Principles of Corporate Finance_ 12th Edition

Chapter 21 Valuing Options 567

bre44380_ch21_547-572.indd 567 10/05/15 12:53 PM

In this chapter we introduced the basic principles of option valuation by considering a call option
on a stock that could take on one of two possible values at the option’s maturity. We showed that
it is possible to construct a package of the stock and a loan that would provide exactly the same
payoff as the option regardless of whether the stock price rises or falls. Therefore, the value of the
option must be the same as the value of this replicating portfolio.
We arrived at the same answer by pretending that investors are risk-neutral, so that the expected
return on every asset is equal to the interest rate. We calculated the expected future value of the
option in this imaginary risk-neutral world and then discounted this figure at the interest rate to
find the option’s present value.
The general binomial method adds realism by dividing the option’s life into a number of subpe-
riods in each of which the stock price can make one of two possible moves. Chopping the period
into these shorter intervals doesn’t alter the basic method for valuing a call option. We can still
replicate the call by a package of the stock and a loan, but the package changes at each stage.
Finally, we introduced the Black–Scholes formula. This calculates the option’s value when the
stock price is constantly changing and takes on a continuum of possible future values.
An option can be replicated by a package of the underlying asset and a risk-free loan. Therefore,
we can measure the risk of any option by calculating the risk of this portfolio. Naked options are
often substantially more risky than the asset itself.
When valuing options in practical situations there are a number of features to look out for. For
example, you may need to recognize that the option value is reduced by the fact that the holder is
not entitled to any dividends.

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SUMMARY

Three readable articles about the Black–Scholes model are:

F. Black, “How We Came up with the Option Formula,” Journal of Portfolio Management 15 (1989),
pp. 4–8.

F. Black, “The Holes in Black–Scholes,” RISK Magazine 1 (1988), pp. 27–29.

F. Black, “How to Use the Holes in Black–Scholes,” Journal of Applied Corporate Finance 1 (Winter
1989), pp. 67–73.

There are a number of good books on option valuation. They include:

J. Hull, Options, Futures and Other Derivatives, 9th ed. (Englewood Cliffs, NJ: Prentice-Hall, Inc., 2014).

R. L. McDonald, Derivatives Markets, 3rd ed. (Reading, MA: Pearson Addison Wesley, 2012).

P. Wilmott, Paul Wilmott on Quantitative Finance, 2nd ed. (New York: John Wiley & Sons, 2006).

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FURTHER

READING

Select problems are available in McGraw-Hill’s Connect.

Please see the preface for more information.

BASIC

1. Binomial model The stock price of Heavy Metal (HM) changes only once a month: either
   it goes up by 20% or it falls by 16.7%. Its price now is $40. The interest rate is 1% per month.

a. What is the value of a one-month call option with an exercise price of $40?

b. What is the option delta?

c. Show how the payoffs of this call option can be replicated by buying HM’s stock and
borrowing.

d. What is the value of a two-month call option with an exercise price of $40?

e. What is the option delta of the two-month call over the first one-month period?

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PROBLEM

SETS