## Info

This table shows the probability [N(d)] of observing a value less than or equal to d. For example, as illustrated, if d is -.24, then N(d) is .4052.

This table shows the probability [N(d)] of observing a value less than or equal to d. For example, as illustrated, if d is -.24, then N(d) is .4052.

The formula for d1 looks a little intimidating, but it is mostly a matter of plug-and-chug with a calculator. To illustrate, suppose we have the following:

Ross et al.: Fundamentals VIII. Topics in Corporate 24. Option Valuation © The McGraw-Hill of Corporate Finance, Sixth Finance Companies, 2002

Edition, Alternate Edition

CHAPTER 24 Option Valuation 815

R = 4% per year, continuously compounded ct = 60% per year t = 3 months

With these numbers, d1 is:

d1 = [ln(S/E) + (R + ct2/2) X t]/(a X Jt) = [ln(70/80) + (.04 + .62/2) X V4]/(.6 X //4) = -.26 d2 = d1 - ct X Jt

Referring to Table 24.3, the values of N(d1) and N(d2) are .3974 and .2877, respectively. Plugging all the numbers in:

C = S X N(d1) - E X e-Rt X N(d2) = \$70 X .3974 - \$80 X e- 04(1/4) X .2877 = \$5.03

If you take a look at the Black-Scholes formula and our examples, you will see that the price of a call option depends on five, and only five, factors. These are the same factors that we identified earlier: namely, the stock price, the strike price, the time to maturity, the risk-free rate, and the standard deviation of the return on the stock.

Call Option Pricing

Suppose you are given the following:

R = 4% per year, continuously compounded ct = 70% per year t = 3 months

What's the value of a call option on the stock?

We need to use the Black-Scholes OPM. So, we first need to calculate d1 and d2:

d1 = [ln(S/E) + (R + ct2/2) X t]/(ct X Jt) = [ln(40/36) + (.04 + .72/2) X 1A]/(7 X J/4) = .50 d2 = d - CT X Jt = .50 - .7 X /a = .15

Referring to Table 24.3, the values of N(d1) and N(d2) are .6915 and .5597, respectively. To get the second of these, we averaged the two numbers on each side, (.5557 + .5636)/2 = .5597.

Plugging all the numbers in:

C = S X N(d1) - E X e-Rt X N(d2) = \$40 X .6915 - \$36 X e- 04(1/4) X .5597 = \$7.71

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

VIII. Topics in Corporate Finance

24. Option Valuation