We can visualize the above points in a manner that will let us have some insight into the path of the price process. One plus the rate of return on a stock, see footnote 18, has a derministic part, , which is multiplied by a random part, . Consider, for example, a stock such that at , its price is , its expected return per unit of time is , and its standard deviation per unit of time is . Its price at time will be

,

(5.33)

where is a random number, drawn from the standard normal distribution with and . The stock price at time can also be expressed in terms of its price at time . This however will require two independent drawings; ~ drawn at time zero and ~ drawn at time . At time the stock price will be

(5.34)

for some realization of . At time it will be

for yet another independent drawing of . Thus,

,

(5.35)

where and are independent identically distributed random variables. In the absence of the stochastic part, that is, if is set equal to zero, we would have = .

In general we can divide the time interval into parts and draw numbers from the standard normal distribution. This will result in the expression of as the product of terms times the current price, that is, as

,

(5.36)

where denotes the product of the 's, i.e., * *....* .

The Procedure Gmbm simulates this process. It takes the folowing parameters: the price of the stock at the initial time, the value of (which we assumed to be zero until now), the value of , the value of , and the values of and reported per the unit of time. The procedure animates the random price process against the deterministic price process, , by linear interpolation.

Thus if we choose, for example, and and as in equation (5.33), the procedure simulates the price of the stock at time , as in Figure 5.6.

> Gmbm(100,0,1,1,.15,.23);

Figure 5.6: Deterministic Process vs. Stochastic Process for n=1

The graphs generated are thus linear. One graph connects the value of the stock at to its deterministic value at , that is, to . The other line connects it to a realization of the price of the stock at time . Every time you run this procedure a random number is drawn to generate the price of the stock at time . The deterministic price of course stays the same as long as the parameter stays the same. The reader is invited to run this procedure for different values of the parameters.

When is chosen, the procedure also simulates the price of the stock at time . Thus the graph of the price of the stock is no longer linear. It has a knot point at and its value there is based on equation (5.34). The graph of the deterministic price of the stock is of course the same. This is demonstrated in Figure 5.7 for .

> Gmbm(100,0,1,2,.15,.23);

Figure 5.7: Deterministic Process vs. Stochastic Process for n=100

Running this procedureure a few times, the reader can appreciate the random effect on the price of the stock, by comparing it to the deterministic value of the stock price. The deterministic value of the stock price behaves like a risk-free asset when the interest rate . Perhaps a better way of appreciating this risk is to run the procedure for a higher value of with relatively low and high values of . For example, keep the value of at 0.15 and try two values for . Let us start with, say, , as in Figure 5.8.

> Gmbm(100,0,1,50,.15,.03);

Figure 5.8: Deterministic Process vs. Stochastic Process for n=100

The reader should run this procedure a few times before moving to the next value of . Keep in mind that each time that it is run, the result is different. Each time random numbers are drawn from the normal distribution. Yet in most of the graphs the stochastic sample path is fairly close to the deterministic value graph. Let us now change the vlaue of to be 0.63, as in Figure 5.9.

> Gmbm(100,0,1,50,.15,.63);

Figure 5.9: Deterministic Process vs. Stochastic Process for n=100 and a "High'' Value of

Footnotes

For the pricing of the European option, the path that the stock price follows is irrelevant. However, for a more complex type option it is very relevant as we will see in next chapters.