468 Part IV: Additional Considerations 
TABLE 28-6. 
Computation of expected profit. 
Chance of Being Profit at Expected 
XYZ Price at at That Price That Price Profit: 
Expiration (A) (B) (A) x (8) 
Below 30 20% -$200 -$ 40 
31 10% - 100 - 10 
32 10% 0 0 
33 10% + 100 + 10 
34 10% + 200 + 20 
Above 35 40% + 300 + 120 
Total expected profit $100 
As is readily observable, the selection of what percentages to assign to the pos­
sible outcomes in the stock price is a crucial choice. In the example above, if one 
altered his assumption slightly so that XYZ had a 30% chance of being below 30 and 
a 30% chance of being above 35 at expiration, the expected return would drop con­
siderably, to 25%. Thus, it is important to have a reasonably accurate and consistent 
method of assigning these percentages. Furthermore, the example above was too sim­
plistic, in that it did not allow for the stock to close at any fractional prices, such as 
32½. A correct expected return computation must take into account all possible out­
comes for the stock. 
Fortunately, there is a straightforward method of computing the expected per­
centage chance of a given stock being at a certain price at a certain point in time. This 
computation involves using the distribution of stock prices. As mentioned earlier, the 
Black-Scholes model assumes a lognormal distribution for stock prices, although 
many modelers today use nonstandard (empirical or heuristic) distributions. No mat­
ter what the distribution, the area under the distribution curve between any two 
points gives the probability of being between those two points. 
Figure 28-1 is a graph of a typical lognormal distribution. The peak always lies 
at the "mean," or average, of the distribution. For stock price distributions, under the 
random walk assumption, the "mean" is generally considered to be the current stock 
price. The graph allows one to visualize the probability of being at any given price. 
Note that there is a fairly great chance that the stock will be relatively unchanged; 
there is no chance that the stock will be below zero; and there is a bullish bias to the 
graph - the stock could rise infinitely, although the chances of it doing so are 
extremely small.