Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionV. Risk and Return 14. Options and Corporate
Finance(^492) © The McGraw−Hill
Companies, 2002
VALUING A CALL OPTION
We now investigate the value of a call option when there is the possibility that the op-
tion will finish out of the money. We will again examine the simple case of two possible
future stock prices. This case will let us identify the remaining factor that determines an
option’s value.
A Simple Model: Part II
From our previous example, we have a stock that currently sells for $100. It will be
worth either $110 or $130 in a year, and we don’t know which. The risk-free rate is 20
percent. We are now looking at a different call option, however. This one has an exer-
cise price of $120 instead of $105. What is the value of this call option?
This case is a little harder. If the stock ends up at $110, the option is out of the money
and worth nothing. If the stock ends up at $130, the option is worth $130 120 $10.
Our basic approach to determining the value of the call option will be the same. We
will show once again that it is possible to combine the call option and a risk-free in-
vestment in a way that exactly duplicates the payoff from holding the stock. The only
complication is that it’s a little harder to determine how to do it.
For example, suppose we bought one call and invested the present value of the exer-
cise price in a riskless asset as we did before. In one year, we would have $120 from the
riskless investment plus an option worth either zero or $10. The total value would be ei-
ther $120 or $130. This is not the same as the value of the stock ($110 or $130), so the
two strategies are not comparable.
Instead, consider investing the present value of $110 (the lower stock price) in a risk-
less asset. This guarantees us a $110 payoff. If the stock price is $110, then any call op-
tions we own are worthless, and we have exactly $110 as desired.
When the stock is worth $130, the call option is worth $10. Our risk-free investment
is worth $110, so we are $130 110 $20 short. Because each call option is worth
$10, we need to buy two of them to replicate the value of the stock.
Thus, in this case, investing the present value of the lower stock price in a riskless as-
set and buying two call options exactly duplicates owning the stock. When the stock is
worth $110, we have $110 from our risk-free investment. When the stock is worth $130,
we have $110 from the risk-free investment plus two call options worth $10 each.
Because these two strategies have exactly the same value in the future, they must
have the same value today, or else arbitrage would be possible:
S 0 $100 2 C 0 $110/(1 Rf)
2 C 0 $100 110/1.20
C 0 $4.17
Each call option is thus worth $4.17.
CONCEPT QUESTIONS
14.2a What is the value of a call option at expiration?
14.2bWhat are the upper and lower bounds on the value of a call option anytime be-
fore expiration?
14.2c Assuming that the stock price is certain to be greater than the exercise price
on a call option, what is the value of the call? Why?
464 PART FIVE Risk and Return
14.3
The Philadelphia Stock
Exchange has a good
discussion of options:
http://www.phlx.com/products.