Introduction to Corporate Finance

Ross et al.: Fundamentals

of Corporate Finance, Sixth

Edition, Alternate Edition

V. Risk and Return 14. Options and Corporate

Finance

© The McGraw−Hill^493

Companies, 2002

The Fifth Factor

We now illustrate the fifth (and last) factor that determines an option’s value. Suppose
everything in our example is the same as before except that the stock price can be $105
or $135 instead of $110 or $130. Notice that the effect of this change is to make the
stock’s future price more volatile than before.
We investigate the same strategy that we used previously: invest the present value of
the lowest stock price ($105 in this case) in the risk-free asset and buy two call options.
If the stock price is $105, then, as before, the call options have no value and we have
$105 in all.
If the stock price is $135, then each option is worth S 1
E$135
120 $15. We
have two calls, so our portfolio is worth $105  2  15 $135. Once again, we have
exactly replicated the value of the stock.
What has happened to the option’s value? More to the point, the variance of the re-
turn on the stock has increased. Does the option’s value go up or down? To find out, we
need to solve for the value of the call just as we did before:

S 0 $100  2 C 0 $105/(1 Rf)

2 C 0 $100
 105/1.20

C 0 $6.25

The value of the call option has gone up from $4.17 to $6.25.

CHAPTER 14 Options and Corporate Finance 465

Don’t Call Us, We’ll Call You

We are looking at two call options on the same stock, one with an exercise price of $20 and

one with an exercise price of $30. The stock currently sells for $35. Its future price will be ei-

ther $25 or $50. If the risk-free rate is 10 percent, what are the values of these call options?

The first case (with the $20 exercise price) is not difficult because the option is sure to fin-

ish in the money. We know that the value is equal to the stock price less the present value of

the exercise price:

C 0 S 0 
 E/(1 Rf)

$35
 20/1.1

$16.82

In the second case, the exercise price is $30, so the option can finish out of the money. At

expiration, the option is worth $0 if the stock is worth $25. The option is worth $50
 30 

$20 if it finishes in the money.

As before, we start by investing the present value of the lowest stock price in the risk-free

asset. This costs $25/1.1 $22.73. At expiration, we have $25 from this investment.

If the stock price is $50, then we need an additional $25 to duplicate the stock payoff. Be-

cause each option is worth $20 in this case, we need $25/20 1.25 options. So, to prevent

arbitrage, investing the present value of $25 in a risk-free asset and buying 1.25 call options

must have the same value as the stock:

S 0 1.25 C 0 $25/(1 Rf)

$35 1.25 C 0 $25/(1 .10)

C 0 $9.82

Notice that this second option had to be worth less because it has the higher exercise price.

EXAMPLE 14.2