Introduction to Corporate Finance

ParT 2: ValuaTION, rISk aNd reTurN

What about the term Xe-rt? Recall from our discussion about continuous compounding, in Chapter 3,

that the term e-rt reflects the present value of $1, discounted at r per cent for t years. Therefore, Xe-rt simply

equals the present value of the option’s strike price.^15 With this in mind, look again at Equation 8.2.

The first term equals the share price multiplied by a quantity labelled N(d 1 ). The second term is the

present value of the strike price multiplied by a quantity labelled N(d 2 ). Therefore, we can say that the

call option value equals the ‘adjusted’ share price minus the present value of the ‘adjusted’ strike price,

where N(d 1 ) and N(d 2 ) represent adjustment factors. Earlier in this chapter, we saw that call option

values increase as the difference between the share price and the strike price, S–X, increases. The same

relationship holds here, although we must now factor in the terms N(d 1 ) and N(d 2 ).

In the Black–Scholes equation, d 1 and d 2 are simply numerical values (calculated using Equation 8.3)

that depend on the model’s inputs: the share price, the strike price, the interest rate, the time to

expiration and volatility. The expressions N(d 1 ) and N(d 2 ) convert the numerical values of d 1 and d 2 into

probabilities using the standard normal distribution.^16 Figure 8.9 shows that the value N(d 1 ) equals the

area under the standard normal curve to the left of value d 1. For example, if we calculate the value of

d 1 and find that it equals 0, then N(d 1 ) equals 0.5, because half of the area under the curve falls to the

left of zero. The higher the value of d 1 , the closer N(d 1 ) gets to 1.0. The lower the value of d 1 , the closer

N(d 1 ) gets to zero. The same relationship holds between d 2 and N(d 2 ). Given a particular value of d 1 (or

d 2 ), to calculate N(d 1 ) you need a table showing the cumulative standard normal probabilities, or you can

plug d 1 into the Excel function =normsdist(d 1 ). A common intuitive interpretation of N(d 1 ) and N(d 2 )

is that they represent the risk-adjusted probabilities that the call will expire in the money. Therefore, a

verbal description of Equation 8.2 is as follows: The call option price equals the share price, minus the

15 Remember, this expression can be written in two ways: Xe X

e

–rt

= rt. Assuming that the continuously compounded risk-free rate of interest

equals r and the amount of time before expiration equals t, this is simply the present value of the strike price.

16 Recall from the statistics that the standard normal distribution has a mean equal to zero and a standard deviation equal to 1.

FIGure 8.9 STANDARD NORMAL DISTRIBUTION

The expression N(d 1 ) equals the probability of drawing a particular value, d 1 , or a lower value, from the standard normal

distribution. In the figure, N(d 1 ) is represented by the shaded portion under the bell curve. Because the normal distribution

is symmetric about the mean, we can write, N(d 1 ) = 1 − N(−d 1 ).

Probability

N (d 1 ) = Shaded region

0 d 1