Introduction to Corporate Finance

Ross et al.: Fundamentals

of Corporate Finance, Sixth

Edition, Alternate Edition

VIII. Topics in Corporate

Finance

(^838) 24. Option Valuation © The McGraw−Hill
Companies, 2002
THE BLACK-SCHOLES OPTION
PRICING MODEL
We’re now in a position to discuss one of the most celebrated results in modern finance,
the Black-Scholes Option Pricing Model (OPM). The OPM is a sufficiently important
discovery that it was the basis for the Nobel Prize in Economics in 1997. The underly-
ing development of the Black-Scholes OPM is fairly complex, so we will focus only on
the main result and how to use it.
The Call Option Pricing Formula
Black and Scholes showed that the value of a European-style call option on a non-
dividend paying stock, C, can be written as follows:
CSN(d 1 )
Ee
RtN(d 2 ) [24.5]
where S, E, and e
Rtare as we previously defined them and N(d 1 ) and N(d 2 ) are proba-
bilities that must be calculated. More specifically, N(d 1 ) is the probability that a stan-
dardized, normally distributed random variable (widely known as a “z” variable) is less
than or equal to d 1 , and N(d 2 ) is the probability of a value less than or equal to d 2. De-
termining these probabilities requires a table such as Table 24.3.
To illustrate, suppose we are given the following information:
S$100
E$90
Rf4% per year, continuously compounded
d 1 .60
d 2 .30
t9 months
Based on this information, what is the value of the call option, C?
To answer, we need to determine N(d 1 ) and N(d 2 ). In Table 24.3, we first find the row
corresponding to a dof .60. The corresponding probability N(d) is .7258, so this is
N(d 1 ). For d 2 , the associated probability N(d 2 ) is .6179. Using the Black-Scholes OPM,
we calculate that the value of the call option is:
CSN(d 1 )
Ee
RtN(d 2 )
$100 .7258
$90 e
.04(3/4).6179
$18.61
Notice that t, the time to expiration, is 9 months, which is 9/12 3/4 of one year.
As this example illustrates, if we are given values for d 1 and d 2 (and the table), then
using the Black-Scholes model is not difficult. Generally, however, we would not be
given the values of d 1 and d 2 , and we must calculate them. This requires a little extra ef-
fort. The values for d 1 and d 2 for the Black-Scholes OPM are given by:
d 1 [ln(S/E) (R^2 /2) t]/( )
[24.6]
d 2 d 1

In these expressions, is the standard deviation of the rate of return on the underlying
asset. Also, ln(S/E) is the natural logarithm of the current stock price divided by the ex-
ercise price.
t
t
CHAPTER 24 Option Valuation 813

24.2

There’s a Black-Scholes

calculator (and a lot more)

at http://www.cboe.com.