Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionVIII. Topics in Corporate
Finance(^850) 24. Option Valuation © The McGraw−Hill
Companies, 2002
If we plug all this information into the Black-Scholes formula, we would be left with
one unknown, the standard deviation ( ). However, it’s not possible to directly solve for
, so trial and error must be used. In other words, we just start plugging in values for
until we find one that produces the call price of $4.59.
For a stock option, .50 is a good place to start. If you plug this in, you will see that the
calculated call value is $4.38, which is too low. Recall that option values increase as we
increase , so we might try .60. Now the option value is $4.52, so we’re getting close, but
we’re still low. At .65, the calculated value is $4.61, which is just a little too high. After
a little more work, we discover that the implied volatility is .64, or 64 percent.
VALUATION OF EQUITY AND DEBT IN
A LEVERAGED FIRM
In our earlier chapter on options, we pointed out that the equity in a leveraged corpora-
tion (i.e., a corporation that has borrowed money) can be viewed as a call option on the
assets of the business. The reason is that, when a debt comes due, the stockholders have
the option to pay off the debt, and thereby acquire the assets free and clear, or else de-
fault. The act of paying off the debt amounts to exercising an in-the-money call option
CONCEPT QUESTIONS
24.3a What are an option’s delta, rho, theta, and vega?
24.3bWhat is an ISD?
CHAPTER 24 Option Valuation 825
The options calculator at
http://www.numa.commakes it
easy to calculate ISDs.
ISD
Here is an actual example. At the end of September 2001, common stock in network hardware
manufacturer Cisco was trading for $12.18. A call option expiring in January 2002 with a
strike price of $12.50 traded for $1.75. Treasury bills maturing in late January were paying
2.35 percent. Based on this information, how volatile is the return on Cisco predicted to be?
Just to summarize, the relevant numbers we have are:
S$12.18
E$12.50
R2.35% per year, compounded annually
?
t4 months
C$1.75
From here, it’s plug and chug. As you have probably figured out by now, it’s easier to use
an options calculator to solve this problem. That’s what we did; the implied standard deviation
is about 66 percent. Our nearby Work the Webbox shows you how to do this.
In principle, to solve this problem, we need to convert the interest rate of 2.35 percent to
a continuously compounded rate. If we do, we get 2.323 percent. However, we’ve seen that
option values are not very sensitive to small changes in interest rates, and, in this case, it ac-
tually makes almost no difference. For this reason, in practice, the continuous compounding
issue is often ignored, particularly when rates are low.
EXAMPLE 24.8