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[World Scientific Handbook in Financial Economics Series] The World Scientific Handbook of Futures...
[World Scientific Handbook in Financial Economics Series] The World Scientific Handbook of Futures Markets Volume 5  Proof that Properly Anticipated Prices Fluctuate Randomly
Malliaris, Anastasios G, Ziemba, William Tكم أعجبك هذا الكتاب؟
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المجلد:
10.1142/89
عام:
2015
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english
DOI:
10.1142/9789814566926_0002
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July 3, 2015 8:27 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 CHAPTER 2 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. PROOF THAT PROPERLY ANTICIPATED PRICES FLUCTUATE RANDOMLY Paul A. Samuelson Abstract: By positing a rather general stochastic model of price change, I shall deduce a fairly sweeping theorem in which nextperiod’s price differences are shown to be uncorrelated with (if not completely independent of) previous period’s price diﬀerences. This martingale property of zero expected capital gain will then be replaced by the slightly more general case of a constant mean percentage gain per unit time. Keywords: Martingale property, stochastic behavior of prices. The Enigma Posed “In competitive markets, there is a buyer for every seller. If one could be sure that a price will rise, it would have already risen.” Arguments like this are used to deduce that competitive prices must display price changes over time, Xt+1 − Xt , that perform a random walk with no predictable bias. Is this a correct fact about wellorganized wheat or other commodity markets? About stock exchange prices for equity shares? About futures markets for wheat or other commodities, as contrasted to the movement of actual “spot prices” for the concrete commodity? Or is it merely an interesting (refutable) hypothesis about actual markets that can somehow be put to empirical testing? Or is it a valid deduction (like the Pythagorean theorem applicable to Euclidean triangles) whose truth is as immutable as 2 + 2 = 4? Does its truth follow from the very deﬁnition of “free, competitive markets?” (If so, can there fail to exist in New York and London actual stock and commodity markets with those properties; and must any failures of the “truism” that turn up be attributable to “manipulation,” “thinness of markets,” or other market imperfections?) The more one thinks about the problem, the more one wond; ers what it is that could be established by such abstract argumentation. Is the fact that American stocks have shown an average annual rise of more than 5% 25 page 25 July 3, 2015 8:27 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. 26 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 P. A. Samuelson over many decades compatible with the alleged “fair game” (or martingale property) of an unbiased random walk? Is it an exception that spot wheat prices generally rise (presumably because of storage costs) from the July harvest time to the following spring and drop during June? Is the fact that the price of next July’s future shows much less strong seasonal patterns a conﬁrmation of the alleged truism? If so, what about the alleged Keynes– Hicks–Houthakker–Cootner pattern of “normal backwardation,” in which next July’s wheat future could be expected to rise in price a little from July harvest to, say, the following March (as a result of need of holders of the crop to coax out, at a cost, riskdisliking speculators with whom to make shorthedging transactions); and what about the Cootner pattern in which, once wheat stocks become low in March, processors wishing to be sure of having a minimum of wheat to process, seek shortselling speculators with whom to make longhedging transactions, even at the cost of having the July quotation dropping a little in price in months like April and May? Consideration of such prosaic and mundane facts raises doubt that there is anything much in celestial a priori reasoning from the axiom that what can be perceived about the future must already be “discounted” in current price quotations. Indeed, suppose that all the participants in actual markets are necessarily boobs when it comes to foreseeing the unforeseeable future. Why should “afterthefact” price changes show any systematic pattern, such as nonbias? Are the very mathematical notions of probability of any relevance to actual market quotations? If so, how could we decide that this is indeed so? Whatever the answers to these questions, I think we can suspect that there is no a priori necessity for actual Board of Trade grain prices to act in accordance with speciﬁc probability models. Perhaps it is a lucky accident, a boon from Mother Nature so to speak, that so many actual price time series do behave like uncorrelated or quasirandom walks. Thus, Maurice Kendall (1953, 1964) almost proves too much when he funds negligible serial correlation in spot grain prices. For reasons that I shall discuss, we would not be too surprised to ﬁnd this property in futures price changes. But surely spot prices ought to vary with shifts in such supply and demand factors as weather, crop yields, and crop plantings; or population, income, and taste changes. Who says that the weather must itself display no serial correlation? A dry month does tend to the followed by a dryerthanaverage month because of persistence of pressure patterns, etc. Perhaps it is true that prices depend on a summation of so many small and somewhat independent page 26 July 3, 2015 8:27 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. Proof that Properly Anticipated Prices Fluctuate Randomly 27 sources of variation that the result is like a random walk. But there is no necessity for this. And the fact, if it is one, is not particularly related to perfect competition or market anticipations. For example, consider a monopolist who sells (or buys) at ﬁxed price. If the demand (or supply) curve he faces is the resultant of numerous independent, additive sources of variation each of which is limited or small, his resulting quantity {qt } may well behave like a random walk, showing variations like the normal curve of error. At this point, the reader may feel inclined to doubt that the arguments of my ﬁrst paragraph have even a germ of interest for the economist. But I hope to show that such a rejection goes too far. By positing a rather general stochastic model of price change, I shall deduce a fairly sweeping theorem in which nextperiod’s price diﬀerences are shown to be uncorrelated with (if not completely independent of) previous period’s price diﬀerences. This martingale property of zero expected capital gain will then be replaced by the slightly more general case of a constant mean percentage gain per unit time. You never get something for nothing. From a nonempirical base of axioms you never get empirical results. Deductive analysis cannot determine whether the empirical properties of the stochastic model I posit come at all close to resembling the empirical determinants of today’s realworld markets. That question I shall not here investigate. I shall be content if I can, for once, ﬁnd deﬁnite and unambiguous content to the arguments of the opening paragraph — arguments which have long haunted economists’ discussions of competitive markets. A General Stochastic Model of Price Let {. . . , Xt−1 , Xt , Xt+1 , . . . , Xt+T , . . .} represent the time sequence of prices — as for example, the price of spot #2 wheat in Chicago (or it could be a vector of prices or even quantities of several diﬀerent goods). Given knowledge of today’s price and of past prices [Xt , Xt−1 , . . .], suppose we cannot know with certainty tomorrow’s price Xt+1 , or any future price Xt+T . Suppose there is at best a probability distribution for any future price, whose form depends solely on the number of periods ahead over which we are trying to forecast prices, given by Prob{Xt+T ≤ XXt = x0 , Xt−1 = x1 , . . .} = P (X, X0 , x1 , . . . ; T ). (2.1) page 27 July 3, 2015 8:27 28 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 P. A. Samuelson The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. These P ’s are assumed not to depend on, or change with, historical calendar time. In that sense, I posit a “stationary” process (but one consistent with, say, chronic inﬂation). From the fundamental logic of probability, we can relate these probabilities relevant to diﬀerent future forecast spans; T = 1, 2, . . . . All are ultimately deducible from the basic oneperiod P (X, x0 , x1 , . . . ; 1). Thus, always P (X, x0 , x1 , . . . ; 2) ∞ = P (X, y, x0 , x1 , . . . ; 1)dP (y, x0 , x1 , . . . ; 1) (2.2) −∞ P (X, x0 , x1 , . . . ; T ) ∞ P (X, y, x0 , x1 , . . . ; T − 1)dP (y, x0 , x1 , . . . ; 1). = −∞ These general Stieltjes integrals include summation of ﬁnite (or countablyinﬁnite) probabilities and ordinary integrals of probability densities; no real ambiguity concerning the variable over which dP is being integrated will usually arise. In words, (2.2) merely says that the probability of an event’s happening is the sum of the probabilities of the diﬀerent mutuallyexclusive ways by which it could happen. If P (X, x0 , x1 , . . . ; 1) had the Markov property of being quite independent of (x1 , x2 , . . .), (2) would be the socalled Chapman– Kolmogorov equation. But I do not assume any special Markov property. The generality of (1) and (2) must be emphasized. Nothing necessarily Gaussian or normal is assumed about any P (X, x0 , x1 , . . . ; T ). It is possible, but not necessarily assumed, that an ergodic state for P will emerge in the limit as T goes to inﬁnity. Thus, lim P (X, x0 , x1 , . . . ; T ) T →∞ = P (X) independently of (x0 , x1 , . . .) (2.3) would involve such an ergodic property. Here are some examples of possible processes that deﬁne the P ’s. Suppose {Xt } satisﬁes a Yule–Wold autoregressive linear equation of the type Xt+1 = aXt + {ut }, a < 1, (2.4) with {ut } independent drawings from the same table of random digits or of Gaussian variates. page 28 July 3, 2015 8:27 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. Proof that Properly Anticipated Prices Fluctuate Randomly 29 Or suppose the Xt price can take on only the ﬁnite discrete values [Q1 , Q2 , . . . , Qn ]; and let aij , the nonnegative coeﬃcients of a Markov transitionalprobability matrix, represent the probability that a price now observed to be Qi will one period later be observed to be Qj , where 2 j aij = 1 for all i. If [aij ] = A, the reader can verify that A deﬁnes P (X, X0 ; 2) and AT deﬁnes P (X, x0 ; T ). If all aij > 0, A∞ is a welldeﬁned ergodic state, composed of the matrix [aj ] with identical rows and hence independent of the initial observed Qi value. As a third possible model let {Xt } take a random walk in the sense of doubling in the next period with probability 1/3 and halving with probability 2/3. This deﬁnes P (X, x0 ; 1), and the reader can verify that (2.2) gives for P (X, x0 ; 2) the property that Xt+2 /Xt will be (4, 1, 1/4) with respective probabilities (1/9, 4/9, 4/9), and so on with the usual binomial distribution for each T and with the centrallimit theorem showing that a normal distribution is approached for log(Xt+T /Xt ) as T → ∞. This is an instance of a multiplicative Brownian motion applied to prices. Unlike the absolute or additive Brownian motion, it has the grace to avoid negative prices. Specification of a Model Defining Behavior of a Futures Price Now consider today’s “futures price quotation” for the actual spot price that will prevail T periods from now — i.e., the price quoted today at t for a contract requiring delivery of actual physical goods at t + T . If the present time is t, with present spot price Xt , the relevant spot price that is to prevail later is given by Xt+T . The newly deﬁned futures price, quoted today, for that future Xt+T , we might denote by Y (T, t). When another period passes, we shall know (Xt+1 , Xt , Xt−1 , . . .) instead of merely (Xt , Xt−1 , . . .); and the new quotation for the same futures price we have been talking about will be written as Y (T −1, t+1). It in turn will be succeeded by the sequence [Y (T − 2, T +2), . . . , Y (t−n, t+n), . . . , Y (1, t+T −1), Y (0, t + T )]. After t + T , there is no problem of pricing this particular futures contract. Thus, the July, 1964. Chicago Wheat Contract for delivery of wheat became closed, ancient history after July, 1964; but for more than 12 months prior to that data, its quotation oscillated from period to period and could be read in the newspaper. What relationship shall we posit between the sequence {Y (T − n, t + n)} and the sequence {Xt+1 }? When the due data for the futures contract arrives, arbitrage will ensure that Y (0, t + T ) = Xt+T , commissions aside. (2.5) page 29 July 3, 2015 8:27 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. 30 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 P. A. Samuelson A period earlier no one can know what Xt+T will turn out to be. If interest and riskaversion can be ignored, it is tempting to assume that people in the market place make as full use as they can of the posited probability distribution P (Xt+T , Xt+T −1 , Xt+T −2 , . . . ; 1) of nextperiod’s price and bid by supply and demand Y (1, t + T − 1) to the mean or, mathematicallyexpected level of tomorrow’s price. That way neither shortsellers nor longbuyers stand to make a positive gain or loss. This constitutes the rationale of my ﬁrst model of futures price, which is based on the following: Axiom of Mathematically Expected Price Formation. If spot prices {Xt } are subject to the probability distribution of (2.1), a futures price is to be set by competitive bidding at the nowexpected level of the terminal spot price. That is, Y (T, t + T ) = E[Xt+T Xt , Xt−1 , . . .], (T = 1, 2, . . .) ∞ XdP (X, Xt , Xt−1 , . . . ; T ). = (2.6) −∞ The Basic Theorem Equations (2.1) and (2.6) completely determine the properties of the model. I can now derive from them the basic theorem at which I hinted in the usual vague arguments expressed in the opening paragraph. Let us observe numerous sequences of futures prices generated by this model, up until their terminal data. They will turn out, on the average, to have no upward or downward drift anywhere! (This will be true, regardless of the systematic seasonal patterns in Xt . This would be true, under our axiom, even in time of severe inﬂation or deﬂation of Xt itself — only remember, please, that the neglect of interest underlying our axiom would seem unwarranted in time of conﬁdentlyanticipated extreme inﬂation). Theorem of FairGame Futures Pricing. If spot prices {Xt } are subject to the probability laws of (1) and (2), and the futures price sequence {Y (T, t + T ), Y (T −1, t+T −1), . . . , Y (1, t+T −1), Y (0, t+T )} is subject to the axiom of expected price as formulated in (5) and (6), then the latter sequence is a fair game (or martingale) in the sense of having unbiased price changes, or E[Y (T − 1, t + T − 1)Xt , Xt−1 , . . .] ≡ Y (T, t), (2.7) or writing Y (T − 1, t + T − 1) − Y (T, t) = ∆Y (T, t), E[∆Y (T, t)] ≡ 0; (2.8) page 30 July 3, 2015 8:27 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 Proof that Properly Anticipated Prices Fluctuate Randomly 31 from which it follows inductively that The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. E[∆n Y (T, t)] ≡ 0 (n = 1, 2, . . . , T ). (2.9) This means that there is no way of making an expected proﬁt by extrapolating past changes in the futures prices, by chart or any other esoteric devices of magic or mathematics. The market quotation Y (T, t) already contains in itself all that can be known about the future and in that sense has discounted future contingencies as much as in humanly possible (or inhumanly possible within the axiom of the model). The theorem does not imply that the sequence of Y ’s perform a Brownian motion. It does not imply that ∆Y (T, t) is statistically independent of ∆Y (T + 1, t − 1); it implies only that given knowledge of Y (T, t) the Pearsonian correlation coeﬃcient between the above two ∆’s will be zero. It is a source of comfort to the economist, rather than otherwise, that wheat prices should not perform a Brownian random walk. A Brownian walk, like the walk of a drunken sailor, wanders indeﬁnitely far, listing with the wind. Surely, economic law tells us that the price of wheat — whether it be spot {Xt } or futures {Y (T, t)} — cannot drift skyhigh or groundlow. It does have a rendezvous with its destiny of supply and demand, albeit our knowledge of future supply and demand trends becomes dimmer as the envisaged date recedes farther into the future. We would expect people in the market place, in pursuit of avid and intelligent selfinterest, to take account of those elements of future events that in a probability sense may be discerned to be casting their shadows before them. (Because past events cast their shadows after them, future events can be said to cast their shadows before them.) Although the sequence {∆Y (T − n, t + n)} has a zero ﬁrst moment at all time periods T −n, there is no reason to suppose that the riskiness of holding a futures — in the sense of the second movement or variance, as measured by E[{∆Y (T − n, t + n)}] — should be the same when T is large and the terminal data far away as when T −n is small and the futures contract about to expire. It is a wellknown rule of thumb that nearness to expiration date involves greater variability or riskiness per hour or per day or per month than does farness. Partly this is due, I think, to factors not encompassed in the present model — for example, the reallife complications that make arbitrage equality shown in (2.5) hold only approximately. However, the present theory can contribute an elegant explanation of why we should expect fardistant futures to move more sluggishly than near ones. Its explanation does not lean at all on the undoubted fact that, during certain preharvest periods when stocks are normally low, changes in spot prices {∆Xt } can themselves be expected to experience great volatility and second moment. page 31 July 3, 2015 8:27 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. 32 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 P. A. Samuelson Instead, it uses the contrary minor premise of posited uniformity through time of the distribution of {∆Xt } — so that it constitutes a stationary time series — to deduce the law of increasing volatility of a maturing futures contract. Before deriving the result, I present a proof of the martingale theorem. By deﬁnition ∞ Y (T, t) = XdP (X, Xt , Xt−1 , . . . ; T ), axiom of (6) −∞ Y (T − 1, t + 1) ∞ XdP (X, Xt+1 , Xt , Xt−1 , . . . ; T − 1), axiom of (6) = −∞ = f (Xt+1 , Xt , Xt−1 , . . .) E[Y (T − 1, t + 1)Xt , Xt−1 , . . .] ∞ f (Z, Xt , Xt−1 , . . .)dP (Z, Xt , Xt−1 , . . . ; 1) = −∞ ∞ −∞ = −∞ ∞ = −∞ ∞ = −∞ ∞ Xd XdP (X, Z, Xt , . . . ; T − 1) dP (Z, Xt , . . . ; 1) ∞ −∞ P (X, Z, Xt , . . . ; T − 1)dP (Z, Xt , . . . ; 1) XdP (X, Xt , . . . ; T ), by (2) = Y (T, t) Q.E.D. (2.10) In thus proving (2.7), I have made permissable interchanges in the order of integration of the double Stieltjes integrals. The content of this general theorem can be illustrated by some speciﬁc stochastic processes. (a) Thus, suppose {Xt } represents a simple Brownian motion without bias. Then Y (T, t) becomes nothing but Xt itself. Since {Xt } is a fairgame with E[∆X] = 0, then so must E[∆Y ] = 0. (b) Going beyond this simple case, suppose {Xt } is a random walk with biased drift so that E[∆X] = µ = 0. Then Y (T, t) = Xt + µT and E[∆Y ] = E[∆X] + µ(T − 1 − T ) = µ − µ = 0, as the theorem requires. (c) Now let {Xt } satisfy an autoregressive stochastic relation Xt+1 = aXt + {ut }, E(ut ) = µ, page 32 July 3, 2015 8:27 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 Proof that Properly Anticipated Prices Fluctuate Randomly 33 where {ut } is an unserially correlated random variable with distribution P (U ). Then Y (T, t) = aT Xt + µT ∞ aT −1 (aXt + Ut )dP (Ut ) + µ(T − 1) − aT Xt − µT E[∆Y ] = −∞ The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. = 0 + µ − µ = 0. (d) Finally, let A = [a(i, j)] be a ﬁnite Markov transitionprobability matrix giving Prob{Xt+1 = Qi Xt = Qj }. Write AT = [aTij ], and Y (T, t) = [Q1 , . . . , Qn ] times the j th column of AT , by Axiom Qi aTij , denoted by Y (T Qj ). = 1 Then E[Y (T − 1, t + 1)Xt = Qj ] = Y (T − 1)Qk )akj , by deﬁnition k = = T −1 akj Qi aij Qi aTij , = Y (T, t) since AT −1 A = AT Q.E.D. The theorem is so general that I must confess to having oscillated over the years in my own mind between regarding it as trivially obvious (and almost trivially vacuous) and regarding it as remarkably sweeping. Such perhaps is characteristic of basic results. And actually the empirical question of the applicability of the model to economic reality must be kept distinct from the logical problem of what is the model’s implied content. Figure 1 should help to explain the theorem. It supposes {Xt } is generated by (2.4)’s autoregressive model with a = 1/2. Random shocks {ut } aside, if price is once perturbed above (or below) its normal value (set at zero by convention), in each period it will return onehalf way back to zero. Generalizing the Theorem If money has to be tied up in holding the Y (T, t) contract and if there is a positive safe rate of interest, the process — being a fair game — does page 33 July 3, 2015 8:27 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. 34 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 P. A. Samuelson Figure 1A. At time t, we know Xt = 80 with certainty. We can expect, from (2.4) with a = 1/2 and {ut } approximately normally distributed, that Xt+1 be distributed as shown, with E[Xt+1 Xt = 80] = 40; for Xt+2 , Xt+3 the distributions are becoming slightly more dispersed, with means drifting downward like 80(2−2 ), 80(2−3 ); because a = 1/2 < 1, there is a limiting ergodic state for t + ∞, much as shown, with E[Xt+∞ Xt ] = 0 independently of Xt . At time t, the futures price Xt+3 is evaluated at 10, as shown by the white arrow at A, which merely reﬂects what can now be known about the distribution and mean of the spot price three periods from now. Now ask yourself the question. What do I think at this time t will be the likely distribution of this futures price Y (3, t) at the next period when it will have become Y (2, t + 1)? It will be distributed around 1/4 of next period’s spot price Xt+1 in exactly the same way that C shows Xt+2 distributed around 1/4 of the period’s spot price Xt . You do not today know where tomorrow’s Xt+1 will fall; but B does give you its probability distribution. Combining these bits of information, Figure 1B below plots (with enlarged vertical scale) what you now are entitled to regard as the probability distribution for Y (3, t), Y (2, t + 1), Y (1, t + 2), Y (0, t + 3) = Xt+3 . not stand to earn even the opportunitycost of foregone safe interest. If, in addition, people have risk aversion (so that the utility whose expected value they seek to maximize is a strictly concave function of money wealth or income), the contract will probably have to promise a positive percentage yield per unit time, R, where 1 + R = λ > 1 + safe interest. Therefore, we replace the simple axiom of mean expected value by a slightly more general axiom. Axiom of PresentDiscounted Expected Value. At each point of time t for a future price, we posit Y (T, t) = λ−T E[Xt+T Xt , Xt−1 , . . .], or, slightly more generally, since the variability of Y may be slightly diﬀerent for each value of T , we may expect a diﬀerent (λ1 , λ2 , . . . , λT ) yield for which people hold out if they are to hold for the next period a futures contract with page 34 July 3, 2015 8:27 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. Proof that Properly Anticipated Prices Fluctuate Randomly 35 Figure 1B. A , B , C , and D give the probability distributions you are entitled to envisage on the basis of today’s certain knowledge of [Xt , Xt+1 , . . . and Y (3, t)] for next periods’ [Y (2, t+1), Y (1, t+2), Y (0, t+3) = Xt+3 ]. Note that all have the same mean of 10 but that your cone of uncertainly widens as the time in which new unknown, independent disturbances {ut+1 } can intervene. Note that after one period passes, both diagrams will have to be redrawn: A and A will be irrelevant; a particular point on B and B will become the new present, and the new (C, C ) and (D, D ) will bear the same relation to (B, B ) that now (C, C ) and (B, B ) bear respectively to (A, A ). If the stochastic variables ut+1 , ut+2 , and ut+3 were identically zero, the futures price would show no variability, staying always at 10. Y (3, t) and Y (2, t + 1) are likely to be more alike with a less volatile diﬀerence ∆Y than can Y (2, t + 1) and Y (1, t + 2). (For Equation (2.4), the variance of ∆Y (t − n, t + n) is proportional to aT −n , diminishing as T − n becomes large.) Why? Because the ﬁrst pair stand to have (ut+2 , ut+3 ) in common and diﬀer only in that the ﬁrst of the pair also has ut as a source of variation. The fact that this pair has two out of three elements of variation in common is stabilizing on their diﬀerence ∆Y in comparison with the second pair which have only ut+3 out of the possible (ut+2 , ut+3 ) in common. A fardistant future will not change much in the next month since so few of the disturbances upon which its fate depends will change in this month; it stays close to the general level given by the socalled law of averages. At the other extreme, Y (1, t + 2) and Y (0, t + 3) = Xt+3 will diﬀer because of the single unknowable ut+3 , and this suddendeath kind of situation will subject their diﬀerence to great variance without any cushioning from the law of large numbers. A striking way of seeing this is to disregard the varying means and to think of the passage of one period as leaving us at the left of the diagram subtracting the last (rather than the ﬁrst, as when we shift the present from A to B) of the vertical axes. Precisely because D approaches an ergodic state, C and D diﬀer less than do A and B in Figure 1A. So losing the last axis, which is like making C become the new D, gives less of a change than would shifting B to A. T years to go. The presentdiscounted expectation becomes −1 −1 Y (T, t) = λ−1 1 λ2 . . . λT E[Xt+T Xt , Xt−1 , . . .]. Just as the simple axiom of expected gain led to the theorem of unbiased price change, the new general axiom leads to the theorem on “normal backwardation”. page 35 July 3, 2015 8:27 36 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 P. A. Samuelson The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. Theorem of Mean Percentage Price Drift. If spot prices {Xt } are determined by the stipulated general stochastic process P (Xt+T , Xt , Xt−1 , . . . ; T ) and we deﬁne the sequence {Y (T, t), Y (T − 1, t + 1), . . . , Y (1, t + T − 1), Y (0, t + T )} by −1 Y (T, t) = λ−1 T . . . λ1 E[Xt+T Xt , Xt−1 , . . .]. ∞ −1 . . . λ XdP (X, Xt , Xt−1 , . . . ; T ), = λ−1 1 T −∞ it follows that E[Y (T − n, t + n)Xt , Xt−1 , . . .] = λT . . . λT −n+1 Y (T, t) = λn Y (T, t) if λ1 ≡ λ. In words, this says that the futures price will rise in each period by the percentage λ − 1 (or λT − 1, where presumably λ1 > λ2 > · · · > λT if people abhor the extra riskiness inherent in a futures contract as it matures). This theorem provides rational account of the Keynes–Houthakker–Cootner (Cootner, 1960a, 1960b; Houthakker, 1961; Keynes, 1930) doctrine of “normal backwardation”. It says that, within the deﬁned model, all chart methods attempting to read out of the past sequence of known prices [Xt , Xt−1 , . . . , Y (T, t), Y (T +1, t−1), . . .] any proﬁtable pattern of prediction is doomed to failure. So to speak, the market has already, by our axiom, discounted all knowable future information so that the presentdiscounted variable (λT . . . λ1 )−1 Y sequence is itself a fairgame martingale. Reexamining the proof of the simple theorem, we see that the same interchange of order of integration in the relevant double integral gives an immediate proof of the theorem. Finally, examining the proof shows that a still more general theorem is valid. Wherever we have had Xt+T in E[Xt+T Xt , Xt−1 , . . .] or X in the double integral, we could have put in any function of X, say g(X) or g(Xt+T ) and still preserved the martingale properties. This is important because, as mentioned, we may want to interpret Xt as a vector — as e.g., with components (soft wheat price, hard wheat price, corn price, midwest rainfall, national income, etc.). Then the scalar g(Xt+T ) might be the selection of soft wheat price, the deliverable goods for the futures contract in question. Only a deluded chartist would think that extraneous variables, such as rainfall, can be excluded from an optimal probability description of future soft wheat prices; and yet it is true that, after speculators have taken such variables into account in implementing the axiom of expected gain, the resulting Y sequence tells its own (simple!) story. page 36 July 3, 2015 8:27 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. Proof that Properly Anticipated Prices Fluctuate Randomly 37 Still one more bargain in cheap generality can be garnered. As far as the theorem’s proof is concerned, g(Xt+T ) could just as well be a function also of all present known data and be written as g(Xt+T , Xt , Xt−1 , . . .), without aﬀecting its general martingale property. It would even be possible to put nowknown values of Y (T + k, t − k) as variables in g, but there would seem to be no need to do this since ultimately all known Y ’s should be reducible to functions of the Xt−k inputs. Still there is no harm in so including the Y ’s, and in some cases the form of the function might be much simpliﬁed by including such Y ’s (and even the Y ’s of other terminal data). To illustrate how one might want Xt in g(Xt+T , Xt , . . .), suppose that either number 1 or number 2 grades of soft wheat are deliverable on the July, 1965, contract. Let the vector Xt contain as elements price data on both grades; and suppose that if one of these is now known to be much cheaper than the other, one can be pretty sure that this cheaper grade will be the grade actually delivered on the contract at the terminal data; and hence for certain value of the elements of the vector Xt , g(. . .) will be the expected price of one grade rather than of the other. Conclusion A result of some generality has now been established. Anyone who thinks it obvious should reﬂect on the following fact: if instead of taking an expected value or mean value for Xt+T , we set Y (T, t) at, say, its median value, then it will not necessarily be true that expected value of ∆Y is zero. Nor is it true that the median of ∆Y so deﬁned is zero. For example, let P (Xt+1 , Xt , 1) = P (Xt+1 /Xt ) with P (1) > 1/2 = P (m), 0 < m < 1. Then Y (T, t) = mXt , Y (T + 1, t + 1) = mXt+1 . For median ∆Y = 0, we require 1 Prob{Yt+1 Yt } = Prob{mXt+1 mXt } = Prob{Xt+1 − Xt } = . 2 But actually, Xt+1 1 ≤ 1 = P (1) > . Prob{Xt+1 ≤ Xt } = Prob Xt 2 Hence, ∆Y is more likely to be negative than positive — even if P (X/x) deﬁnes a fair game with E(X/x) = x. One should not read too much, into the established theorem. It does not prove that actual competitive markets work well. It does not say that speculation is a good thing or that randomness of the price changes would be good thing. It does not prove that anyone who makes money in speculation is ipso facto deserving of the gain or even that he has accomplished something page 37 July 3, 2015 8:27 The World Scientific Handbook of Futures Markets Downloaded from www.worldscientific.com by LA TROBE UNIVERSITY on 05/21/16. For personal use only. 38 World Scientific Handbook of Futures Markets. . . 9.75in x 6.5in b1892ch02 P. A. Samuelson good for society or for anyone but himself. All or none of these may be true, but that would require a diﬀerent investigation. I have not here discussed where the basic probability distributions are supposed to come from. In whose minds are they ex ante? Is there any ex post validation of them? Are they supposed to belong to the market as a whole? And what does that mean? Are they supposed to belong to the “representative individual,” and who is he? Are they some defensible or necessitous compromise of divergent expectation patterns? Do price quotations somehow produce a Paretooptimal conﬁguration of ex ante subjective probabilities? This chapter has not attempted to pronounce on these interesting questions. References Cootner, PH (1960a). Returns to speculators: Telser vs. Keynes. Journal of Political Economy 68, 396–404. Cootner, PH (1960b). Rejoinder. Journal of Political Economy, 68, 415–418. Houthakker, H (1959). The scope and limits of futures trading. In Abramovitz, M. (editor), Allocation of Economic Resources, pp. 134–159. Houthakker, H (1961). Systematic and random elements in shortterm price movements. American Economic Review 51, 164–172. Kendall, MG (1953). The analysis of economic timeseries — Part I: Prices. Journal of the Royal Statistical Society, 96 (Part I) 11–25; also pp. 85–99. In The Random Character of Stock Market Prices, RH Cootner (ed.). Cambridge, MA: The M.I.T. Press, 1964. Keynes, JM (1930). A Treatise on Money, Vol. II: The Applied Theory of Money. London, UK: Macmillan & Company, 1930. page 38