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Stock price
$60 $80



5. The covered call strategy would consist of a straight bond with a call written on the bond. The
value of the covered call position at option expiration as a function of the value of the straight
bond is given in the figure following, and is virtually identical to the value of the callable bond
in Figure 14.12.


Value of straight bond


Payoff of covered call


Value of straight bond
X


Call written


6. The call option is worth less as call protection is expanded. Therefore, the coupon rate need not be
as high.
7. Lower. Investors will accept a lower coupon rate in return for the conversion option.
www.mhhe.com/bkm
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




15
OPTION VALUATION



AFTER STUDYING THIS CHAPTER
YOU SHOULD BE ABLE TO:


> Identify the features of an option that affect its market
value.


> Compute an option value in a two-scenario model of the
economy.

> Compute the Black-Scholes value of an option.


> Compute the proper relationship between call and put
prices.

> Compute the hedge ratio of an option.


> Formulate a portfolio insurance plan using option hedge
ratios.




530
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




Related Websites that are used to estimate sensitivity of option values to
changes in parameters.
http://www.options.about.com/money/options
http://www.numa.com/derivs/ref/calculat/option/
This site has extensive links to many other sites. It
calc-opa.htm
contains sections on education, exchanges, research,
http://www.optionsstrategist.com
and quotes, as well as extensive sources related to
futures markets. http://www.schaefferresearch.com
http://www.optionscentral.com http://www.fintools.net/options/optcalc.html
This site offers extensive educational material, including The sites listed above offer options analysis and
access to the freely available Options Toolbox. The calculators.
toolbox is an excellent source that allows you to
http://www.numa.com
simulate different options positions and examine
http://www.phlx.com
option pricing.
These sites have extensive links to options and other
http://www.hoadley.net/options/BS.htm
derivative websites.
This site contains discussion of Black-Scholes and other
pricing models. It also has a discussion of the “Greeks”




n the previous chapter, we examined option markets and strategies. We ended by

I noting that many securities contain embedded options that affect both their
values and their risk-return characteristics. In this chapter, we turn our attention to
option valuation issues. Understanding most option valuation models requires
considerable mathematical and statistical background. Still, many of the ideas and
insights of these models can be demonstrated in simple examples, and we will con-
centrate on these.
We start with a discussion of the factors that ought to affect option prices. After
this qualitative discussion, we present a simple “two-state” quantitative option valua-
tion model and show how we can generalize it into a useful and accurate pricing tool.
Next, we move on to one particular valuation formula, the famous Black-Scholes
model, one of the most significant breakthroughs in finance theory in the past three
decades. Finally, we look at some of the more important applications of option pric-
ing theory in portfolio management and control.
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




532 Part FIVE Derivative Markets


15.1 OPTION VALUATION: INTRODUCTION
Intrinsic and Time Values
Consider a call option that is out of the money at the moment, with the stock price below the
exercise price. This does not mean the option is valueless. Even though immediate exercise
today would be unprofitable, the call retains a positive value because there is always a chance
the stock price will increase sufficiently by the expiration date to allow for profitable exercise.
If not, the worst that can happen is that the option will expire with zero value.
The value S0 X is sometimes called the intrinsic value of an in-the-money call option
intrinsic value
because it gives the payoff that could be obtained by immediate exercise. Intrinsic value is set
Stock price minus
equal to zero for out-of-the-money or at-the-money options. The difference between the actual
exercise price, or
call price and the intrinsic value is commonly called the time value of the option.
the profit that
could be attained Time value is an unfortunate choice of terminology because it may confuse the option™s
by immediate time value with the time value of money. Time value in the options context simply refers to
exercise of an in-the-
the difference between the option™s price and the value the option would have if it were expir-
money call option.
ing immediately. It is the part of the option™s value that may be attributed to the fact that it still
has positive time to expiration.
Most of an option™s time value typically is a type of “volatility value.” As long as the op-
tion holder can choose not to exercise, the payoff cannot be worse than zero. Even if a call op-
tion is out of the money now, it still will sell for a positive price because it offers the potential
for a profit if the stock price increases, while imposing no risk of additional loss should the
stock price fall. The volatility value lies in the right not to exercise the option if that action
would be unprofitable. The option to exercise, as opposed to the obligation to exercise, pro-
vides insurance against poor stock price performance.
As the stock price increases substantially, it becomes more likely that the call option will
be exercised by expiration. In this case, with exercise all but assured, the volatility value be-
comes minimal. As the stock price gets ever larger, the option value approaches the “adjusted”
intrinsic value”the stock price minus the present value of the exercise price, S0 PV(X).
Why should this be? If you know the option will be exercised and the stock purchased for
X dollars, it is as though you own the stock already. The stock certificate might as well be sit-
ting in your safe-deposit box now, as it will be there in only a few months. You just haven™t
paid for it yet. The present value of your obligation is the present value of X, so the present
value of the net payoff of the call option is S0 PV(X).1
Figure 15.1 illustrates the call option valuation function. The value curve shows that when
the stock price is low, the option is nearly worthless because there is almost no chance that it
will be exercised. When the stock price is very high, the option value approaches adjusted in-
trinsic value. In the midrange case, where the option is approximately at the money, the option
curve diverges from the straight lines corresponding to adjusted intrinsic value. This is be-
cause, while exercise today would have a negligible (or negative) payoff, the volatility value
of the option is quite high in this region. The option always increases in value with the stock
price. The slope is greatest, however, when the option is deep in the money. In this case, exer-
cise is all but assured, and the option increases in price one-for-one with the stock price.


1
This discussion presumes the stock pays no dividends until after option expiration. If the stock does pay dividends
before maturity, then there is a reason you would care about getting the stock now rather than at expiration”getting
it now entitles you to the interim dividend payments. In this case, the adjusted intrinsic value of the option must sub-
tract the value of the dividends the stock will pay out before the call is exercised. Adjusted intrinsic value would more
generally be defined as S0 PV(X) PV(D), where D represents dividends to be paid before option expiration.
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




533
15 Option Valuation




F I G U R E 15.1
Option value
Call option value
before expiration

Value of call option




Value of option if
now at expiration =
Time value intrinsic value

S0
X

Out of the In the
money money




Determinants of Option Values
We can identify at least six factors that should affect the value of a call option: the stock price,
the exercise price, the volatility of the stock price, the time to expiration, the interest rate, and
the dividend rate of the stock. The call option should increase in value with the stock price and
decrease in value with the exercise price because the payoff to a call, if exercised, equals
ST X. The magnitude of the expected payoff from the call increases with the difference
S0 X.
Call option value also increases with the volatility of the underlying stock price. To see
why, consider circumstances where possible stock prices at expiration may range from $10 to
$50 compared to a situation where stock prices may range only from $20 to $40. In both cases,
the expected, or average, stock price will be $30. Suppose the exercise price on a call option
is also $30. What are the option payoffs?
High-Volatility Scenario
Stock price $10 $20 $30 $40 $50
Option payoff 0 0 0 10 20

Low-Volatility Scenario
Stock price $20 $25 $30 $35 $40
Option payoff 0 0 0 5 10

If each outcome is equally likely, with probability 0.2, the expected payoff to the option under
high-volatility conditions will be $6, but under the low-volatility conditions, the expected pay-
off to the call option is half as much, only $3.
Despite the fact that the average stock price in each scenario is $30, the average option pay-
off is greater in the high-volatility scenario. The source of this extra value is the limited loss
an option holder can suffer, or the volatility value of the call. No matter how far below $30 the
stock price drops, the option holder will get zero. Obviously, extremely poor stock price per-
formance is no worse for the call option holder than moderately poor performance.
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




534 Part FIVE Derivative Markets


In the case of good stock performance, however, the call option will expire in the money,
and it will be more profitable the higher the stock price. Thus, extremely good stock outcomes
can improve the option payoff without limit, but extremely poor outcomes cannot worsen the
payoff below zero. This asymmetry means volatility in the underlying stock price increases
the expected payoff to the option, thereby enhancing its value.


>
1. Should a put option increase in value with the volatility of the stock?
Concept
Similarly, longer time to expiration increases the value of a call option. For more distant
CHECK expiration dates, there is more time for unpredictable future events to affect prices, and the
range of likely stock prices increases. This has an effect similar to that of increased volatility.
Moreover, as time to expiration lengthens, the present value of the exercise price falls, thereby

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