Investments, tenth edition

 Figure 21.6 

A standard normal curve  

bod61671_ch21_722-769.indd   738

bod61671_ch21_722-769.indd   738

7/27/13   1:45 AM

7/27/13   1:45 AM

Final PDF to printer

---

  C H A P T E R  

2 1

 Option 

Valuation 

739

money, and ln(105/100)  5  .049. Similarly, if  S  

0

   5  95, the option is 5% out of the money, 

and ln(95/100)  5   2 .051.  The  denominator,     s

"T,  adjusts the amount by which the option 

is in or out of the money for the volatility of the stock price over the remaining life of the 

option. An option in the money by a given percent is more likely to stay in the money 

if both stock price volatility and time to expiration are low. Therefore,  N ( d  

1

 ) and  N ( d  

2

 ) 

increase with the probability that the option will expire in the money. 

 You can use the Black-Scholes formula fairly easily. Suppose you want to value a call 

option under the following circumstances:

   Stock price:   

   S  

0

   5  100  

  Exercise price:   

   X   5  95  

  Interest rate:      

r   5  .10 (10% per year)  

  Time to expiration:   

   T   5  .25 (3 months or one-quarter of a year)  

  Standard deviation:   

   s   5  .50 (50% per year)  

 

First calculate 

  

ln(100 / 95) 1 (.10 1 .5

2

/ 2).25

.43

.5√.25

d

1

 5

5

 

d

2

5

.43 2 .5

".25 5 .18 

Next find  N ( d  

1

 ) and  N ( d  

2

 ). The values of the normal distribution are tabulated and may 

be found in many statistics textbooks. A table of  N ( d ) is provided here as  Table  21.2 . 

The normal distribution function,  N ( d ), is also provided in any spreadsheet program. In 

Microsoft Excel, for example, the function name is NORMSDIST. Using either Excel or 

 Table 21.2  we find that 

N(.43) 5 .6664

N(.18) 5 .5714

 Thus the value of the call option is 

   C 5 100 3 .6664 2 95e

2

.103.25

3

.5714

 5

66.64 2 52.94 5 $13.70

  

 Example  21.4 

Black-Scholes Valuation 

 Recalculate the value of the call option in Example 21.4 using a standard deviation of .6 instead of .5. 

Confirm that the option is worth more using the higher stock-return volatility. 

 CONCEPT CHECK 

21.6 

   What if the option price in Example 21.4 were $15 rather than $13.70? Is the option 

mispriced? Maybe, but before betting your fortune on that, you may want to reconsider 

the valuation analysis. First, like all models, the Black-Scholes formula is based on some 

simplifying abstractions that make the formula only approximately valid. 

bod61671_ch21_722-769.indd   739

bod61671_ch21_722-769.indd   739

7/27/13   1:45 AM

7/27/13   1:45 AM

Final PDF to printer

---

740

P A R T   V I

  Options, Futures, and Other Derivatives