Chapter 28: Mathematical Applications 459 
Now, returning to the formula for theoretical option price, we can complete the 
calculation of the July 50 call's theoretical value, called value here for short: 
value = 45 x N(d1) - 50 x e-·1 x ·16438 x N(d2) 
= 45 X .25134 - 50 X .9837 X .21421 
= .7746 
Thus, the theoretical value of the July 50 call is just slightly over¼ of a point. 
Note that the delta of the call was calculated along the way as N(d1) and is equal to 
just over .25. That is, the July 50 call will change price about¼ as fast as the stock 
for a small price change by the stock. 
This example should answer many of the questions that readers of the first edi­
tion have posed. The reader interested in a more in-depth description of the model, 
possibly including the actual derivation, should refer to the article "Fact and Fantasy 
in the Use of Options." 1 One of the less obvious relationships in the model is that call 
option prices will increase (and put option prices will decrease) as the risk-free inter­
est rate increases. It may also be observed that the model correctly preserves rela­
tionships such as increased volatility, higher stock prices, or more time to expiration, 
which all imply higher option prices. 
CHARACTERISTICS Of THE MODEL 
Several aspects of this model are worth further discussion. First, the reader will 
notice that the model does not include dividends paid by the common stock. As has 
been demonstrated, dividends act as a negative effect on call prices. Thus, direct 
application of the model will tend to give inflated call prices, especially on stocks that 
pay relatively large dividends. There are ways of handling this. Fisher Black, one of 
the coauthors of the model, suggested the following method: Adjust the stock price 
to be used in the formula by subtracting, from the current stock price, the present 
worth of the dividends likely to be paid before maturity. Then calculate the option. 
price. Second, assume that the option expires just prior to the last ex-dividend date 
preceding actual option expiration. Again adjust the stock price and calculate the 
option price. Use the higher of the two option prices calculated as the theoretical 
price. 
Another, less exact, method is to apply a weighting factor to call prices. The 
weighting factor would be based on the dividend payment, with a heavier weight 
being applied to call options on high-yielding stock. It should be pointed out that, in 
1Fisher Black, Financial Analysts Journal, July-August 1975, pp. 36-70.