuch general statements. For the particular case of the single mass on thespring, or simple harmonic motion, the absolute value of the second differenceof p(t), | A@p(e)| is in fact largest at the maxima and minima, but this is nottrue of all physical systems in general.
In summary therefore, for a price model in which the underlying mechanismgives prices which are at all times continuous with continuous derivative func-tions of the time, with dynamical periods or “proper values” longer than the
 330 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1967
interval of observation (day, week or month), J and S events will avoid coinci-dence with | Ap and FR events. There is no indication, from our arguments asto where |A@p and volume events will occur, so for this model we assumethese occur independently of J and S.
It should also be noted that the continuous and analytic model assumes ob-servations at intervals (day, week, month) smaller than the natural dynamical“periods” of the underlying mechanism. If these day, week or month intervalsare larger rather than smaller, one could well have the observations fitting arandom walk model. “Large”, rather than “small” is precisely the conditionunder which Einstein originally derived the properties of Brownian motionfrom the continuous motion of small dust particles in collision with molecules.
6. COINCIDENCES EXPECTED FROM THE DISCRETE TRANSIENT MODEL
The third model is the one we have already proposed [7]. The price sequenceis a set of transients of the form Ae™ for ¢ taking on discrete values only, thetransients being initiated by large volume. For this model, as we indicated inour previous paper (see Fig. 3, ref. 7), we expect more coincidences, relative tothe hypotheses of independence, of I and S events with |Ap|, [Ap], R, V,i.e., just the coincidences implied by the folklore statement concerning the endof “important moves” (Section 3 and Table 2). This third model also impliesexcess binary coincidences among V, |Ap|, [A®p| and R. In this respect itspredictions are qualit