Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionVIII. Topics in Corporate
Finance- Option Valuation © The McGraw−Hill^851
Companies, 2002
to acquire the assets. Defaulting amounts to letting an out-of-the-money call option ex-
pire. In this section, we expand on the idea of equity as a call option in several ways.Valuing the Equity in a Leveraged Firm
Consider a firm that has a single zero-coupon bond issue outstanding with a face value
of $10 million. It matures in six years. The firms’ assets have a current marketvalue of
$12 million. The volatility (standard deviation) of the return on the firm’s assets is 40
percent per year. The continuously compounded risk-free rate is 6 percent. What is the
current market value of the firm’s equity? Its debt? What is its continuously com-
pounded cost of debt?
What this case amounts to is that the stockholders have the right, but not the obliga-
tion, to pay $10 million in six years. If they do, they get the assets of the firm. If they
don’t, they default and get nothing. So, the equity in the firm is a call option with a strike
price of $10 million.
Using the Black-Scholes formula in this case can be a little confusing because now
we are solving for the stock price. So, the symbol “C” is the value of the stock and the
symbol “S” is the value of the firm’s assets. With this in mind, we can value the equity
of the firm by plugging the numbers into the Black-Scholes OPM with S $12 million
and E $10 million. When we do so, we get $6.516 million as the value of the equity,
with a delta of .849.826 PART EIGHT Topics in Corporate Finance
From our discussionof implied standard deviations (ISDs), you
know that solving for an ISD when you know the option price is simply
trial and error. Fortunately, most option calculators will do the work for
you. To illustrate, suppose you have a call option with a strike price of $90
that matures in 62 days. The stock currently sells for $87.10, the option sells for
$5.80, and the interest rate is 4.5 percent per year, compounded continuously. What
is the ISD? To find out, we went to the options calculator at http://www.numa.com. After
entering all this information, here is what we got:Notice the calculator changes the days to maturity to 0.17, which is 62/365 of a year.
So, the underlying stock has an ISD of 47.36 percent per year.Work the Web