630 CHAPTER 17 Option Pricing with Applications to Real Options
5.Pricing the call option. To this point, we have not mentioned the price of the call
option that was sold to create the riskless hedge. How much should it sell for?
Obviously, the seller would like to get a high price, but the buyer would want a
low price. What is the fair, or equilibrium, price? To find this price, we proceed as
follows:
a. The value of the portfolio will be $22.50 at the end of the year, regardless of
what happens to the price of the stock. This $22.50 is riskless.
b. The risk-free rate is 8 percent, so the present value of the riskless $22.50 year-
end value is
PV $22.50/(1.08) $20.83.
c. Since Western’s stock is currently selling for $40, and since the portfolio con-
tains 0.75 share, the cost of the stock in the portfolio is
0.75($40) $30.00.
d. If one paid $30 for the stock, and if the present value of the portfolio is $20.83,
the option would have to sell for $9.17:
Price of option Cost of stock PV of portfolio
$30 $20.83 $9.17.
If this option sold at a price higher than $9.17, other investors could create risk-
less portfolios as described above and earn more than the riskless rate. Investors
(especially the large investment banking firms) would create such portfolios—and
options—until their price fell to $9.17, at which point the market would be in
equilibrium. Conversely, if the options sold for less than $9.17, investors would
create an “opposite” portfolio by buying a call option and selling short the stock.^6
The resulting supply shortage would drive the price up to $9.17. Thus, investors
(or arbitrageurs) would buy and sell in the market until the options were priced at
their equilibrium level.Clearly, this example is unrealistic—Western’s stock price could be almost anything
after one year, and you could not purchase 0.75 share of stock (but you could do so in ef-
fect by buying 75 shares and selling 100 options). Still, the example does illustrate that
investors can, in theory, create riskless portfolios by buying stocks and selling call op-
tions against those stocks, and the return on such portfolios should be the risk-free rate.
If call options are not priced to reflect this condition, arbitrageurs will actively trade
stocks and options until option prices reflect equilibrium conditions. In the next sec-
tion, we discuss the Black-Scholes Option Pricing Model, which is based on the general
premise we developed here—the creation of a riskless portfolio—but which is applica-
ble to “real-world” option pricing because it allows for a complete range of ending stock
prices.Describe how a risk-free portfolio can be created using stocks and options.
How can such a portfolio be used to help estimate a call option’s value?(^6) Suppose an investor (or speculator) does not now own any IBM stock. If the investor anticipates a rise in
the stock price and consequently buys IBM stock, he or she is said to have gone long in IBM. On the other
hand, if the investor thinks IBM’s stock is likely to fall, he or she could go short, or sell IBM short. Because the
short seller has no IBM stock, he or she would have to borrow the shares sold short from a broker. If the
stock price falls, the short seller could, later on, buy shares on the open market and pay back the ones bor-
rowed from the broker. The short seller’s profit, before commissions and taxes, would be the difference be-
tween the price received from the short sale and the price paid later to purchase the replacement stock.