erations research March-Aprll 1959 BROWNIAn MOTION IN THE STOCK MARKETt M. F. M. Osborne U8 Naval Research Laboratory, Washington 26, D C (ReceIved February 6, 1958) that common- stock pnces, and the value of money can be re- garded as an ensemble statistical equilbrium, with properti of partie Y=log P(t+)/P()l, where P(t+r)and Po(t)are the price of the same random choice stock at random times t+r and t, then the steady state dis- tribution function of Y 18 (Y)=exp(y/2o)/v2no't, which 1s pr sely the probability distribution for a particle in Brownian motion the dispersion developed at the end of unit time A similar distribution holds for the value of money, measured approxmately by stock-market Indices Sufficent, but not necessary conditions to derive this distribu tion quantitatively are given by the conditions of trading, and the Weber Fechner law A consequence of the distribution function is that the ex ectation values for prce itself &(P)=fo Pp(r)(dr/dp)dP increase actuation, or dispersion, of P This secular increase has noth ine to do wtn long-term infiation, or the growth of assets n economy, since the expected reciprocal of price, or number of shares pur chasuble in the future per dollar, Increases with r in an identical fashion T IS THE PURPOSE of this paper to show that the logarithms of com- mon-stock prices can be regarded as an ensemble of decisions in a statisti cal steady state, and that thus ensemble of logarithms of prices, each varying wIth the time has a close analogy with the ensemble of coordinates of a large number of molecules We wish to show that the methods of statisti mechanics, normally applied to the latter problem, may also be appled to the former Although the results of this paper were first reached inductively from direct examination of the data on prices, for the sake of clarty we shall present them, at least In part, n a deductive fashion, and compare the t Read before the US Naval Research Laboratory Sold State uary28,1958 145
146 M. F. M. Osborne deductions wIth observations on prices The fundamental facts, as sumptions, and critical points in the derivation of the properties of this ensemble, are given in the numbered paragraphs below These facts and assumptions are sufficient, but not necessary to obtain agreement of theory with observation 1. Prees move in discrete unIts of 1 of a dollar From this it immedate follows that the natural logarithms of prices also move in discrete unIts, approxi mately 1/(8xprce) We shall call loge-unts ratio units A ratio unt of +100 orresponds to a ratio of +2 817/1 We point out for the convenience of the reader, that percentage changes of less than +15 per cent, expressed as fractions from unIty, are very nearly natural logarithms of the same ratio Thus logr(100+15)/1001≈+015 2. There 1s a finIte, Integral number of transactions, or decisions, per unIt me This number may vary from zero to a thousand or more per day for a single stock Hence the decisions also are separated by discrete units of time Observationally, the number of decisions per day may be estimated to be not more than the volume in round lots It may be less, for there may be more than one round lot per trade 3. The stimulus of price in dollars, and the subjective sensation of value in the mind of the trader or investor, are related in accordance with the weber-Fechner aw As this assumption has engendered some controversy, let us specify pre- Isely its meanng The Weber-Fechner law states that equal ratios of physical tumulus, for example, of sound frequency in vibrations /second, or of lght or sound intensty in watts per unt area, correspond to equal intervals of subgectue ensation, such as pitch, bnghtness, or noise The value of a subjective sensation, lke absolute position in physical space, Is not measurable, but changes or differences In sensation are, since by experiment they can be equated and reproduced thus fulfilling the criteria of measurablity The Weber- Fechner law Is best applicable when there is a single dor nant, or prmary stimulus Thus in comparing two sounds, or two light es, the frequency distrbution(pitch or color) must be nearly the same, or the errors of comparison are large Thus, in assuming the Weber Fechner law, we are relegating earnings, dividends, management, etc and their future outlook to postions of secondary Importance These factors may be Important, Just as the intermolecular force law 1s Important In molecular problems, In determining departures from the steady state For determining the steady state, we ignore them in the molecular problem and we ignore their analogs in this problem also The hypothesis that price and value are related by the Weber-Fechner law can be reached Inductively from the raw data by the following rather sImple-minded argument Let us imagine that a statistician, trained perhaps in astronomy and totally unfamiliar wnth finance, Is handed a page of the Wall Street Journal containing the n y Stock Exchange
Brownian Moton in the Stock Market 147 (NYSE)transaction for a gIven day. He 1s told that these data consti- tute a sample of approximately 1000 from some unknown population, together wnth some of their more important attributes or varables, eleven In all The fact that these eleven were the most important, out of a much larger number obtainable, from annual reports, for example, might be ferred from the fact that this choice of eleven was published every day Our statistician 1s asked to investigate this population, to determine If It 1s a homogeneous sample, and what relations (in the probabIlty sense) exist between the varables or attributes lsted for each member The methods of attacking the raw data on such a problem al nown, especially to biologists, we quote an astronomical reference l4 primarly because of personal famIliarty A common first step 1s the determination of distrbution functions Casual Inspection of the data eveals that of eleven attributes or varables lsted for each member of the population, Six, exclusive of the change, are devoted to something called prce, 'evidently a dominant variable even among those so important 120 Fig 1 Distribution function of prices as to be published every day On learning that close'was the most recent data, our statistician would plot the distrbution function of closing price alone for the 1000 members of the sample(Fig 1) Inspection of Fig shows that closing prices on that day were certainly not normally dis tributed, but the shape suggests that logarithms of prices might beE 1 suggests a loganthmuc-normal distrbution Figure 2 gives the Identical data of Fig 1 wth loge price as independent variable( see refer ence 1, page 9 for numerical methods) At this point our statistician will make a'discovery'and answer one th he questions posed to him A subsIdiary maximum around log Pa4 5 PA$100) In Fig 2 suggests that the population contains at least two sub-groups, 1e, It Is not homogeneous Re-examination of the raw dat around Pas100 reveals an excessive number of our sample wnth the attrbute'pfd'(preferred), and plotting the distribution function of these only gives Fig 3 The remaining members of the population, the common or ordinary ones is plotted n Fig 4 This appears normal, and as a rough
M. F. M. Osborne test, the cumulated distrbution 1s plotted in Fig 6 at thus stage our statistician can say that the population, insofar as the distribution wth respect to price 18 concerned, appears to be divded into perhaps three 与auoaooz 200 RATIO UNIT SCALE (LOGe P L 030405060800o 200 Fig 2 Distribution function for log, on July 31, 1956(all items NYSE) classes, If one can regard the two subdivisions of Fig 3 as signIficant a more searching examination of the data may reveal others(cf Fig 10) 5Hu 10o 2325 RATIO UNIT SCALE (LOGe P 星52004056°的160200 Fig 3. Distribution function of logeP for preferred stocks Our statistIcian lkes to choose an independent variable (log. in this case)that renders the data approximately normally distributed The testing of a statistical hypothesis 1s thereby greatly facilitated, and analo
Brownian Motion In the stock Market 149 gies wIth many other populations, also normally distributed, may be successfully exploited in trying to understand the new one A rationale for the use of loge P in preference to P as independent var able 1s also given by the general statistical precept that equal intervals of the argument chosen as independent varable should have equal physical or in this case psychological, signIficance, for the data to be most reveal (reference 1, p 6) This choice was confirmed by the resulting discover of the preferred stocks This equal-interval'argument imphes that the difference in subjective sensation of profit (or loss), or change in value, for example, between a $10 and an Sll price for a given stock, Is equal to 300一 2325 RATIO UNIT SCALE (LOG. P) 20304050607000150 Fug. 4 Distribution function of log, P for common stocks (NYSE, July 31, 1956) that for a change from $100 to $110 Thus our statistician 1s led to hypoth cate the Weber-Fechner law, and the dominance of a single variable in the stImulus, from the observational procedure outlined above Figure 5 gives the closing prices of common stocks on the american Stock Exchange(ASE)and Fig 6 its cumulated distribution, as a rough test of normality The introduction of the Weber-Fechner law as a working hypothesIs now leads our statistician to examine price changes that occur in indivdual stocks, since by hypothesis the absolute level of price 1s of no signIficance only changes in prices(specIfically Aloge P or the loge of price ratios)can measured by traders or investors Histograms of th hese for Interval for interve th are published n The Exchange Accumulated distrbutions of o of a month and a year are given n Figs 7 and 8 Note that
F. M. Osborne RATIO UNIT SCALE (LOGe P) Fig 5 Distrbution function of log P for common stocks 8 SCALE FOR NYSE DISTRIBUTION 55=6396 95 70 60 B SCALE FOR ASE DISTRIBUTION Fig. 6. Cumulated distrbutions of log P for NYSE and asE
Brownian Moton in the Stock Market l51 for both intervals the distributions are nearly normal in ratio unIts ThIs Is slightly less true for percentage unIts in which the data was orginally published The effect Is less noticeable in the monthly data, where the percentage changes are small, and hence nearly equivalent to ratio unIts his nearly normal distribution in the changes of logarithm of prices uggests that It may be a consequence of many independent random vanables contributing to the changes in values(as defined by the Weber Fechner The normal dIstribution arses In many stochastIc proc o0o d03 Fig 7 Cumulated distrbutions of Aloge P=logeIP(+r)/P(e] for 1=1 (NYSE common stocks) These, and also Fig distrIbutions of S()for fixed M*() sold line is the distrbution of Z)M(), transcrIbed from Fig 12 for comparison esses involving large numbers of ndependent variables, and certaInly the market place should fulfill this condition, at least 4. As a fourth element in our analy sIs, we would lke to define alog Ision As an elementary example let us suppose we must make a decision urse of action A, and course of action b We know, or can estimate ense)that course of action A has possible outcomes Yal, yt with probabilties p(Yan),(Ya2), ete, whIle a decision for b has possble outcomes YBl, YB2, wIth probabIlties p(YB,P(Y 2), ete Then the logieal choice 1s to make a decision for A, or B, for whuch the expectation value, & of the outcome, &(YA)-E, YA, P(Yad) or 8(rB)=X,YB, (YB, is the larger
152 M. F. M. Osborne Evidently decision problems can be much more complicated than this xample They may involve several alternatives and sequences of de cisions in which the estimate of the probabilities and payoffs (the ys) are interrelated The general approach is the same to maximize-using 995 8 90 98 08-06-04-020+02+4+06+08 △ LOG. F=LOGe -50 十50 100 scAle ng 8 Cumulated dis cons of Alog P=log[P(t+r)/P(t)l for r=l Data from NYSE Year Book, 1956, an the given nformation, estimates of probabilties, payoffs, and restraints- the expectation value of the end result In view of our previous remarks we might illustrate the above example with a stock - market decision a trader has sufficent capital to bi hundred shares of a corporation, now(time t) sellIng at Po(t) He wIshes to ncrease his capital and can choose between A, buying for future sale at some time t+T, or not buying, b The Y's refer to possable changes In
Motion in the Stock Market 153 the logarithm of the price of 100 shares, le, Ya(T)=Aloge[100 P(t) loge[P(t-+r)/Po(O)l, since by hypothesis it is this quantity that is measur able In the trader's mind There is only one YB, zero, the null chang wIth probabity one, a certainty The logical de cislon'to buy or not buy 1s thus determined by whether the estimated expectation value of Ya(T) Is positive or negatIve We do not claim that the trader sits down and consciously estimates the y's and p(y),'s, any more than one could claim that a baseball player consciously computes the trajectory of a baseball, and then runs to inter cept It The net result, or decision to act, Is the same as If they did In both cases the mind acts unconsciously as a storehouse of information and a computer of probabilties, and acts accordingly Now let us examine the nature of the decisions, of which the published prices gives a numerical measure, concerning the common-stock lstings of NYsE These prices represent decisions at which a buyer is willing to acquire stock (and sell money )and a seller Is wling to dispose of stock and hence buy money There are, therefore, in each tran nsaction two types of decisions being made by each participant From what has beer sald about the anatomy of logical decisions(they need not be consciously logical, but this 1s the supposition as to how they are reached ), we must suppose that for the buyer, his estimate of the expectation value for the change m value(Aloge P) for the stock Is positive, while the seller,s estI mate of the same quantity Is negative Presumably, the reverse situation holds in the minds of buyers and sellers for the estimated expectation value for changes in the value of (their second decision), though we have not yet specIfied how changes in the value of money are measured In this situation In view of the equalty of opportunIty in bidding between buyers and rs, In accordance with the regulations of the Exchange, It would appear that the most probable condition under which a transaction is consum- mated,and a price or decision Is recorded, 1s obtaned when these two stomates are equal and opposite, or E8(△logP)g+E8(△logP)B=0, (1) where P denotes price per share, and E the estimate of the expectation value Hence we can say that for the market as a whole, consisting of buyers and sellers F8(△logP)M=B+8=0 Is the condition under which transactions are most probably recorded A few moments later another transaction may be recorded for the same stock at a slightly different prce, and again equation(2)will most probably be applicable, and so on for succeeding transactions One might even
M. F. M. Osborne argue that in equation(2)the symbol E for estimate could be dropped since in such buying and selling the decisive estimates are definitive of actual value Or to put It differently, If enough people decide and act on the belief that something is valuable it Is valuable at that time 5. The above contains a critical point n our argument, Ie, the most probable ondition under which a transaction 1s recorded is given by equation(2) In words, this states that the contestants are unlkely to trade unless there is equalty of opportunIty to profit, w hether an indivdual happens at the moment to be a buyer or a seller, of stock, or of money The Exchanges are certainly governed but we also feel that this condtion must have obtained prior to any regulation, since every buyer, once having consummated his trade now finds himself as a potential seller in the virtual postion of his opponent wIth whom he was so recently hagghng The converse sltuation apples to the seller, now a potential buyer Under these circumstances It 1s difficult to see how trading could persist unless pnces moved in such a way that equalty of op- tunIty most probably prevailed, and equation(2)expresses this quantitatively perhaps less as an assumption than as a consequence of assumptions 3 and 4 We now ask, what Is the effect of the condition(2)on the distribution function ultimately developed for Alog P? Our argument follows closely one originally given by giBBs for an ensemble of molecules in equilbrium The actual distrbution function Is determined by the conditions of maxI- mum probabilIty (reference 3, p 79) 6. Assuming the decisions for each transaction in the sequence of transactions In a single stock are made independently (n the probabilty sense), then under fairly general condItions outlined below, we can expect that the distnbution func tion for y()logP(t+T/Po() will be normal, of zero mean wIth a dispersion or( which increases as the square root of the number of transactiong If these numbers of transactions (the volume )are fairly unIformly distributed in time, then or( will increase as the square root of the time interval, Ie, oy() will be of the form oVT, where a is the dispersion at the end of unIt time T. Mathematically we may express this as follows Suppose we have kin dependent random varables y(),飞=1,,配, y(o)=A. logo P=log(P(t+10)/P(+(1-1 8)J where P(e) Is the priee of a single stock at time t and 8 is the small time interval between trades t Assume that each y(i) has the same dispersion o(i)=o, then after k trades, a time T= k8 lat define Y() Y()=F(6)=∑=iy()=logP(t+)/P()=4logP() e8 1, g In parentheses will refer to independent random variables In a time As subscripts, 1, wlll refer to independent varables at the same rent stocks