5.5 The Distribution of the Rate of Return
Let us utilize MAPLE to visulize some conseqences of our assumptions about . The normal density function with expcted value of and standard deviaiton of at a point is given by
.
(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);
>
In our case, the random varible (we will omit the suscript in the MAPLE calculation) is the continuously compounded rate of return for a period of length . We assumed that is normally distributed such that its expected value and standard deviation are dependent on time, and are given by and for some constant and . Thus the density function of at a point is given by
> Normalpdf(y,mu*t,sigma*sqrt(t));
>
When , , and the value of Normalpdf at a point can be calculated using the MAPLE command.
> Normalpdf(y,.15*2,.23*sqrt(2));
>
For a specific value of , say = 0.15 x 2, is simply replaced with a numerical value:
> Normalpdf(.15*2,.15*2,.23*sqrt(2));
>
To get MAPLE to evaluate it in a decimal form we utilize the evalf command:
> evalf(Normalpdf(.15*2,.15*2,.23*sqrt(2)));
>
Let us look at Figure 5.3, the density function of where and .
> plot(Normalpdf(y,.15*2,.23*sqrt(2)),y=-3..4,title =`Figure 5.3: The Normal Density Function`, titlefont=[TIMES, BOLD, 10]);
>
This is the famous bell-shaped graph of the normal density function.
In order to visualize the effect of time on we run the animation below. The animation can be used to demonstrate how this graph changes as the time interval 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]);
>
Figure 5.4 presents a static graph of the distribution for , and 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])});
>
Recall that the normal distribution has a peak at its expected value, at in our case, and that the larger the standard deviation, in our case, the flatter the graph and the riskier the situation. The area under the graph between two points and is the probability that and . Hence, as decreases, decreases, the graph gets steeper, and the probablity of being close to its mean, , increases. Of course is also getting smaller as 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 approaches zero, the probability of 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 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]);
>
It is now apparent that as approches zero, the mass of probability, the area under the graph, is concentrated around zero. Thus the probability that the instantaneous rate of return is close to zero as approaches zero is close to one. Hence, as approaches zero, is very likely to be one. The situation therefore becomes less and less risky. If at time the price of the stock is , its price at time , , will be the random variable . For a small , the probability of being one is close to one, and thus, the probablity of being close to is also close to one. Hence for short time intervals the realization of is very likely to be close to .
The normal cumulative distribution function of a random variable at a point is defined as the probabilty that will have a value less than or equal to . Thus, it is the area under the graph of Normalpdf from negative infinity to . In other words it is
,
(5.27)
or in our notation, as in equation (5.26),
.
(5.28)
We define this function in MAPLE and call it Normalcdf .
> Normalcdf:=(x,mu,sigma)->int(Normalpdf(v,mu,sigma),v=-infinity..x);
>
The effect of time on the distribution of 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]);
> 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]);
>
The probability that the random variable will have a value between and 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 and 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)));
>
However, consider the effect on the interval , when approaches zero. The interval clearly approches zero, but the probability of the interval remains the same as it is independent of . Hence the likelihood of 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 approaches zero, remains a random variable for every , and hence the ``limit of " as 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
. Both the expected value and the variance of
are linearly increasing functions of the length
, i.e.,
and
, respectively. The
volatility
of the return is measured by the standard deviation of
,
, and depends on time via
. 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
.
Footnote 17
As in the case of the risk-free rate, the stochastic return is additive across disjoint time intervals. If is the return over , , and , are the returns over and , respectively; then
(5.29)
for every . 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,
= = ,
(5.30)
and since and are independent,
= =
(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 . Our assumptions above pertain to the real-life probability. We shall soon see that we actually assume that under both distributions is normally distributed with , and , commonly denoted by ~ . However, the values of the parameters is different under the risk-neutral probability and under the real-life probability.
Assumption I ( on page 193) and equation (5.25) imply that
= = = ,
(5.32)
where ~ . The in equation (5.32) shows that the rate of return on a stock is the sum of the deterministic part, , and the stochastic part Footnote 18. The deterministic part is a linear function of , as is the risk-free return. The stochastic part however, depends on the time , via the square root of . These last two points are visualized in the next section.
Footnote
In this animation the parameter, time, shrinks from 6 to 0.1. Later we will speak the situation where approaches zero.
The use of ''limit of '' 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 and it is possible to ask MAPLE to generate the sequence of the intervals as approaches zero from 1 to, say, .
> seq(evalf([.15*(1/t)-2*.23*sqrt((1/t)),.15*(1/t)+2*.23*sqrt(1/t)]),t=1..10);
>
Alternatively it is possible to ask MAPLE to calculate the limit of the length of the interval as approaches zero.
> limit((mu*t +2*sigma*sqrt(t))-(mu*t-2*sigma*sqrt(t)),t=0);
>
These implications will be mainly dealt with in Chapter 15.
Thus, when t shrinks to zero, will approach zero at a different speed than . This phenomenon can be illustrated by looking at the graph of and t plotted on the same axis. Hence, if we refer to the per unit time return and volatility by and , respectively, as approaches zero the first expression is while the second diverges to infinity. We will return to implications of this observation when we discuss the prices process more rigorously.
Or putting it differently, one plus the rate of return on a stock, , has a deterministic part which is multiplied by a random part .