Under the martingale model, asset returns have zero mean and are uncorre- lated. However, the framework we used to derive this model is a simple one and its assumptions are unrealistic in many respects. In particular, we assumed that the price of an asset is based only on the expectation of the asset’s value.
An alternative, and more realistic, approach is based on the assumption that the current price of a stock is based on current beliefs regarding the statistical properties of future earnings of the firm; in particular, the price may be based on the expected value of the future earnings as well as their variability. Including variability in the analysis is important because investors are generally “risk averse.” Under risk aversion, more risky investments are worth less than less risky investments with the same expected return.
For example, consider two investments. Stock 1 has a guaranteed return of awhile Stock 2 has a return of zero with probability 1/2 and a return ofbwith probability 1/2. Stock 1 may be preferable, that is, it may be worth more to an investor than Stock 2, even ifa < b/2. It follows that the argument given in the previous sections, which uses only expected values, is, at best, only a rough approximation of the process used to determine asset prices. Under a model that recognizes risk aversion, the martingale property no longer holds.
Therefore, although the assumption of an efficient market suggests certain properties of asset prices, it does not necessarily follow that the martingale model holds in practice.
Hence, in this section, we consider models for asset prices that have many, but not all, of the general features of the martingale model. For instance, based on the models considered in this section, returns are statistically unrelated,
but they do not necessarily have zero mean. These models are based on the concept of arandom walk.
Consider a sequence of random variablesY0, Y1, Y2, . . . .A simple model for such a process is one in which the changes in process,Yt−Yt−1, are “random”
in the sense that they have no discernible pattern and no relationships among them. Such a process is known as a random walk because it can be viewed as a model for the location on the real line of an “individual” who, at each time point, moves randomly along the line. The statistical properties of a process of this type depend on the interpretation of the term “random” used to describe the movements of the individual. Thus, there are several different technical definitions of a random walk, corresponding to different interpretations.
For a given stochastic process {Yt:t= 0,1,2, . . .}, let Zt=Yt−Yt−1, t= 1,2, . . . denote the changes in the values ofYt; theZt are known as the increments of the process. Note that when discussing random walks, we will often includeY0, the value at timet= 0, in the process; this random variable is needed to define the first incrementZ1. Thus, Y0 represents the “starting point” of the process. The increment process{Zt:t= 1,2, . . .}will generally start at timet= 1.
Note that, givenY0,
Yt=Y0+Z1+ã ã ã+Zt, t= 1,2, . . . (3.4) so that the random variables Y1, Y2, . . . are equivalent to the increments Z1, Z2, . . .together with the starting pointY0.
In a random walk process, the increments are “noise” in the sense that they are statistically unrelated. We have seen one example of such a noise process, weak white noise; this definition, as well as some others, are used in defining random walks.
Definitions of a Random Walk
We now consider three specific definitions of a random walk. Consider a stochastic process {Yt:t= 0,1,2, . . .}, and let Zt=Yt−Yt−1, t= 1,2, . . . denote the increments of the process.
Suppose thatZ1, Z2, . . .are independent and identically distributed (i.i.d.) random variables each with meanμand standard deviationσand suppose that Y0 is independent ofZ1, Z2, . . . .This is the strongest version of the random walk model, known as Random Walk 1 (RW1). Thus, in RW1, the changes in the position of the process,Yt−Yt−1, are i.i.d. random variables.
Note that, using (3.4), under RW1,
E(Yt|Y0) = E(Y0+Z1+ã ã ã+Zt|Y0)
= E(Y0|Y0) + E(Z1|Y0) +ã ã ã+ E(Zt|Y0)
=Y0+ E(Z1) +ã ã ã+ E(Zt)
=Y0+μt, t= 1,2, . . . .
A similar property holds for the conditional variance of Yt given Y0, defined as
Var(Yt|Y0) = E
[Yt−E(Yt|Y0)]2Y0}= E(Yt2|Y0)−E(Yt|Y0)2. Note that
Yt−E(Yt|Y0) = (Y0+Z1+ã ã ã+Zt)−(Y0+μ+ã ã ã+μ)
= (Z1−μ) + (Z2−μ) +ã ã ã+ (Zt−μ).
Using this expression, together with the assumption that Y0 and (Z1, Z2, . . .) are independent, it follows that
E
[Yt−E(Yt|Y0)]2|Y0
= E
[(Z1−μ) + (Z2−μ) +ã ã ã+ (Zt−μ)]2|Y0
= E
[(Z1−μ) + (Z2−μ) +ã ã ã+ (Zt−μ)]2
= Var(Z1) + Var(Z2) +ã ã ã+ Var(Zt)
=σ2t.
Here,μ andσare called thedrift andvolatility, respectively, of the process.
Because Yt=Y0+Z1+ã ã ã+Zt, it follows thatYt andZt+1 are indepen- dent; using the fact thatYt+1=Yt+Zt+1,
E(Yt+1|Yt) = E(Yt|Yt) + E(Zt+1|Yt) =Yt+ E(Zt+1) =Yt+μ.
That is, the best predictor of the position of the random walk at timet+ 1, given knowledge of the position at timet, isYt+μ.
In fact, the same basic argument can be used to show that E(Yt+1|Y0, Y1, . . . , Yt) = E(Yt+Zt+1|Y0, Y1, . . . , Yt) =Yt+μ;
that is, the best predictor ofYt+1, given previous knowledge of all past values of the process, isYt+μ.
Weaker forms of the random walk model are based on weaker assumptions regarding the distribution of the increments. For instance, Random Walk 2 (RW2) assumes that the increments Z1, Z2, . . ., together with the initial valueY0, are independent random variables, as in RW1; Z1, Z2, . . .are each assumed to have meanμand standard deviationσ, but they are not necessarily identically distributed.
Under this model, the result
E(Yt+1|Y0, Y1, . . . , Yt) =Yt+μ
still holds. This generalization is useful because the assumption of identical distributions of the increments is difficult to verify in practice.
The weakest form of random walk that is commonly used is a weaker ver- sion of RW2 in which the independence assumption forZ1, Z2, . . .is replaced
by the assumption that the increments are uncorrelated, Cov(Zt, Zs) = 0 for any s=t, and they are uncorrelated withY0,
Cov(Zt, Y0) = 0, t= 1,2, . . .;
this is known as Random Walk 3 (RW3). That is, in RW3, the increment process{Zt:t= 1,2, . . .} is a weak white noise process.
In this case, we can no longer say that the best predictor of Yt+1 based onY0, Y1, . . . , YtisYt+μ. However, thebest linear predictor ofYt+1based on Y0, Y1, . . . , YtisYt+μ.
The best linear predictor ofYt+1 based onY0, Y1, . . . , Ytis defined as the function of the form
a+b0Y0+b1Y1+ã ã ã+btYt that minimizes
E{(Yt+1−a−b0Y0− ã ã ã −btYt)2}. (3.5) Proposition 3.3. Consider a stochastic process {Yt:t= 0,1,2, . . .} and let Zt=Yt−Yt−1,t= 1,2, . . . denote the increments of the process. If{Zt:t= 1,2, . . .}is weak white noise and Cov(Y0, Zt) = 0fort= 1,2, . . ., then the best linear predictor ofYt+1 based onY0, Y1, . . . , Yt isYt+μ.
Proof. Consider a linear function of Y0, Y1, . . . , Yt of the form a+b0Y0+ b1Y1+ã ã ã+btYt for some constantsa, b0, b1, . . . , bt.
According to (3.4), each of Y1, Y2, . . . , Yt is a linear function of Y0, Z1, . . . , Zt; hence, any function of the form
a+b0Y0+b1Y1+ã ã ã+btYt=a+b0Y0+b1Y1+ã ã ã+ (bt−1)Yt+Yt may be written as
Yt+c+d0Y0+d1Z1+ã ã ã+dtZt (3.6) for some constantsc, d0, d1, . . . , dt.
Therefore, using the fact thatYt+1−Yt=Zt+1, the best linear predictor ofYt+1 is given by the constantsc, d0, d1, . . . , dt that minimize
E{(Zt+1−c−d0Y0−d1Z1− ã ã ã −dtZt)2}, (3.7) which is simply 3.5 written in terms ofY0, Z1, . . . , Zt.
Recall that, for any random variableX, E(X2) = E(X)2+ Var(X). Note thatZt+1−c−d0Y0−d1Z1− ã ã ã −dtZthas the expected value
⎛
⎝1− t j=1
dj
⎞
⎠μ−c−d0E(Y0) (3.8)
and variance ⎛
⎝1 +t
j=1
d2j
⎞
⎠σ2+d20Var(Y0)
where here we have used the fact that any pair ofY0, Z1, . . . , Zt, Zt+1is uncor- related. Clearly, the variance is minimized byd0=d1=ã ã ã=dt= 0. Because, for these choices of d0, d1, . . . , dt, the expected value in (3.8) is 0 for c=μ, it follows that the best linear predictor of Yt+1 is an expression of the form 3.6 withd0=d1=ã ã ã=dt= 0 andc=μ, yielding the expressionYt+μ, as
claimed in the proposition.
Note that the random walk models are related: RW1 implies RW2, which implies RW3. Thus, if RW3 does not hold, then neither RW2 nor RW1 holds, and if RW2 does not hold, then RW1 does not hold.
Geometric Random Walk
The same ideas can be applied after a log-transformation of the random vari- ables; such a transformation is useful if we believe that the “random changes”
in the process are multiplicative rather than additive. In this case, the original untransformed process is said to follow ageometric random walk model.
Let{Ut:t= 0,1,2, . . .}denote a stochastic process such thatYt= logUt, t= 0,1,2, . . . follows a random walk model. For instance, if {Yt:t= 0,1,2, . . .}follows RW1, then logUt=Y0+Z1+ã ã ã+Zt, whereZ1, Z2, . . .are i.i.d. random variables. In this case, we say that{Ut:, t= 0,1,2, . . .} follows a geometric RW1 model. Note that, under this model,
Ut= exp(Y0) exp(Z1+ã ã ã+Zt)
≡U0exp(Z1)ã ã ãexp(Zt)
whereU0= exp(Y0). These ideas may also be applied to RW2 and RW3.
Application of Random Walk Models to Asset Prices
The martingale model for asset prices suggests that the stochastic process corresponding to the prices of an asset,{Pt:t= 0,1,2, . . .}, might be usefully modeled as a random walk. Under this model, the increments of the process Pt−Pt−1, corresponding to changes in the price of the asset, are “noise”;
for instance, if{Pt:t= 0,1,2, . . .} follows RW3, then the price changes form a weak white noise process.
However, empirical analyses suggest that changes in prices are often roughly proportional to the price, in the sense that stocks with higher prices tend to exhibit larger price changes than stocks with lower prices, generally speaking.
This behavior may be modeled by assuming that Pt−Pt−1=WtPt−1, t= 1,2, . . . ,
whereWtis a random variable representing the proportional change in price.
Under this assumption, the conditional expectation of Xt=Pt−Pt−1 given Pt−1 depends onPt−1 in general. On the other hand,
pt−pt−1= log Pt
Pt−1 = log (Wt+ 1)
so that, letting Zt= log(Wt+ 1), pt−pt−1=Zt, t= 1,2, . . . . Thus, if price changes are proportional to the price, we might expect log-prices to follow a random walk so that{Pt:t= 0,1,2, . . .} follows a geometric random walk.
Note that the increments of the process {pt:t= 0,1,2, . . .} are simply the log-returns. However, a basic argument showing that market efficiency implies that log-returns are uncorrelated, along the lines of the one we used in Proposition 3.2 for returns, is not available.
To see why such an argument fails, consider the framework of Section 3.3, in whichPt= E(V|Ωt). Then
pt= log E(V|Ωt) so that
rt=pt−pt−1= log E(V|Ωt) E(V|Ωt−1). It follows that
E(rt+1rt) = E
logE(V|Ωt+1)
E(V|Ωt) logE(V|Ωt+1) E(V|Ωt)
.
Because of the presence of the log(ã) function, this expression cannot be usefully simplified.
However, suppose that the returnsRt= (Pt−Pt−1)/Pt−1are small. Then rt=pt−pt−1= log Pt
Pt−1 = log
1 +Pt−Pt−1 Pt−1
=. Pt−Pt−1 Pt−1 =Rt.
For example, if Rt= 0.02, a fairly large value for a monthly return, then Pt/Pt−1= 1.02 and rt= log(1.02) = 0.01980 and approximating rt by Rt yields an error of 0.0002, or about 1%.
Thus, if two returns are uncorrelated, it is reasonable to expect that the corresponding log-returns will be approximately uncorrelated. Therefore, although the martingale model for log-prices does not follow directly from the efficient market assumption, the argument used in Section 3.3 suggests that it may be reasonable to model{pt:t= 0,1,2, . . .} as a random walk.
The random walk hypothesis in finance generally refers to a geometric random walk for asset prices. Suppose{Pt:t= 0,1,2, . . .}follows a geometric random walk model; then the stochastic process corresponding to the log- prices,{pt:t= 0,1,2. . .} follows a random walk model.
Specifically, under RW1,r1, r2, . . .are i.i.d. random variables, under RW2, r1, r2, . . . are independent, each with mean μ and standard deviationσ, but they are not necessarily identically distributed; RW3 weakens the indepen- dence condition of RW2 to the condition that r1, r2, . . . are uncorrelated random variables. The key idea in the random walk hypothesis is that the past valuesr1, r2, . . . , rt do not provide any useful information aboutrt+1.
As noted in the introduction to this chapter, it is important to keep in mind that the random walk hypothesis does not imply that the information provided byr1, r2, . . . , rtis useless in understanding the properties ofrt+1. For instance, under any of the random walk models we have considered,r1, . . . , rt, rt+1each has mean μ; hence, the sample mean of r1, . . . , rt is an unbiased estimator ofμ, which is the expected value ofrt+1. Thus, the past values do provide a type of indirect information about the future.