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