Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionVIII. Topics in Corporate
Finance(^834) 24. Option Valuation © The McGraw−Hill
Companies, 2002
What does this strategy accomplish? Once again, we will create a table to illustrate
your gains and losses. Notice that in Table 24.2 your $100 grows to $105 based on a
5 percent interest rate. If you compare Table 24.2 to our previous Table 24.1, you will
make an interesting discovery. No matter what the stock price is one year from now, the
two strategies alwayshave the same value in one year!
The fact that the two strategies always have exactly the same value in one year ex-
plains why they have the same cost today. If one of these strategies were cheaper than
the other today, there would be an arbitrage opportunity involving buying the one that’s
cheaper and simultaneously selling the one that’s more expensive.
The Result
Our example illustrates a very important pricing relationship. What it shows is that a
protective put strategy can be exactly duplicated by a combination of a call option (with
the same strike price as the put option) and a riskless investment. In our example, notice
that the investment in the riskless asset, $100, is exactly equal to the present value of the
strike price on the option calculated at the risk-free rate, $105/1.05 $100.
Putting it all together, what we have discovered is the put-call parity (PCP)condi-
tion. It says that:
Share of stock a put option Present value of strike price a call option [24.1]
In symbols, we can write:
SPPV(E) C [24.2]
where Sand Pare stock and put values, and PV(E) and Care the present value of the ex-
ercise price and the value of the call option.
Because the present value of the exercise price is calculated using the risk-free rate,
you can think of it as the price of a risk-free, pure discount instrument (i.e., a T-bill) with
a face value equal to the strike price. In our experience, the easiest way to remember the
PCP condition is to remember that “stock plus put equals T-bill plus call.”
The PCP condition is an algebraic expression, meaning that it can be rearranged. For
example, suppose we know that the risk-free rate is .5 percent per month. A call with a
strike price of $40 sells for $4, and a put with the same strike price sells for $3. Both
have a three-month maturity. What’s the stock price?
To answer, we just use the PCP condition to solve for the stock price:
CHAPTER 24 Option Valuation 809
TABLE 24.2
Gains and Losses
in One Year.
Original investment:
purchase a one-year
call option with a strike
price of $105 for $15.
Invest $100 in risk-free
asset paying 5 percent.
Total cost is $115.Value of Total Gain
Call Option Value of or Loss
Stock Price (Strike Price Risk-Free Combined (Combined Value
in One Year $105) Asset Value Less $115)
$125 $20 $105 $125 $10
120 15 105 120 5
115 10 105 115 0
110 5 105 110 5
105 0 105 105 10
100 0 105 105 10
95 0 105 105 10
90 0 105 105 10put-call parity (PCP)
The relationship
between the prices
of the underlying
stock, a call option,
a put option, and a
riskless asset.