Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionV. Risk and Return 14. Options and Corporate
Finance(^490) © The McGraw−Hill
Companies, 2002
we don’tknow the odds associated with these two prices. In other words, we know the
possible values for the stock, but not the probabilities associated with those values.
Because the exercise price on the option is $105, we know that the option will be
worth either $110 105 $5or $130 105 $25, but, once again, we don’t know
which. We do know one thing, however: Our call option is certain to finish in the money.
The Basic Approach Here is the crucial observation: It is possible to exactly dupli-
cate the payoffs on the stock using a combination of the option and the risk-free asset.
How? Do the following: buy one call option and invest $87.50 in a risk-free asset (such
as a T-bill).
What will you have in a year? Your risk-free asset will earn 20 percent, so it will be
worth $87.50 1.20 $105. Your option will be worth $5or $25, so the total value
will be either $110or $130, just like the value of the stock:
As illustrated, these two strategies—buying a share of stock or buying a call and invest-
ing in the risk-free asset—have exactly the same payoffs in the future.
Because these two strategies have the same future payoffs, they must have the same
value today or else there would be an arbitrage opportunity. The stock sells for $100 to-
day, so the value of the call option today, C 0 , is:
$100 $87.50 C 0
C 0 $12.50
Where did we get the $87.50? This is just the present value of the exercise price on the
option, calculated at the risk-free rate:
E/(1 Rf) $105/1.20 $87.50
Given this, our example shows that the value of a call option in this simple case is
given by:
S 0 C 0 E/(1 Rf)
[14.5]
C 0 S 0 E/(1 Rf)
In words, the value of the call option is equal to the stock price minus the present value
of the exercise price.
A More Complicated Case Obviously, our assumption that the stock price in one
year will be either $110 or $130 is a vast oversimplification. We can now develop a
more realistic model by assuming that the stock price in one year can be anything
greater than or equal to the exercise price. Once again, we don’t know how likely the
different possibilities are, but we are certain that the option will finish somewhere in
the money.
We again let S 1 stand for the stock price in one year. Now consider our strategy of in-
vesting $87.50 in a riskless asset and buying one call option. The riskless asset will
again be worth $105 in one year, and the option will be worth S 1 $105, the value of
which will depend on what the stock price is.
Stock Risk-Free Call Total
Value vs. Asset Value Value Value
$110 $105 $ 5 $110
130 105 25 130
462 PART FIVE Risk and Return