460 Part IV: Additional Considerations 
many of the applications that are going to be prescribed, it is not necessary to know 
the exact theoretical price of the call. Therefore, the dividend "correction" might not 
have to be applied for certain strategy decisions. 
The model is based on a lognormal distribution of stock prices. Even though the 
normal distribution is part of the model, the inclusion of the exponential functions 
makes the distribution lognormal. For those less familiar with statistics, a normal dis­
tribution has a bell-shaped curve. This is the most familiar mathematical distribution. 
The problem with using a normal distribution is that it allows for negative stock 
prices, an impossible occurrence. Therefore, the lognormal distribution is generally 
used for stock prices, because it implies that the stock price can have a range only 
between zero and infinity. Furthermore, the upward (bullish) bias of the lognormal 
distribution appears to be logically correct, since a stock can drop only 100% but can 
rise in price by more than 100%. Many option pricing models that antedate the 
Black-Scholes model have attempted to use empirical distributions. An empirical 
distribution has a different shape than either the normal or the lognormal distribu­
tion. Reasonable empirical distributions for stock prices do not differ tremendously 
from the lognormal distribution, although they often assume that a stock has a 
greater probability of remaining stable than does the lognormal distribution. Critics 
of the Black-Scholes model claim that, largely because it uses the lognormal distri­
bution, the model tends to overprice in-the-money calls and underprice out-of-the­
money calls. This criticism is true in some cases, but does not materially subtract 
from many applications of the model in strategy decisions. True, if one is going to buy 
or sell calls solely on the basis of their computed value, this would create a large prob­
lem. However, if strategy decisions are to be made based on other factors that out­
weigh the overpriced/underpriced criteria, small differentials will not matter. 
The computation of volatility is always a difficult problem for mathematical 
application. In the Black-Scholes model, volatility is defined as the annual standard 
deviation of the stock price. This is the regular statistical definition of standard devi­
ation: 
where 
P = average stock price of all P/s 
Pi = daily stock price 
n 
~ (Pi -P)2 
cr2 = _1=_1 __ _ 
n-1 
v = a!P