5.5 The Distribution of the Rate of Return

Let us utilize MAPLE to visulize some conseqences of our assumptions about [Maple Math] . The normal density function with expcted value of [Maple Math] and standard deviaiton of [Maple Math] at a point [Maple Math] is given by

[Maple Math] .

(5.26)

We defined this function in MAPLE and called it Normalpdf.

> Normalpdf:=(x,mu,sigma)->exp(-(x-mu)^2/2/(sigma)^2 )/sqrt(2*Pi*(sigma)^2);

[Maple Math]

>

In our case, the random varible [Maple Math] (we will omit the suscript [Maple Math] in the MAPLE calculation) is the continuously compounded rate of return for a period of length [Maple Math] . We assumed that [Maple Math] is normally distributed such that its expected value and standard deviation are dependent on time, and are given by [Maple Math] and [Maple Math] for some constant [Maple Math] and [Maple Math] . Thus the density function of [Maple Math] at a point [Maple Math] is given by

> Normalpdf(y,mu*t,sigma*sqrt(t));

[Maple Math]

>

When [Maple Math] , [Maple Math] , and [Maple Math] the value of Normalpdf at a point [Maple Math] can be calculated using the MAPLE command.

> Normalpdf(y,.15*2,.23*sqrt(2));

[Maple Math]

>

For a specific value of [Maple Math] , say [Maple Math] = 0.15 x 2, [Maple Math] is simply replaced with a numerical value:

> Normalpdf(.15*2,.15*2,.23*sqrt(2));

[Maple Math]

>

To get MAPLE to evaluate it in a decimal form we utilize the evalf command:

> evalf(Normalpdf(.15*2,.15*2,.23*sqrt(2)));

[Maple Math]

>

Let us look at Figure 5.3, the density function of [Maple Math] where [Maple Math] and [Maple Math] .

> plot(Normalpdf(y,.15*2,.23*sqrt(2)),y=-3..4,title =`Figure 5.3: The Normal Density Function`, titlefont=[TIMES, BOLD, 10]);

[Maple Plot]

>

This is the famous bell-shaped graph of the normal density function.

In order to visualize the effect of time on [Maple Math] we run the animation below. The animation can be used to demonstrate how this graph changes as the time interval [Maple Math] Footnote 14 shrinks from 6 to 0.1. To ensure that the animation is run in the right direction click the arrow ( <-- ) in the animation bar.

> plots[animate](Normalpdf(Y,.15*t,.23*sqrt(t)),Y=-3..5,t=0.1..6,color=green,title=`The Density Function of Y(t) as t Approaches zero`,titlefont=[TIMES, BOLD, 10]);

[Maple Plot]

>

Figure 5.4 presents a static graph of the distribution for [Maple Math] , and [Maple Math] in the same plane.

> plots[display]({
plot(Normalpdf(y,.15*1,.23*sqrt(1)),y= -3..5, colour=green),
plot(Normalpdf(y,.15*3,.23*sqrt(3)),y= -3..5, colour=yellow),
plot(Normalpdf(y,.15*6,.23*sqrt(6)),y= -3..5, colour=red, title =`Figure 5.4: The density function of Y(t) as t approaches zero`,titlefont=[TIMES, BOLD, 10])});

[Maple Plot]

>

Recall that the normal distribution has a peak at its expected value, at [Maple Math] in our case, and that the larger the standard deviation, [Maple Math] in our case, the flatter the graph and the riskier the situation. The area under the graph between two points [Maple Math] and [Maple Math] is the probability that [Maple Math] and [Maple Math] . Hence, as [Maple Math] decreases, [Maple Math] decreases, the graph gets steeper, and the probablity of [Maple Math] being close to its mean, [Maple Math] , increases. Of course [Maple Math] is also getting smaller as [Maple Math] is getting smaller; thus the peak of the graph is getting close to zero. Hence the area under the graph, the mass of the probability, is mostly concentrated close to zero. Thus, as [Maple Math] approaches zero, the probability of [Maple Math] being equal to zero approaches one.

The same point can be illustrated by looking at a three-dimensional picture, Figure 5.5. We observe the graph in a static version for each [Maple Math] between 0 and 6.

> plot3d(Normalpdf(y,10*t,15*sqrt(t)),y=-110..200,t=0..6,axes=framed,orientation=[-124,57],title =`Figure 5.5: Three-Dimensional View of the Density of Yt as a Function of t and y`,titlefont=[TIMES, BOLD, 10]);

[Maple Plot]

>

It is now apparent that as [Maple Math] approches zero, the mass of probability, the area under the graph, is concentrated around zero. Thus the probability that the instantaneous rate of return [Maple Math] is close to zero as [Maple Math] approaches zero is close to one. Hence, as [Maple Math] approaches zero, [Maple Math] is very likely to be one. The situation therefore becomes less and less risky. If at time [Maple Math] the price of the stock is [Maple Math] , its price at time [Maple Math] , [Maple Math] , will be the random variable [Maple Math] . For a small [Maple Math] , the probability of [Maple Math] being one is close to one, and thus, the probablity of [Maple Math] being close to [Maple Math] is also close to one. Hence for short time intervals the realization of [Maple Math] is very likely to be close to [Maple Math] .

The normal cumulative distribution function of a random variable [Maple Math] at a point [Maple Math] is defined as the probabilty that [Maple Math] will have a value less than or equal to [Maple Math] . Thus, it is the area under the graph of Normalpdf from negative infinity to [Maple Math] . In other words it is

[Maple Math] ,

(5.27)

or in our notation, as in equation (5.26),

[Maple Math] .

(5.28)

We define this function in MAPLE and call it Normalcdf .

> Normalcdf:=(x,mu,sigma)->int(Normalpdf(v,mu,sigma),v=-infinity..x);

[Maple Math]

>

The effect of time on the distribution of [Maple Math] point can also be demonstrated with the cumulative function Normalcdf . This is performed by replacing the pdf with cdf as in the commands below. The graphs are shown only in the on-line version of the book.

> plot3d(Normalcdf(y,10*t,15*sqrt(t)),y=-110..200,t=0..6,title =`The Probability Function of Y as a function of t and y`,axes=framed,orientation=[118,67],style=PATCH,titlefont=[TIMES, BOLD, 10]);

[Maple Plot]

> plots[animate](Normalcdf(Y,.15*t,.23*sqrt(t)),Y=-3..5,t=0.1..6,color=green, title=`The probability Function of Y as t Approaches zero`,thickness=3,titlefont=[TIMES, BOLD, 10]);

[Maple Plot]

>

The probability that the random variable [Maple Math] will have a value between [Maple Math] and [Maple Math] is given by the area under the graph between these two points.We can calculate it in MAPLE with the aid of the function Normalcdf . However, we must let MAPLE know that [Maple Math] and [Maple Math] are both positive.

> assume(sigma>0,t>0);

> evalf(Normalcdf(mu*t+2*sigma*sqrt(t),mu*t,sigma*sqrt(t))-Normalcdf(mu*t-2*sigma*sqrt(t),mu*t,sigma*sqrt(t)));

[Maple Math]

>

It is well known that the probability of a normal random variable having a value between two standard deviations above or below the expected value is around 0.95. This is indeed confirmed above. As you can see, MAPLE was able to calculate the probability without requiring us to specify the values of [Maple Math] , [Maple Math] , and [Maple Math] . We see therefore that the probablity does not depend on the actual values of these parameters.

However, consider the effect on the interval [Maple Math] , when [Maple Math] approaches zero. The interval clearly approches zero, but the probability of the interval remains the same as it is independent of [Maple Math] . Hence the likelihood of [Maple Math] being close to zero approaches one. Keep in mind that as the interval shrinks the tail ends of the probability shrink as well. In addition, most of the probability mass is now centred around zero. As [Maple Math] approaches zero, [Maple Math] remains a random variable for every [Maple Math] , and hence the ``limit of [Maple Math] " as [Maple Math] approaches zero will still be a random variable. Footnote 15

There are a few implications of these assumptions, including the instantaneous behaviour of the price process. Some of these implications will be discussed shortly and some will be deferred to later chapters. Our next task is to investigate the risk-neutral probability which is induced by the above assumptions. We would like to summarize and highlight a few issues demonstrated by the above graphic presentation. Footnote 16

The larger the time interval t , the larger the variance and the expected value of [Maple Math] . Both the expected value and the variance of [Maple Math] are linearly increasing functions of the length [Maple Math] , i.e., [Maple Math] and [Maple Math] , respectively. The volatility of the return is measured by the standard deviation of [Maple Math] , [Maple Math] , and depends on time via [Maple Math] . Hence when t gets bigger, say, by a factor of 2, the expected return also increases by the same factor of 2. However, the volatility or the "riskiness'' of the return, increases only by a factor of [Maple Math] . Footnote 17

As in the case of the risk-free rate, the stochastic return is additive across disjoint time intervals. If [Maple Math] is the return over [Maple Math] , [Maple Math] , and [Maple Math] , are the returns over [Maple Math] and [Maple Math] , respectively; then

[Maple Math]

(5.29)

for every [Maple Math] . This equality, however, is between random variables, and indeed being held in the sense that the sum of two normal random variables is again a normal random variable. Moreover,

[Maple Math] = [Maple Math] = [Maple Math] ,

(5.30)

and since [Maple Math] and [Maple Math] are independent,

[Maple Math] = [Maple Math] = [Maple Math]

(5.31)

The reader must be alerted to the distinction between the risk-neutral probability and the "real-life'' probability. The former is only an artificial consequence of the no-arbitrage condition, and the latter is the real-life probability of the continuously compounded rate of return [Maple Math] . Our assumptions above pertain to the real-life probability. We shall soon see that we actually assume that under both distributions [Maple Math] is normally distributed with [Maple Math] , and [Maple Math] , commonly denoted by [Maple Math] ~ [Maple Math] . However, the values of the parameters [Maple Math] is different under the risk-neutral probability and under the real-life probability.

Assumption I ( on page 193) and equation (5.25) imply that

[Maple Math] = [Maple Math] = [Maple Math] = [Maple Math] ,

(5.32)

where [Maple Math] ~ [Maple Math] . The [Maple Math] in equation (5.32) shows that the rate of return on a stock is the sum of the deterministic part, [Maple Math] , and the stochastic part [Maple Math] Footnote 18. The deterministic part is a linear function of [Maple Math] , as is the risk-free return. The stochastic part [Maple Math] however, depends on the time [Maple Math] , via the square root of [Maple Math] . These last two points are visualized in the next section.

Footnote

Footnote 14

In this animation the [Maple Math] parameter, time, shrinks from 6 to 0.1. Later we will speak the situation where [Maple Math] approaches zero.

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Footnote 15

The use of ''limit of [Maple Math] '' should and could be made precise. However, it is used here in an intuitive way. See Appendix 15.11.2 for a further explanation and [13] for a more formal treatment.

For [Maple Math] and [Maple Math] it is possible to ask MAPLE to generate the sequence of the intervals as [Maple Math] approaches zero from 1 to, say, [Maple Math] .

> seq(evalf([.15*(1/t)-2*.23*sqrt((1/t)),.15*(1/t)+2*.23*sqrt(1/t)]),t=1..10);

[Maple Math]
[Maple Math]
[Maple Math]

>

Alternatively it is possible to ask MAPLE to calculate the limit of the length of the interval as [Maple Math] approaches zero.

> limit((mu*t +2*sigma*sqrt(t))-(mu*t-2*sigma*sqrt(t)),t=0);

[Maple Math]

>

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Footnote 16

These implications will be mainly dealt with in Chapter 15.

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Footnote 17

Thus, when t shrinks to zero, [Maple Math] will approach zero at a different speed than [Maple Math] . This phenomenon can be illustrated by looking at the graph of [Maple Math] and t plotted on the same axis. Hence, if we refer to the per unit time return and volatility by [Maple Math] and [Maple Math] , respectively, as [Maple Math] approaches zero the first expression is [Maple Math] while the second diverges to infinity. We will return to implications of this observation when we discuss the prices process more rigorously.

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Footnote 18

Or putting it differently, one plus the rate of return on a stock, [Maple Math] , has a deterministic part [Maple Math] which is multiplied by a random part [Maple Math] .

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