Investments, tenth edition

   The Black-Scholes Formula 

 Financial economists searched for years for a workable option-pricing model before 

Black and Scholes  

7

    and  Merton  

8

   derived a formula for the value of a call option. Scholes 

and Merton shared the 1997 Nobel Prize in Economics for their accomplishment.  

9

   Now 

widely used by options market participants, the    Black-Scholes  pricing  formula    for a 

call option is  

 

  C

0

5 S

0

N(d

1

) 2 Xe

2rT

N(d

2

)  

(21.1)  

where

      d

1

5

ln(S

0

/X) 1 (r 1 s

2

/2)T

s

"T

   

     d

2

5 d

1

2 s

"T    

  

7

 Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,”  Journal of Political 

Economy  81 (May–June 1973). 

   

8

 Robert C. Merton, “Theory of Rational Option Pricing,”  Bell Journal of Economics and Management Science   4 

(Spring 1973).  

   

9

 Fischer Black died in 1995.  

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P A R T   V I

  Options, Futures, and Other Derivatives

and

C  

0

   5  Current call option value.  

S  

0

   5  Current stock price.  

N ( d )  5   The probability that a random draw from a standard normal distribution will 

be less than  d.  This equals the area under the normal curve up to  d,  as in the 

shaded area of  Figure 21.6 . In Excel, this function is called NORMSDIST( ).

X   5  Exercise  price.  

e   5   The base of the natural log function, approximately 2.71828. In Excel,  e  

 x 

   can 

be evaluated using the function EXP( x ).  

r   5   Risk-free interest rate (the annualized continuously compounded rate on a safe 

asset with the same maturity as the expiration date of the option, which is to be 

distinguished from  r  

 f 

 , the discrete period interest rate).  

T   5  Time to expiration of option, in years.  

   

ln  5  Natural logarithm function. In Excel, ln( x ) can be calculated as LN( x ).  

s   5   Standard deviation of the annualized continuously compounded rate of return 

of  the  stock.    

 Notice a surprising feature of Equation 21.1: The option value does  not  depend on the 

expected rate of return on the stock. In a sense, this information is already built into the 

formula with the inclusion of the stock price, which itself depends on the stock’s risk 

and return characteristics. This version of the Black-Scholes formula is predicated on the 

assumption that the stock pays no dividends. 

 Although you may find the Black-Scholes formula intimidating, we can explain it at 

a somewhat intuitive level. The trick is to view the  N ( d  ) terms (loosely) as risk-adjusted 

probabilities that the call option will expire in the money. First, look at Equation 21.1 

assuming both  N ( d  ) terms are close to 1.0, that is, when there is a very high probability 

the option will be exercised. Then the call option value is equal to  S  

0

   2   Xe  

 2  rT 

 ,  which 

is what we called earlier the adjusted intrinsic value,  S  

0

   2  PV( X ). This makes sense; if 

exercise is certain, we have a claim on a stock with current value  S  

0

 , and an obligation 

with present value PV( X  ), or, with continu-

ous compounding,  Xe  

 2  rT 

 . 

 Now look at Equation 21.1 assuming the 

 N ( d  

) terms are close to zero, meaning the 

option almost certainly will not be exercised. 

Then the equation confirms that the call is 

worth nothing. For middle-range values of 

 N ( d  ) between 0 and 1, Equation 21.1 tells us 

that the call value can be viewed as the present 

value of the call’s potential payoff adjusting 

for the probability of in-the-money expiration. 

 

How do the  

N ( d 

) terms serve as risk-

adjusted probabilities? This question quickly 

leads us into advanced statistics. Notice, 

however, that ln( S  

0

 / X ), which appears in the 

numerator of  d  

1

  and  d  

2

 , is approximately the 

percentage amount by which the option is cur-

rently in or out of the money. For example, if 

 S  

0

   5  105 and  X   5  100, the option is 5% in the 

N(d) = 

Shaded area

d

0