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