Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionVIII. Topics in Corporate
Finance- Option Valuation © The McGraw−Hill^853
Companies, 2002
bonds wish to eliminate the risk of default. In other words, the holders want to turn their
risky bonds into risk-free bonds. How can they do this?
The answer is that the bondholders can do a protective put along the lines we de-
scribed earlier in the chapter. In this case, the bondholders want to make sure that their
bonds will never be worth less than their face value of $10 million, so the bondholders
need to purchase a put option with a six-year life and a $10 million face value. The put
option is an option to sell the assets of the firm for $10 million.
Remember that if the assets of the firm are worth more than $10 million in six years,
the shareholders will pay the $10 million. If the assets are worth less than $10 million,
the stockholders will default, and the bondholders will receive the assets of the firm. At
that point, however, the bondholders will exercise their put and sell the assets for $10
million. Either way, the bondholders get their $10 million.
So, what we have discovered is that a risk-free bond is the same thing as a combina-
tion of a risky bond and a put option on the assets of the firm with a matching maturity
and a strike price equal to the face value of the bond:
Value of risky bond put option Value of risk-free bond [24.7]
In our example, the face value of the debt is $10 million, and the risk-free rate is 6 per-
cent, so the value of the bonds if they were risk free is:
Value of risk-free bonds $10 million e.06(6)
$6.977 million
If we compare this to the value of the risky bonds, $5.484 million, we see that the put
option is worth $6.977 5.484 $1.493 million. Notice that the value of the risk-free
bonds is also the present value of the strike price at the risk-free rate.
We can check that this put value is correct. We know the value of the underlying as-
sets is $12 million, value of the call option (the stock) is $6.516 million, and the present
value of the strike price is $6.977 million. Using the PCP condition:
P$6.977 6.516 $12
$1.493 million
which is exactly what we calculated.
We can restate our result here as follows:
Value of risky bond Value of risk-free bond put option
[24.8]
EeRtP
This shows us that anything that increases the value of the put option decreasesthe
value of the firm’s bonds. With this in mind, we can use the PCP condition to bring to-
gether and unite a lot of our discussion in this chapter (and this book!).
Using the PCP condition, we can write:
SCEeRtP
Remember that, in this case, the stock is the underlying asset. Now, if we are thinking
of the stock in a firm as being a call option on the assets of the firm, here is how we
would interpret this:
Value of assets (S) Value of stock (C) (EeRtP) [24.9]
where E, the strike price, is the face value of the firm’s debt. Notice that, as we have just
seen, the term in parentheses is the value of the firm’s risky bonds, so this expression is
really just the balance sheet identity:828 PART EIGHT Topics in Corporate Finance