Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 20 docx

For example, at an Lose if stock price falls Lose if stock price falls Win if stock price rises Your payoff Future stock price $55 a Win if stock price rises Your payoff Future stock pri

U N D E R S T A N D I N G

O P T I O N S

FIGURE 20.1(A) SHOWSyour payoff if you buy AOL Time Warner (AOL) stock at $55 You gain for-dollar if the stock price goes up and you lose dollar-for-dollar if it falls That’s trite; it doesn’t take

dollar-a genius to drdollar-aw dollar-a 45-degree line

Look now at panel (b), which shows the payoffs from an investment strategy that retains the

up-side potential of AOL stock but gives complete downup-side protection In this case your payoff stays

at $55 even if the AOL stock price falls to $50, $40, or zero Panel (b)’s payoffs are clearly better than panel (a)’s If a financial alchemist could turn panel (a) into (b), you’d be willing to pay for the service.

Of course alchemy has its dark side Panel (c) shows an investment strategy for masochists You

lose if the stock price falls, but you give up any chance of profiting from a rise in the stock price If

you like to lose, or if somebody pays you enough to take the strategy on, this is the strategy for you.

Now, as you have probably suspected, all this financial alchemy is for real You really can do all thetransmutations shown in Figure 20.1 You do them with options, and we will show you how

But why should the financial manager of an industrial company be interested in options? There areseveral reasons First, companies regularly use commodity, currency, and interest-rate options to re-duce risk For example, a meatpacking company that wishes to put a ceiling on the cost of beef mighttake out an option to buy live cattle A company that wishes to limit its future borrowing costs mighttake out an option to sell long-term bonds And so on In Chapter 27 we will explain how firms em-ploy options to limit their risk

Second, many capital investments include an embedded option to expand in the future For stance, the company may invest in a patent that allows it to exploit a new technology or it may pur-chase adjoining land that gives it the option in the future to increase capacity In each case the com-pany is paying money today for the opportunity to make a further investment To put it another way,

in-the company is acquiring growth opportunities.

Here is another disguised option to invest: You are considering the purchase of a tract of desertland that is known to contain gold deposits Unfortunately, the cost of extraction is higher than thecurrent price of gold Does this mean the land is almost worthless? Not at all You are not obliged tomine the gold, but ownership of the land gives you the option to do so Of course, if you know thatthe gold price will remain below the extraction cost, then the option is worthless But if there is un-certainty about future gold prices, you could be lucky and make a killing.1

If the option to expand has value, what about the option to bail out? Projects don’t usually go onuntil the equipment disintegrates The decision to terminate a project is usually taken by manage-ment, not by nature Once the project is no longer profitable, the company will cut its losses and ex-ercise its option to abandon the project Some projects have higher abandonment value than others.Those that use standardized equipment may offer a valuable abandonment option Others may ac-tually cost money to discontinue For example, it is very costly to decommission an offshore oil rig

We took a peek at these investment options in Chapter 10, and we showed there how to use cision trees to analyze Magna Charter’s options to expand its airline operation or abandon it In Chap-

de-ter 22 we will take a more thorough look at these real options.

The other important reason why financial managers need to understand options is that they are ten tacked on to an issue of corporate securities and so provide the investor or the company with theflexibility to change the terms of the issue For example, in Chapter 23 we will show how warrants and

of-continued

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1 In Chapter 11 we valued Kingsley Solomon’s gold mine by calculating the value of the gold in the ground and then subtracting

the value of the extraction costs That is correct only if we know that the gold will be mined Otherwise, the value of the mine is

in-creased by the value of the option to leave the gold in the ground if its price is less than the extraction cost.

The Chicago Board Options Exchange (CBOE) was founded in 1973 to allow vestors to buy and sell options on shares of common stock The CBOE was an al-most instant success and other exchanges have since copied its example In addi-tion to options on individual common stocks, investors can now trade options onstock indexes, bonds, commodities, and foreign exchange.

in-Table 20.1 reproduces quotes from the CBOE for June 22, 2001 It shows theprices for two types of options on AOL stock—calls and puts We will explain each

in turn

Call Options and Position Diagrams

A call option gives its owner the right to buy stock at a specified exercise or strike

price on or before a specified exercise date If the option can be exercised only on

one particular day, it is conventionally known as a European call; in other cases

convertibles give their holders an option to buy common stock in exchange for cash or bonds Then inChapter 25 we will see how corporate bonds may give the issuer or the investor the option of earlyrepayment

In fact, we shall see that whenever a company borrows, it creates an option The reason is that the

borrower is not compelled to repay the debt at maturity If the value of the company’s assets is less

than the amount of the debt, the company will choose to default on the payment and the holders will get to keep the company’s assets Thus, when the firm borrows, the lender effectively ac-quires the company and the shareholders obtain the option to buy it back by paying off the debt.This is an extremely important insight It means that anything that we can learn about traded optionsapplies equally to corporate liabilities.2

bond-In this chapter we use traded stock options to explain how options work, but we hope that ourbrief survey has convinced you that the interest of financial managers in options goes far beyondtraded stock options That is why we are asking you to invest here to acquire several important ideasfor use later

If you are unfamiliar with the wonderful world of options, it may seem baffling on first encounter

We will therefore divide this chapter into three bite-sized pieces Our first task is to introduce you tocall and put options and to show you how the payoff on these options depends on the price of theunderlying asset We will then show how financial alchemists can combine options to produce the in-

teresting strategies depicted in Figure 20.1 (b) and (c).

We conclude the chapter by identifying the variables that determine option values Here you willencounter some surprising and counterintuitive effects For example, investors are used to thinkingthat increased risk reduces present value But for options it is the other way around

2 This relationship was first recognized by Fischer Black and Myron Scholes, in “The Pricing of Options

and Corporate Liabilities,” Journal of Political Economy 81 (May–June 1973), pp 637–654.

20.1 CALLS, PUTS, AND SHARES

(such as the AOL options shown in Table 20.1), the option can be exercised on or at

any time before that day, and it is then known as an American call.

The third column of Table 20.1 sets out the prices of AOL Time Warner call

tions with different exercise prices and exercise dates Look at the quotes for

op-tions maturing in October 2001 The first entry says that for $10.50 you could

re-quire an option to buy one share3of AOL stock for $45 on or before October 2001

Moving down to the next row, you can see that an option to buy for $5 more

($50 vs $45) costs $3.75 less, that is $6.75 In general, the value of a call option goes

down as the exercise price goes up

Now look at the quotes for options maturing in January 2002 and 2003 Notice

how the option price increases as option maturity is extended For example, at an

Lose

if stock price falls

Lose

if stock price falls

Win if stock price rises

Your

payoff

Future stock price

$55 (a)

Win if stock price rises

Your payoff

Future stock price

$55 (b)

Protected on downside

No upside

Your payoff

Future stock price

$55 (c)

F I G U R E 2 0 1

Payoffs to three investment strategies for AOL stock (a) You buy one share for $55 (b) No downside If

stock price falls, your payoff stays at $55 (c) A strategy for masochists? You lose if stock price falls, but

you don’t gain if it rises.

3

You can’t actually buy an option on a single share Trades are in multiples of 100 The minimum order

would be for 100 options on 100 AOL shares.

exercise price of $60, the October 2001 call option costs $2.10, the January 2002 tion costs $3.75, and the January 2003 option costs $8.80.

op-In Chapter 13 we met Louis Bachelier, who in 1900 first suggested that securityprices follow a random walk Bachelier also devised a very convenient shorthand

to illustrate the effects of investing in different options.4We will use this shorthand

to compare three possible investments in AOL—a call option, a put option, and thestock itself

The position diagram in Figure 20.2(a) shows the possible consequences of investing

in AOL January 2002 call options with an exercise price of $55 (boldfaced in Table20.1) The outcome from investing in AOL calls depends on what happens to the stockprice If the stock price at the end of this six-month period turns out to be less than the

$55 exercise price, nobody will pay $55 to obtain the share via the call option Your callwill in that case be valueless, and you will throw it away On the other hand, if thestock price turns out to be greater than $55, it will pay to exercise your option to buythe share In this case the call will be worth the market price of the share minus the

$55 that you must pay to acquire it For example, suppose that the price of AOL stockrises to $100 Your call will then be worth That is your payoff, but

of course it is not all profit Table 20.1 shows that you had to pay $5.75 to buy the call

Put Options

Now let us look at the AOL put options in the right-hand column of Table 20.1.

Whereas the call option gives you the right to buy a share for a specified exercise price, the comparable put gives you the right to sell the share For example, the

Prices of call and put options on

AOL Time Warner stock on

June 22, 2001 The closing

stock price was $53.10.

*Long-term options are called

“LEAPS.”

Source: Chicago Board Options

Exchange Average of bid and asked

boldfaced entry in the right-hand column of Table 20.1 shows that for $6.55 you

could acquire an option to sell AOL stock for a price of $55 anytime before January

2002 The circumstances in which the put turns out to be profitable are just the

op-posite of those in which the call is profitable You can see this from the position

di-agram in Figure 20.2(b) If AOL’s share price immediately before expiration turns

out to be greater than $55, you won’t want to sell stock at that price You would do

better to sell the share in the market, and your put option will be worthless

Con-versely, if the share price turns out to be less than $55, it will pay to buy stock at the

low price and then take advantage of the option to sell it for $55 In this case, the

value of the put option on the exercise date is the difference between the $55

pro-ceeds of the sale and the market price of the share For example, if the share is

worth $35, the put is worth $20:

 $55
$35  $20 Value of put option at expiration exercise price
market price of the share

$55 (a)

$55 (b)

$55 (c)

Value

of share

Share price

Share price

Share price

F I G U R E 2 0 2

Position diagrams show how payoffs to owners of AOL calls, puts, and shares (shown by the colored lines)

depend on the share price (a) Result of buying AOL call exercisable at $55 (b) Result of buying AOL put

exercisable at $55 (c) Result of buying AOL share.

Table 20.1 confirms that the value of a put increases when the exercise price is raised However, extending the maturity date makes both puts and calls more valuable.

We have now reviewed position diagrams for investment in calls and puts A

third possible investment is directly in AOL stock Figure 20.2(c) betrays few

se-crets when it shows that the value of this investment is always exactly equal to themarket value of the share

Selling Calls, Puts, and Shares

Let us now look at the position of an investor who sells these investments If you

sell, or “write,” a call, you promise to deliver shares if asked to do so by the callbuyer In other words, the buyer’s asset is the seller’s liability If by the exercisedate the share price is below the exercise price, the buyer will not exercise the calland the seller’s liability will be zero If it rises above the exercise price, the buyerwill exercise and the seller will give up the shares The seller loses the differencebetween the share price and the exercise price received from the buyer Notice that

it is the buyer who always has the option to exercise; the seller simply does as he

or she is told

Suppose that the price of AOL stock turns out to be $80, which is above the option’sexercise price of $55 In this case the buyer will exercise the call The seller is forced tosell stock worth $80 for only $55 and so has a payoff of 5Of course, that $25 loss

is the buyer’s gain Figure 20.3(a) shows how the payoffs to the seller of the AOL call

option vary with the stock price Notice that for every dollar the buyer makes, the

seller loses a dollar Figure 20.3(a) is just Figure 20.2(a) drawn upside down.

In just the same way we can depict the position of an investor who sells, or

writes, a put by standing Figure 20.2(b) on its head The seller of the put has agreed

to pay the exercise price of $55 for the share if the buyer of the put should request

it Clearly the seller will be safe as long as the share price remains above $55 butwill lose money if the share price falls below this figure The worst thing that canhappen is that the stock becomes worthless The seller would then be obliged topay $55 for a stock worth $0 The “value” of the option position would be

Finally, Figure 20.3(c) shows the position of someone who sells AOL stock short.

Short sellers sell stock which they do not yet own As they say on Wall Street:

He who sells what isn’t his’n Buys it back or goes to prison.

Eventually, therefore, the short seller will have to buy the stock back The shortseller will make a profit if it has fallen in price and a loss if it has risen.6You can see

that Figure 20.3(c) is simply an upside-down Figure 20.2(c).

Position Diagrams Are Not Profit Diagrams

Position diagrams show only the payoffs at option exercise; they do not account for

the initial cost of buying the option or the initial proceeds from selling it

This is a common point of confusion For example, the position diagram in

Fig-ure 20.2(a) makes purchase of a call look like a sFig-ure thing—the payoff is at worst

zero, with plenty of “upside” if AOL’s stock price goes above $55 by January 2002.

But compare the profit diagram in Figure 20.4(a), which subtracts the $5.75 cost of the

call in June 2001 from the payoff at maturity The call buyer loses money at all share

prices less than Take another example: The position diagram

in Figure 20.3(b) makes selling a put look like a sure loss—the best payoff is zero But

the profit diagram in Figure 20.4(b), which recognizes the $6.55 received by the

seller, shows that the seller gains at all prices above 7

Profit diagrams like those in Figure 20.4 may be helpful to the options beginner,

but options experts rarely draw them Now that you’ve graduated from the first

options class we won’t draw them either We will stick to position diagrams,

be-cause you have to zero in on payoffs at exercise to understand options and to value

0

0 Share price

Share price

Share price

Value of call

seller's position

Value of put seller's position

Value of stock seller's position

F I G U R E 2 0 3

Payoffs to sellers of AOL calls, puts, and shares (shown by the colored lines) depend on the share price.

(a) Result of selling AOL call exercisable at $55 (b) Result of selling AOL put exercisable at $55 (c) Result of

selling AOL share short.

7 Strictly speaking, the profit diagrams in Figure 20.4 should account for the time value of money, that

is, the interest earned on the seller’s initial proceeds and lost on the call buyer’s outlay.

Now that you understand the possible payoffs from calls and puts, we can startpracticing some financial alchemy by conjuring up the strategies shown in Figure20.1 Let’s start with the strategy for masochists.

Look at row 1 of Figure 20.5 The first diagram shows the payoffs from buying

a share of AOL stock, while the second shows the payoffs from selling a call option

with a $55 exercise price The third diagram shows what happens if you combinethese two positions The result is the no-win strategy that we depicted in panel

(c) of Figure 20.1 You lose if the stock price declines below $55, but, if the stock

price rises above $55, the owner of the call option will demand that you hand overyour stock for the $55 exercise price So you lose on the downside and give up anychance of a profit That’s the bad news The good news is that you get paid for tak-ing on this liability In June 2001 you would have been paid $5.75, the price of a six-month call option

Now, we’ll create the downside protection shown in Figure 20.1(b) Look at row

2 of Figure 20.5 The first diagram again shows the payoff from buying a share ofAOL stock, while the next diagram in row 2 shows the payoffs from buying anAOL put option with an exercise price of $55 The third diagram shows the effect

of combining these two positions You can see that, if AOL’s stock price rises above

$55, your put option is valueless, so you simply receive the gains from your vestment in the share However, if the stock price falls below $55, you can exerciseyour put option and sell your stock for $55 Thus, by adding a put option to yourinvestment in the stock, you have protected yourself against loss.8This is the strat-

in-egy that we depicted in panel (b) of Figure 20.1 Of course, there is no gain without pain The cost of insuring yourself against loss is the amount that you pay for a put

$55 (a) Profit to call buyer (b) Profit to put seller

–$5.75

0

$6.55 Share

Breakeven

is $48.45

F I G U R E 2 0 4

Profit diagrams incorporate the costs of buying an option or the proceeds from selling one In panel

(a), we substract the $5.75 cost of the AOL call from the payoffs plotted in Figure 20.2(a) In panel

(b), we add the $6.55 proceeds from selling the AOL put to the payoffs in Figure 20.3(b).

20.2 FINANCIAL ALCHEMY WITH OPTIONS

8This combination of a stock and a put option is known as a protective put.

option on AOL stock with an exercise price of $55 In June 2001 the price of this put

was $6.55 This was the going rate for financial alchemists

We have just seen how put options can be used to provide downside protection

We will now show you how call options can be used to get the same result This is

illustrated in row 3 of Figure 20.5 The first diagram shows the payoff from placing

the present value of $55 in a bank deposit Regardless of what happens to the price

of AOL stock, your bank deposit will pay off $55 The second diagram in row 3

shows the payoff from a call option on AOL stock with an exercise price of $55, and

Sell call

Your payoff

Future stock price $55

No upside

Your payoff

Future stock price

Buy put

Your payoff

Future stock price $55

Downside protection

Downside protection

Your payoff

Future stock price

Buy call

Your payoff

Future stock price $55

Your payoff

Future stock price

F I G U R E 2 0 5

The first row shows how options can be used to create a strategy where you lose if the stock price falls but do not gain

if it rises [strategy (c) in Figure 20.1] The second and third rows show two ways to create the reverse strategy where you gain on the upside but are protected on the downside [strategy (b) in Figure 20.1].

the third diagram shows the effect of combining these two positions Notice that,

if the price of AOL stock falls, your call is worthless, but you still have your $55 inthe bank For every dollar that AOL stock price rises above $55, your investment inthe call option pays off an extra dollar For example, if the stock price rises to $100,you will have $55 in the bank and a call worth $45 Thus you participate fully

in any rise in the price of the stock, while being fully protected against any fall So

we have just found another way to provide the downside protection depicted in

panel (b) of Figure 20.1.

These last two rows of Figure 20.5 tell us something about the relationship tween a call option and a put option Regardless of the future stock price, both in-vestment strategies provide identical payoffs In other words, if you buy the shareand a put option to sell it after six months for $55, you receive the same payoff asfrom buying a call option and setting enough money aside to pay the $55 exerciseprice Therefore, if you are committed to holding the two packages until the op-tions expire, the two packages should sell for the same price today This gives us afundamental relationship for European options:

be-To repeat, this relationship holds because the payoff of

is identical to the payoff from

This basic relationship among share price, call and put values, and the present

value of the exercise price is called put–call parity.10

The relationship can be expressed in several ways Each expression implies twoinvestment strategies that give identical results For example, suppose that youwant to solve for the value of a put You simply need to twist the put–call parityformula around to give

From this expression you can deduce that

3Buy put,buy share43Buy call,invest present value of exercise price in safe asset94Value of call present value of exercise price  value of put  share price

9

The present value is calculated at the risk-free rate of interest It is the amount that you would have

to invest today in a bank deposit or Treasury bills to realize the exercise price on the option’s tion date.

expira-10

Put–call parity holds only if you are committed to holding the options until the final exercise date It

therefore does not hold for American options, which you can exercise before the final date We discuss

possible reasons for early exercise in Chapter 21 Also if the stock makes a dividend payment before the final exercise date, you need to recognize that the investor who buys the call misses out on this divi- dend In this case the relationship is

Value of call present value of exercise price  value of put share price
present value of dividend.

Default Puts and the Difference between Safe and Risky Bonds

In Chapter 18 we discussed the plight of Circular File Company, which borrowed

$50 per share Unfortunately the firm fell on hard times and the market value of its

assets fell to $30 Circular’s bond and stock prices fell to $25 and $5, respectively

Circular’s market value balance sheet is now

Circular File Company (Market Values)

If Circular’s debt were due and payable now, the firm could not repay the $50 it

originally borrowed It would default, bondholders receiving assets worth $30 and

shareholders receiving nothing The reason Circular stock is worth $5 is that the

debt is not due now but rather is due a year from now A stroke of good fortune

could increase firm value enough to pay off the bondholders in full, with

some-thing left over for the stockholders

Let us go back to a statement that we made at the start of the chapter Whenever

a firm borrows, the lender effectively acquires the company and the shareholders

ob-tain the option to buy it back by paying off the debt The stockholders have in effect

purchased a call option on the assets of the firm The bondholders have sold them

this call option Thus the balance sheet of Circular File can be expressed as follows:

Circular File Company (Market Values)Asset value $30 $25

5

$30 $30 Firm valueStock value asset value value of call

Bond value asset value
value of call

If this still sounds like a strange idea to you, try drawing one of Bachelier’s

po-sition diagrams for Circular File It should look like Figure 20.6 If the future value

of the assets is less than $50, Circular will default and the stock will be worthless

If the value of the assets exceeds $50, the stockholders will receive asset value less

the $50 paid over to the bondholders The payoffs in Figure 20.6 are identical to a

call option on the firm’s assets, with an exercise price of $50

Now look again at the basic relationship between calls and puts:

To apply this to Circular File, we have to interpret “value of share” as “asset value,”

because the common stock is a call option on the firm’s assets Also, “present value

of exercise price” is the present value of receiving the promised payment of $50 to

bondholders for sure next year Thus

Now we can solve for the value of Circular’s bonds This is equal to the firm’s

asset value less the value of the shareholders’ call option on these assets:

 present value of promised payment to bondholders
value of put

Bond value asset value
value of call

 value of put  asset value

Value of call present value of promised payment to bondholders

Value of call present value of exercise price  value of put  value of share

Circular’s bondholders have in effect (1) bought a safe bond and (2) given theshareholders the option to sell them the firm’s assets for the amount of the debt.You can think of the bondholders as receiving the $50 promised payment, but theyhave given the shareholders the option to take the $50 back in exchange for the as-sets of the company If firm value turns out to be less than the $50 that is promised

to bondholders, the shareholders will exercise their put option

Circular’s risky bond is equal to a safe bond less the value of the shareholders’option to default To value this risky bond we need to value a safe bond and thensubtract the value of the default option The default option is equal to a put option

on the firm’s assets Now you can see why bond traders, investors, and financial

managers refer to default puts.

In the case of Circular File the option to default is extremely valuable becausedefault is likely to occur At the other extreme, the value of IBM’s option to default

is trivial compared to the value of IBM’s assets Default on IBM bonds is possiblebut extremely unlikely Option traders would say that for Circular File the put op-tion is “deep in the money” because today’s asset value ($30) is well below the ex-ercise price ($50) For IBM the put option is far “out of the money” because thevalue of IBM’s assets substantially exceeds the value of IBM’s debt

We know that Circular’s stock is equivalent to a call option on the firm’s assets

It is also equal to (1) owning the firm’s assets, (2) borrowing the present value of

$50 with the obligation to repay regardless of what happens, but also (3) buying aput on the firm’s assets with an exercise price of $50

We can sum up by presenting Circular’s balance sheet in terms of asset value,put value, and the present value of a sure $50 payment:

The value of Circular’s common stock is the

same as the value of a call option on the firm’s

assets with an exercise price of $50.

Circular File Company (Market Values)Asset value $30 $25 Bond value  present value of promised

payment
value of default put

5 Stock value  asset value
present

value of promised payment  value of put

$30 $30 Firm value  asset value

Again you can check this with a position diagram The colored line in Figure 20.7

shows the payoffs to Circular’s bondholders If the firm’s assets are worth more than

$50, the bondholders are paid off in full; if the assets are worth less than $50, the firm

defaults and the bondholders receive the value of the assets You could get an

iden-tical payoff pattern by buying a safe bond (the upper black line) and selling a put

op-tion on the firm’s assets (the lower black line)

Spotting the Option

Options rarely come with a large label attached Often the trickiest part of the

prob-lem is to identify the option For example, we suspect that until it was pointed out,

you did not realize that every risky bond contains a hidden option When you are

not sure whether you are dealing with a put or a call or a complicated blend of the

two, it is a good precaution to draw a position diagram Here is an example

The Flatiron and Mangle Corporation has offered its president, Ms Higden,

the following incentive scheme: At the end of the year Ms Higden will be paid

a bonus of $50,000 for every dollar that the price of Flatiron stock exceeds its

cur-rent figure of $120 However, the maximum bonus that she can receive is set at

$2 million

You can think of Ms Higden as owning 50,000 tickets, each of which pays

noth-ing if the stock price fails to beat $120 The value of each ticket then rises by $1 for

each dollar rise in the stock price up to the maximum of

Figure 20.8 shows the payoffs from just one of these tickets The payoffs are not the

same as those of the simple put and call options that we drew in Figure 20.2, but it

is possible to find a combination of options that exactly replicates Figure 20.8

Be-fore going on to read the answer, see if you can spot it yourself (If you are

some-one who enjoys puzzles of the make-a-triangle-from-just-two-match-sticks type,

this one should be a walkover.)

The answer is in Figure 20.9 The solid black line represents the purchase of a

call option with an exercise price of $120, and the dotted line shows the sale of

an-other call option with an exercise price of $160 The colored line shows the payoffs

from a combination of the purchase and the sale—exactly the same as the payoffs

from one of Ms Higden’s tickets

F I G U R E 2 0 7

You can also think of Circular’s bond (the colored line) as equivalent to a risk-free bond (the upper

black line) less a put option on the firm’s assets

with an exercise price of $50 (the lower black line).