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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

© The McGraw-Hill Companies, 2002

PART EIGHT Topics in Corporate Finance

A question that sometimes comes up concerns the probabilities N(d1) and N(d2). Just what are they the probabilities of? In other words, how do we interpret them? The answer is that they don't really correspond to anything in the real world. We mention this because there is a common misconception about N(d2) in particular. It is frequently thought to be the probability that the stock price will exceed the strike price on the expiration day, which is also the probability that a call option will finish in the money. Unfortunately, that's not correct, at least not unless the expected return on the stock is equal to the risk-free rate.

Tables such as Table 24.3 are the traditional means of looking up "z" values, but they have been mostly replaced by computers. They are not as accurate because of rounding, and they also have only a limited number of values. Our nearby Spreadsheet Strategies box shows how to calculate Black-Scholes call option prices using a spreadsheet. Because this is so much easier and more accurate, we will do all the calculations in the rest of this chapter using computers instead of tables.

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