1.Assumptions of the example.The stock of Western Cellular, a manufacturer of
cell phones, sells for $40 per share. Options exist that permit the holder to buy one
share of Western at an exercise price of $35. These options will expire at the end of
one year, at which time Western’s stock will be selling at one of two prices, either
$30 or $50. Also, the risk-free rate is 8.0 percent. Based on these assumptions, we
must find the value of the options.
2.Find the range of values at expiration.When the option expires at the end of the
year, Western’s stock will sell for either $30 or $50, and here is the situation with
regard to the value of the options:
Ending Strike Ending
Stock Price Option
Price Value Value
$50.00 $35.00 $15.00
30.00 35.00 0.00 (Theoptionwillbeworthless.
It cannot have a negative
value.)
Range $20.00 $15.00
3.Equalize the range of payoffs for the stock and the option. As shown above,
the ranges of payoffs for the stock and the option are $20 and $15. To construct the
riskless portfolio, we need to equalize these ranges. We do so by buying 0.75 share
and selling one option (or 75 shares and 100 options) to produce the following sit-
uation, where the range for both the stock and the option is $15:
Ending Ending Ending
Stock 0.75 Value of Value of
Price Stock Option
$50.00 0.75 $37.50 $15.00
30.00 0.75 22.50 0.00
Range $20.00 $15.00 $15.00
4.Create a riskless hedged investment.We can now create a riskless portfolio by
buying 0.75 share of the stock and selling one call option. Here is the situation:
Ending Ending Value of Ending Value of Ending Total
Stock 0.75 Stock in the Option in the Value of
Price Portfolio Portfolio the Portfolio
$50.00 0.75 $37.50 $15.00 $22.50
30.00 0.75 22.50 0.00 22.50
The stock in the portfolio will have a value of either $22.50 or $37.50, depending
on what happens to the price of the stock. The call option that was sold will have
no effect on the value of the portfolio if Western’s price falls to $30, because it will
then not be exercised—it will expire worthless. However, if the stock price ends at
$50, the holder of the option will exercise it, paying the $35 exercise price for stock
that would cost $50 on the open market, so in that case, the option would have a
cost of $15 to the holder of the portfolio.
Now note that the value of the portfolio is $22.50 regardless of whether West-
ern’s stock goes up or down, so the portfolio is riskless. A hedge has been created
that protects against both increases or decreases in the price of the stock.
Introduction to Option Pricing Models 629
624 Option Pricing with Applications to Real Options
