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CHAPTER 23

Options and Corporate Finance:Extensions andApplications

The historically large stock market losses in 2008 dramatically reduced the valueof millions of employee stock options, leaving many of them underwater, meaning thatstock prices were far !elow the corresponding exercise prices" #n fact, some analystsestimated that $2 percent of %ortune &00 employee stock options were underwater, and'( percent of the stock options held !y )E*s of these same companies were in a similarcondition" #n response, a record num!er of companies swapped underwater options forcash, stock, or restricted stock" #n addition, a large num!er of companies repriced,or lowered the exercise prices on their employee stock options" This chapter exploresemployee stock options and a variety of other applications of option principles in corporate+nance"

23.1 Executive Stock Options

hy *ptions-

Executive compensation is usually made up of a !ase salary plus some or all ofthe following elements.

/" ong1term compensation"

2" Annual !onuses"

(" etirement contri!utions"

3" *ptions"

The +nal component of compensation, options, is !y far the !iggest part of totalcompensation for many top executives" Ta!le 2("/ lists the /0 )E*s who received theargest stock option grants during200$42008" The rank is in terms of the face value of the options granted" This is thenum!er of optionstimes the current stock price"

Ta!e 23.1 2""#$2""% Top 1" Option &rants'

5nowing the face value of an option does not automatically allow us to determine the


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market value of the option" e also need to know the exercise price !efore valuing theoption according to either the 6lack47choles model or the !inomial model" owever, theexercise price is generally set e9ual to the market price of the stock on the date theexecutive receives the options" #n the next section, we value options under the assumptionthat the exercise price is e9ual to the market price"

*ptions in the stock of the company are increasingly !eing granted to executives as analternative to increases in !ase pay" 7ome of the reasons given for using options are these.

/" *ptions make executives share the same interests as the stockholders" 6yaligning interests, executives are more likely to make decisions for the !ene+t of the

stockholders"

2" *ptions allow the company to lower the executive:s !ase pay" This removespressures on morale caused !y disparities !etween the salaries of executives and thoseof other employees"

(" *ptions put an executive:s pay at risk, rather than guaranteeing it,regardless of the performance of the +rm"

3" *ptions are a tax1e;cient way to pay employees" , Even though we were a private company, we would grantstock options to every employee company1wide, from the top managers to the !aristas, inproportion to their level of !ase pay" They could then, through their e=orts, help make7tar!ucks more successful every year, and if 7tar!ucks someday went pu!lic, their

options could eventually !e worth a good sum of money"

?aluing Executive )ompensation

#n this section, we value executive stock options" @ot surprisingly, the complexityof the total compensation package often makes valuation a di;cult task" The economicvalue of the options depends on factors such as the volatility of the underlying stock and theexact terms of the option grant"

e attempt to estimate the economic value of the options held !y the executives

listed in Ta!le2("/" To do so, we employ the 6lack47choles option pricing formula from )hapter 2 2" *fcourse, we are missing many features of the particular plans, and the !est we can hope fors a rough estimate" 7imple matters such as re9uiring the executive to hold the option fora +xed period, the free>e1out period, !efore exercising, can signi+cantly diminish the valueof a standard option" E9ually important, the 6lack4 7choles formula has to !e modi+ed if thestock pays dividends and is no longer applica!le if the volatility of the stock is changingrandomly over time" #ntuitively, a call option on a dividend1paying stock is worth less than acall on a stock that pays no dividends. All other things !eing e9ual, the dividends will lowerthe stock price" @evertheless, let us see what we can do"

E(A)P*E 23.2 Options at Capita! One Corporation

According to the proxy statement +led in +scal year 200$, ich %air!ank, the )E* of)apital *ne, was granted /"2 million stock options" The average exercise price of theoptions was B&0"'', and we will assume that all of the options were granted at the money"e:ll also assume that the options expire in +ve years and that the risk1free rate is &percent" This information implies that.


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/" The stock price C7D e9uals the exercise price CED, B&0"''"

2" The risk1free rate, , e9uals "0&"

(" The time interval, t, e9uals &"

#n addition, the stock volatility, , reported in Ta!le 2("2 is given as ($"3 percent peryear, which

mplies that the variance, 2, e9uals C"($3D2 F

/30("

Ta!e 23.2 +a!ue o, 2""#$2""% Top 1" Option&rants'

e now have enough information to estimate the value of ich %air!ank:s options usingthe 6lack4 7choles model.

7ince Gr" %air!ank was granted options on /"2 million shares, and since eachoption is worth

B20"'$, the market value of his options !y the a!ove calculations is /"2 million HB20"'$ F B(3"' million" Ta!le 2("2 shows that the actual value reported !y thecompany, however, was only B/$"0 million" There are a lot of factors that could explain this

di=erence" Any of the parameters we used could !e incorrect and would have a signi+cantmpact on our calculations" Ierhaps most importantly, )apital *ne paid dividends at a rateof a!out $ percent per year" The dividend is a payment that the stockholder receives !utthat the option holder does not" 7ince the shareholders get !oth the dividends and thecapital gains !ut the option holder only !ene+ts from the capital gains, the option isworth less if thestock pays dividends" 7ince the +nal stock price +ve years later on )apital *ne stock is lessthan it would !e if dividends were reinvested, the option value is less as well" edoing the


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computations a!ove to take account of the impact of the dividends gets us a value for theoptions granted to %air!ank of Just a!out the B/$"0 million that the company did, in fact,report"

Ta!le 2("2 shows the grant value of the options as well as the actual option values asreported !y each of the companies" Gost of these companies use the 6lack47choles methodto value the options, !ut they take into account the special features of their plans and theirstock, including whether or not it pays dividends" As can !e seen, these reported values,while large !y ordinary standards, are signi+cantly less than the corresponding grant values"@otice, too, that the ordering !y grant value is not the same as that !y the reported

option value" %or example, whereas Ta!le 2("/ shows that the )E* of yeth received agrant value of B20"$ million, which placed him tenth on the list of the top ten, thereported value of those options in Ta!le 2("2, B/3"' million, was fourth on the list"

The values we have computed in Ta!le 2("2 are the economic values of the options ifthey were to trade in the market" The real 9uestion is this. hose value are we talkinga!out- Are these the costs of the options to the company- Are they the values of the optionsto the executives-

7uppose a company computes the fair market value of the options as we have done inTa!le 2("2" %or illustration, assume that the options are in the money and that they areworth B2& each" 7uppose, too, that the )E* holds / million such options for a total value ofB2& million" This is the amount that the options would trade at in the +nancial markets and

that traders and investors would !e willing to pay for them"/ #f the company were veryarge, it would not !e unreasona!le for it to view this as the cost of granting the options tothe )E*" *f course, in return, the company would expect the )E* to improve the value ofthe company to its shareholders !y more than this amount" As we have seen, perhaps themain purpose of options is to align the interests of management with those of theshareholders of the +rm" e1out period rather than letting theexecutive sell them to reali>e their value"

The implication is that when options are a large portion of an executive:s net worth,the total value of the position to the executive is less than market value" As a purely+nancial matter, an executive might !e happier with B& million in cash rather than B20million in options" At least the executive could then diversify his personal portfolio"

23.2 +a!uin- a Start.

/palph 7immons was not your typical G6A student" 7ince childhood, he:d had one

am!ition. To open a restaurant that sold alligator meat" e went to !usiness school!ecause he reali>ed that although he knew /0/ ways to cook alligator, he didn:t have the!usiness skills necessary to run a restaurant" e was extremely focused, with each courseat graduate school !eing important to him only to the extent that it could further his dream"


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hile taking his school:s course in entrepreneurship, he !egan to develop a !usinessplan for his restaurant, which he now called Alligator Alley" e thought a!out marketingM hethought a!out raising capitalM he thought a!out dealing with future employees" e evendevoted a great deal of time to designing the physical layout of the restaurant" Againstthe professor:s advice in his entrepreneurship class, he designed the restaurant in theshape of an alligator, where the front door went through the animal:s mouth" *f course, his!usiness plan would not !e complete without +nancial proJections" After much thought, hecame up with the proJections shown in Ta!le 2("("

Ta!e 23.3 Financia! Pro0ections ,or A!!i-ator A!!e

The ta!le starts with sales proJections, which rise from B(00,000 in the +rst year to asteady state of

B/ million a year" )ash Nows from operations are shown in the next line, although weeave out the intermediate calculations needed to move from line C/D to line C2D" Aftersu!tracting working capital, the ta!le shows net cash Nows in line C3D" @et cash Nows arenegative initially, as is 9uite common in start1 ups, !ut they !ecome positive !y year ("owever, the rest of the ta!le presents the unfortunate truth" The cash Nows from therestaurant yield a present value of B&82,&/, assuming a discount rate of 20percent"


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that he will need three years of operating the +rst restaurant to C/D get the initialrestaurant running smoothly and C2D have enough information to place an accurate valueon the restaurant" #f the +rst restaurant is successful enough, he will need another yearto o!tain outside capital" Thus, he will !e ready to !uild the (0 additional units around thefourth year"

alph will value his enterprise, including the option to expand, according to the6lack47choles model" %rom Ta!le 2("(, we see that each unit costs B$00,000, implyinga total cost over the (0 additional units of B2/,000,000 CF(0 H B$00,000D" The presentvalue of the cash inNows from these (0 units is B/$,3$,8(0 CF(0 H B&82,&/D, accordingto the ta!le" owever, !ecause the expansion will occur around the fourth year, thispresent value calculation is provided from the point of view of four years in the future"

The present value as of today is B8,328,2&& OFB/$,3$,8(0KC/"20D3P, assuming a discountrate of 20 percent per year" Thus, alph views his potential restaurant !usiness as anoption, where the exercise price is B2/,000,000 and the value of the underlying asset isB8,328,2&&" The option is currently out of the money, a result that follows from thenegative value of a typical restaurant, as calculated in Ta!le 2("(" *f course, alph ishoping that the option will move into the money within four years"

alph needs three additional parameters to use the 6lack47choles model. , thecontinuously compounded interest rateM t, the time to maturityM and , the standarddeviation of the underlying asset" alph uses the yield on a four1year >ero coupon !ond,which is ("& percent, as the estimate of the interest rate" The time to maturity is fouryears" The estimate of standard deviation is a little trickier !ecause there is no historicaldata on alligator restaurants" alph +nds that the average annual standard deviation of thereturns on pu!licly traded restaurants is "(&" 6ecause Alligator Alley is a new venture, hereasons that the risk here would !e somewhat greater" e +nds that the averageannual standard deviation for restaurants that have gone pu!lic in the last few years is "3&" alph:s restaurant is newer still, so he uses a standard deviation of "&0"

There is now enough data to value alph:s venture" The value according to the 6lack47choles model is B/,3&&,/'" The actual calculations are shown in Ta!le 2("3" *f course,alph must start his pilot restaurant !efore he can take advantage of this option" Thus,the net value of the call option plus the negative present value of the pilot restaurant isB/,(($,$&$ CFB/,3&&,/' 4 B//$,3('D" 6ecause this value is large and positive, alphdecides to stay with his dream of Alligator Alley" e knows that the pro!a!ility that therestaurant will fail is greater than &0 percent" @evertheless, the option to expand is

mportant enough that his restaurant !usiness has value" And if he needs outside capital,he pro!a!ly can attract the necessary investors"

Ta!e 23. +a!uin- a Start/p Fir 4A!!i-ator A!!e5 as an Option


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s an alternative andLin some situationsLa superior approach to valuation" The rest ofthis chapter examines two applications of the !inomial model"

eating *il

T8o9ate Exap!e

)onsider Anthony Geyer, a typical heating oil distri!utor, whose !usiness consists of!uying heating oil at the wholesale level and reselling the oil to homeowners at asomewhat higher price" Gost of his revenue comes from sales during the winter" Today,

7eptem!er /, heating oil sells for B2"00 per gallon" *f course, this price is not +xed" ather,oil prices will vary from 7eptem!er / until Recem!er /, the time when his customers willpro!a!ly make their !ig winter purchases of heating oil" et:s simplify the situation !yassuming that Gr" Geyer !elieves that oil prices will either !e at B2"$3 or B/"3 onRecem!er /" %igure 2("/ portrays this possi!le price movement" This potential price rangerepresents a great deal of uncertainty !ecause Gr" Geyer has no idea which of the twopossi!le prices will actually occur" owever, this price varia!ility does not translate intothat much risk !ecause he can pass price changes on to his customers" That is, he willcharge his customers more if he ends up paying B2"$3 per gallon than if he ends up payingB/"3 per gallon"

Fi-ure 23.1 )oveent o, Heatin- Oi! Prices ,ro Septeer 1 to 9eceer 1in a T8o 9ate Exap!e

*f course, Gr" Geyer is avoiding risk !y passing on that risk to his customers" iscustomers accept the risk, perhaps !ecause they are each too small to negotiate a !etterdeal" This is not the case with )E)*, a large electric utility in his area" )E)* approachesGr" Geyer with the following proposition" The utility would like to !e a!le to !uy up to million gallons of oil from him at B2"/0 per gallon on Recem!er /"

Although this arrangement represents a lot of oil, !oth Gr" Geyer and )E)* know thatGr" Geyer can expect to lose money on it" #f prices rise to B2"$3 per gallon, the utility willhappily !uy all million gallons at only B2"/0 per gallon, clearly creating a loss for thedistri!utor" owever, if oil prices decline toB/"3 per gallon, the utility will not !uy any oil" After all, why should )E)* pay B2"/0 pergallon to Gr" Geyer when the utility can !uy all the oil it wants at B/"3 per gallon inthe open market- #n otherwords, )E)* is asking for a call option on heating oil" To compensate Gr" Geyer for the riskof loss, the two parties agree that )E)* will pay him B/,000,000 up front for the right to!uy up to million gallons of oil at B2"/0 per gallon"

#s this a fair deal- Although small distri!utors may evaluate a deal like this !y gutfeel, we can

evaluate it more 9uantitatively !y using the !inomial model descri!ed in the previouschapter" #n that chapter, we pointed out that option pro!lems can !e handled mosteasily !y assuming risk1neutral pricing" #n this approach, we +rst note that oil will eitherrise ($ percent CFB2"$3KB2"00 4 /D or fall 42$ percent CFB/"3KB2"00 4 /D from 7eptem!er /to Recem!er /" e can think of these two num!ers as the possi!le returns on heating oil"n addition, we introduce two new terms, u and d" e de+ne u as /


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S "($ F /"($ and d as /4 "2$ F "$("2 ero is placed in parentheses under the price ofB/"3 in %igure 2("/" #n addition, as mentioned earlier, Gr" Geyer receivesB/,000,000 up front"

Uiven these num!ers, the value of the contract to Gr" Geyer can !e calculated as.

As in the previous chapter, we are valuing an option using risk1neutral pricing" The

cash Nows of 4B"3 CFB2"/0 4 B2"$3D and B0 per gallon are multiplied !y their risk1neutral pro!a!ilities"The entire +rst term in E9uation 2("/ is then discounted at /"02 !ecause the cash Nows inthat term occur on Recem!er/" The B/,000,000 is not discounted !ecause Gr" Geyer receives it today, 7eptem!er /"6ecause the present value of the contract is negative, Gr" Geyer would !e wise to reJect thecontract"

As stated !efore, the distri!utor has sold a call option to )E)*" The +rst term in thepreceding e9uation, which e9uals 4B/,'3,//8, can !e viewed as the value of this calloption" #t is a negative num!er !ecause the e9uation looks at the option from Gr" Geyer:s

point of view" Therefore, the value of the call option would !e SB/,'3,//8 to )E)*" *n aper1gallon !asis, the value of the option to )E)* is.

E9uation 2("2 shows that )E)* will gain B"3 CFB2"$3 4 B2"/0D per gallon in the upstate !ecause )E)* can !uy heating oil worth B2"$3 for only B2"/0 under the contract"


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6y contrast, the contract is worth nothing to )E)* in the down state !ecause the utilitywill not pay B2"/0 for oil selling for only

B/"3 in the open market"


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ow do we value )E)*:s option in this three1date example- e employ the sameprocedure that we used in the two1date example, although we now need an extra step!ecause of the extra date"

Step 1: 9eterinin- t6e Riskeutra!Proai!ities

As we did in the two1date example, we determine what the pro!a!ility of a price risewould !e such that the expected return on heating oil exactly e9uals the riskless rate"owever, in this case we work with an interval of /V months" Assuming an 8 percent

annual rate of interest, which implies a / percent rate over a /V1month interval,& we cansolve for the pro!a!ility of a rise like this.

/ F Iro!a!ility of rise H "2& S C/ 4 Iro!a!ility of riseD HC4"20D

7olving the e9uation, we +nd that the pro!a!ility of a rise here is 3$ percent,mplying that the pro!a!ility of a fall is &( percent" #n other words, if the pro!a!ility of a rises 3$ percent, the expected return on heating oil is / percent per each /V1month interval"Again, these pro!a!ilities are determined under the assumption of risk1neutral pricing"

@ote that the pro!a!ilities of 3$ percent and &( percent hold for !oth the interval from7eptem!er / to *cto!er /& and the interval from *cto!er /& to Recem!er /" This is thecase !ecause the return in the up state is 2& percent and the return in the down state is 420 percent for each of the two intervals" Thus, the preceding e9uation must apply to each ofthe intervals separately"

Step 2: +a!uin- t6e Option as o,Octoer 1;

As indicated in %igure 2("2, the option to )E)* will !e worth B/"02 per gallon onRecem!er / if the price of heating oil has risen to B("/2 on that date" That is, )E)* can

!uy oil from Gr" Geyer at B2"/0 when it would otherwise have to pay B("/2 in the openmarket" owever, the option will !e worthless on Recem!er / if the price of a gallon ofheating oil is either B2 or B/"28 on that date" ere, the option is out of the money !ecausethe exercise price of B2"/0 is a!ove either B2 or B/"28"

ero value on *cto!er /& if the price of heating oil is B/"0 onthat date"


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Step 3: +a!uin- t6e Option onSepteer 1

#n the previous step, we saw that the price of the call on *cto!er /& would !e B"3$3 ifthe price of a gallon of heating oil were B2"&0 on that date" 7imilarly, the price of theoption on *cto!er /& would !eB0 if oil was selling at B/"0 on that date" %rom these values, we can calculate the calloption value on7eptem!er/.

O"3$ H B"3$3 S "&( H B0PK/"0/ FB"220

@otice that this calculation is completely analogous to the calculation of the optionvalue in the previous step, as well as the calculation of the option value in the two1dateexample that we presented earlier" #n other words, the same approach applies regardless ofthe num!er of intervals used" As we will see later, we can move to many intervals, whichproduces greater realism, yet still maintain the same !asic methodology"

The previous calculation has given us the value to )E)* of its option on one gallon ofheating oil" @ow we are ready to calculate the value of the contract to Gr" Geyer" Uiven

the calculations from the previous e9uation, the contract:s value can !e written as.

4B"220 H ,000,000 S B/,000,000 F 4B(20,000

That is, Gr" Geyer is giving away an option worth B"220 for each of the million gallonsof heating oil" #n return, he is receiving only B/,000,000 up front" *verall, he is losingB(20,000" *f course, the value of the contract to )E)* is the opposite, so the value to thisutility is B(20,000"

Extension to )an

9ates

e have looked at the contract !etween )E)* and Gr" Geyer using !oth a two1dateexample and a three1date example" The three1date case is more realistic !ecause morepossi!ilities for price movements are allowed here" owever, why stop at Just threedates- Goving to 3 dates, & dates, &0 dates, &00 dates, and so on should give us evenmore realism" @ote that as we move to more dates, we are merely shortening the interval!etween dates without increasing the overall time period of three months C7eptem!er/ to Recem!er /D"

%or example, imagine a model with '0 dates over the three months" ere, each

nterval is approximately one day long !ecause there are a!out '0 days in a three1monthperiod" The assumption of two possi!le outcomes in the !inomial model is more plausi!leover a one1day interval than it is over a /V1month interval, let alone a three1monthnterval" *f course, we could pro!a!ly achieve greater realism still !y going to an intervalof, say, one hour or one minute"

ow do we adJust the !inomial model to accommodate increases in the num!er ofntervals- #t turns out that two simple formulas relate u and d to the standarddeviation of the return on the underlying asset.

where is the standard deviation of the annuali>ed return on the underlying assetCheating oil, in this

caseD and n is the num!er of intervals over ayear"

hen we created the heating oil example, we assumed that the annuali>ed standarddeviation of the return on heating oil was "( Cor, e9uivalently, ( percentD" 6ecause there


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are four 9uarters in a year, and d F /K/"($ F "$(, as shown in the two1dateexample of %igure 2("/" #n the three1 date example of %igure 2("2, where each interval is/V months long, and d F /K/"2& F80" Thus, the !inomial model can !e applied in practice if the standard deviation of thereturn of the underlying asset can !e estimated"

e stated earlier that the value of the call option on a gallon of heating oil wasestimated to !e

B"282 in the two1date model and B"220 in the three1date model" ow does the value ofthe option change as we increase the num!er of intervals while keeping the time periodconstant at three months Cfrom 7eptem!er / to Recem!er /D- e have calculated the value

of the call for various time intervals inTa!le 2("&"$ The realism increases with the num!er of intervals !ecause the restrictionof only two possi!le outcomes is more plausi!le over a short interval than over a longone" Thus, the value of the call when the num!er of intervals is '' or in+nity is likely morerealistic than this value when the num!er of intervals is, say, / or 2"

Ta!e 23.; +a!ue o, a Ca!! on One &a!!on o,Heatin- Oi!

owever, a very interesting phenomenon can !e o!served from the ta!le" Although thevalue of the call changes as the num!er of intervals increases, convergence occurs 9uiterapidly" The call:s value with intervals is almost identical to the value with '' intervals" Thus, a small num!er of

ntervals appears servicea!le for the !inomial model" 7ix intervals in a three1month periodmplies that each interval is two weeks long" *f course, the assumption that heating oil cantake on only one of two prices in two weeks is simply not realistic" The paradox is that thisunrealistic assumption still produces a realistic call price"

hat happens when the num!er of intervals goes to in+nity, implying that the length ofthe interval goes to >ero- #t can !e proved mathematically that we end up with thevalue of the 6lack47choles model" This value is also presented in Ta!le 2("&" Thus, wecan argue that the 6lack47choles model is the !est approach to value the heating oiloption" #t is also 9uite easy to apply" e can use a calculator to value options with 6lack47choles, whereas we must generally use a computer program for the !inomial model"owever, as shown in Ta!le 2("&, the values from the !inomial model, even withrelatively few intervals, are 9uite close to the 6lack47choles value" Thus, although 6lack4

7choles may save us time, it does not materially a=ect our estimate of value"

At this point it seems as if the 6lack47choles model is prefera!le to the !inomialmodel" ho wouldn:t want to save time and still get a slightly more accurate value-owever, this is not always the case" There are plenty of situations where the !inomialmodel is preferred to the 6lack47choles model" *ne such situation is presented in the nextsection"


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23. S6utdo8n and Reopenin-9ecisions

7ome of the earliest and most important examples of special options have occurred inthe natural resources and mining industries"

?aluing a UoldGine

The oe #s Ge gold mine was founded in /8$8 on one of the richest veins of gold in theest" Thirty years later, !y /'08, the mine had !een played outM !ut occasionally,depending on the price of gold, it was reopened" )urrently, gold is not actively mined atoe #s Ge, !ut its stock is still traded on the exchange under the ticker sym!ol *E" *Ehas no de!t and, with a!out 20 million outstanding shares, its market value Cstock pricetimes num!er of shares outstandingD exceeds B/ !illion" *E owns a!out/0 acres of land surrounding the mine and has a /001year government lease to minegold there" owever, land in the desert has a market value of only a few thousanddollars" *E holds cash securities and other assets worth a!out B(0 million" hat couldpossi!ly explain why a company withB(0 million in assets and a closed gold mine with no cash Now has the market valuethat *E has-

The answer lies in the options that *E implicitly owns in the form of a gold mine"Assume that the current price of gold is a!out B(20 per ounce, and the cost of extractionand processing at the mine is a!out B(&0 per ounce" #t is no wonder that the mine is closed"Every ounce of gold extracted costs B(&0 and can !e sold for only B(20, for a loss of B(0per ounce" Iresuma!ly, if the price of gold were to rise, the mine could !e opened" #t costsB2 million to open the mineM when it is opened, production is &0,000 ounces per year"Ueologists !elieve that the amount of gold in the mine is essentially unlimited, and *Ehas the right to mine it for the next /00 years"


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ounce Cor only ( percentD toB(&0

The estimated volatility of the return on gold is a!out /& percent per year" This meansthat a single annual standard deviation movement in the gold price is /& percent of B(20, orB38 per year" 7urely with this amount of random movement in the gold price, a threshold of,for example, B(&2 is much too low at which to open the mine" 7imilar logic applies to theclosing decision" #f the mine is open, we will clearly keep it open as long as the gold prices a!ove the extraction cost of B(&0 per ounce !ecause we are pro+ting on every ounce ofgold mined" 6ut we also won:t close the mine down simply !ecause the gold price drops!elow B(&0 per ounce" e will tolerate a running loss !ecause gold may later rise !acka!ove B(&0" #f, alternatively, we closed the mine, we would pay the B/ milliona!andonment cost, only to pay another B2 million to reopen the mine if the price rose again"

To summari>e, if the mine is currently closed, then it will !e openedLat a cost of B2millionL whenever the price of gold rises su;ciently a!ove the extraction cost of B(&0 perounce" #f the mine is currently operating, then it will !e closed downLat a cost of B/ millionLwhenever the price of gold falls su;ciently !elow the extraction cost of B(&0 per ounce"*E:s pro!lem is to +nd these two threshold prices at which it opens a closed mine andcloses an open mine" e call these prices popen and pclose, respectively, where.

popen W B(&0Kounce Wpclose

#n other words, *E will open the mine if the gold price option is su;ciently in the moneyand will close it when the option is su;ciently out of the money"

e know that the more volatile the gold price, the further away popen and pclose will!e from B(&0 per ounce" e also know that the greater the cost of opening the mine, thehigher popen will !eM and the greater the cost of a!andoning the mine, the lower pclosewill !e" #nterestingly, we should also expect that popen will !e higher if thea!andonment cost is increased" After all, if it costs more to a!andon the mine, *E willneed to !e more assured that the price will stay a!ove the extraction cost when it decidesto open the mine" *therwise, *E will face the costly choice !etween a!andonment andoperating at a loss if the price falls !elow B(&0 per ounce" 7imilarly, raising the cost ofopening the mine will make *E more reluctant to close an open mine" As a result, pclose

will !e lower"The preceding arguments have ena!led us to reduce the pro!lem of valuing *E to

two stages" %irst, we have to determine the threshold prices, popen and pclose" 7econd,given the !est choices for these thresholds, we must determine the value of a gold optionthat is exercised for a cost of B2 million when the gold price rises a!ove popen and is shutdown for a cost of B/ million whenever the gold price is !elow pclose"

hen the mine is openLthat is, when the option is exercisedLthe annual cash Now ise9ual to the di=erence !etween the gold price and the extraction cost of B(&0 per ouncetimes &0,000 ounces" hen the mine is shut down, it generates no cash Now"

The following diagram descri!es the decisions availa!le at each point in time.

ow do we determine the critical values for popen and pclose and then the value of themine- #t is possi!le to get a good approximation !y using the tools we have currentlydeveloped"


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?aluing the 7imple Uold Gine

ere is what has to !e done !oth to determine popen and pclose and to value the mine"

Step 1 %ind the risk1free interest rate and the volatility" e assume a semiannualnterest rate of ("3 percent and a volatility of /& percent per year for gold"

Step 2 )onstruct a !inomial tree and +ll it in with gold prices" 7uppose, for example,

that we set the steps of the tree six months apart" #f the annual volatility is /& percent, us e9ual to , which is approximately e9ual to /"//" The other parameter, d, is "'0CF/K/"//D" %igure 2("( illustrates the tree" 7tarting at the current price of B(20, the +rst //percent increase takes the price to B(&& in six months" The +rst /0 percent decrease takesthe price to B288" 7u!se9uent steps are up // percent or down /0 percent from theprevious price" The tree extends for the /001year life of the lease, or 200 six1month steps"

Fi-ure 23.3 A 7inoia! Tree ,or &o!d Prices


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n other words, if investors are risk1neutral, they will !e satis+ed with an expected returne9ual to the risk1free rate !ecause the extra risk of gold will not concern them"

Step 3 @ow we turn the computer on and let it simulate, say, &,000 possi!le pathsthrough the tree" At each node, the computer has a "3 pro!a!ility of picking an upmovement in the price and a corresponding "( pro!a!ility of picking a down movementn the price" A typical path might !e represented !y whether the price rose or fell each six1month period over the next /00 yearsM it would !e a list like.

up, up, down, up, down, down, " " " ,

down

where the +rst up means the price rose from B(20 to B(&& in the +rst six months,the next up means it again went up in the second half of the year from B(&& to B('3,and so on, ending with a down move in the last half of year /00"

ith &,000 such paths, we will have a good sample of all the future possi!ilities formovement in the gold price"

Step @ext, we consider possi!le choices for the threshold prices, popen and pclose"%or popen, we let the possi!ilities !e.

popen F B(0 or B($0 or """ or B&00

a total of /& values" %or pclose we let the possi!ilities !e.

pclose F B(30 or B((0 or """ or B/00

a total of 2& values"

e picked these choices !ecause they seemed reasona!le and !ecause increments of

B/0 for each seemed sensi!le" To !e precise, though, we should let the threshold priceschange as we move through the tree and get closer to the end of /00 years" Iresuma!ly,for example, if we decided to open the mine with one year left on the lease, the price ofgold should !e at least high enough to cover the B2 million opening costs in the comingyear" 6ecause we mine &0,000 ounces per year, we will open the mine in year '' only ifthe gold price is at least B30 a!ove the extraction cost, or B('0"

Although this will !ecome important at the end of the lease, using a constantthreshold shouldn:t have too !ig an impact on the value with /00 years to go"Therefore, we will stick with our approximation of constant threshold prices"

Step ; e calculate the value of the mine for each pair of choices of popen and pclose"%or example, if popen F B3/0 and pclose F B2'0, we use the computer to keep track ofthe cash Nows if we opened the mine whenever it was previously closed and the goldprice rose to B3/0, and closed the mine whenever it was previously open and the goldprice fell to B2'0" e do this for each of the &,000 paths we simulated in 7tep ("

%or example, consider the path illustrated in%igure 2("3.

Fi-ure 23. A Possi!e Pat6 ,or t6e Price o, &o!d


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up, up, down, up, up, down, down, down, down

As can !e seen from the +gure, the price reaches a peak of B3($ in 2V years, only tofall to B288 over the following four six1month intervals" #f popen F B3/0 and pclose F B2'0,the mine will !e opened when the price reaches B3($, necessitating a cost of B2million" owever, the +rm can sell 2&,000 ounces of gold at B3($ per ounce at thattime, producing a cash Now of B2"/$& million OF2&,000 H CB3($ 4 B(&0DP" hen the pricefalls to B('3 six months later, the +rm sells another 2&,000 ounces, yielding a cash Nowof B/"/ million OF2&,000 H CB('3 4 B(&0DP" The price continues to fall, reachingB(20 a year later" ere, the +rm experiences a cash outNow !ecause production costsare B(&0 perounce" @ext, the price falls to B288" 6ecause this price is !elow pclose of B2'0, the mine isclosed at a cost of B/ million" *f course, the price of gold will Nuctuate in further years,eading to the possi!ility of future mine openings and closings"

This path is Just a possi!ility" #t may or may not occur in any simulation of &,000 paths"%or each of the &,000 paths that the computer simulated, we have a se9uence ofsemiannual cash Nows using a popen of B3/0 and a pclose of B2'0" e calculate thepresent value of each of these cash Nows, discounting at the interest rate of ("3 percent"7umming across all the cash Nows, we have the present value of the gold mine for one path"

e then take the average present value of the gold mine across all the &,000 simulatedpaths" This num!er is the expected value of the mine from following a policy of opening themine whenever the gold price hits B3/0 and closing it at a price of B2'0"

Step < The +nal step is to compare the di=erent expected discounted cash Nows from7tep & for the range of possi!le choices for popen and pclose and to pick the highest one"This is the !est estimate of the expected value of the mine" The values for pclose andpopen corresponding to this estimate are the points at which to open a closed mine and toshut an open one"

As mentioned in 7tep 3, there are /& di=erent values for popen and 2& di=erent valuesfor pclose, implying ($& CF/& H 2&D di=erent pairs" )onsider Ta!le 2(", which shows thepresent values associated with the 20 !est pairs" The ta!le indicates that the !est pair is


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popen F B300 and pclose F B/30, with a present value of B/"3$ !illion" This num!errepresents the average present value across &,000 simulations, all assuming thepreceding values of popen and pclose" The next !est pair is popen F B30 and pclose FB(00, with a present value of B/"3&' !illion" The third !est pair has a somewhat lowerpresent value, and so on"

Ta!e 23.< +a!uation o, =oe >s )e 4=OE5 &o!d )ine ,or t6e 2" 7est C6oices o,popen and pclose

*f course, our estimate of the value of the mine is B/"3$ !illion, the present value ofthe !est pair of choices" The market capitali>ation CIrice H @um!er of shares outstandingDof *E should reach this value if the market makes the same assumptions that we did"@ote that the value of the +rm is 9uite high using an option framework" owever, as statedearlier, *E would appear worthless if a regular discounted cash Now approach were used"

This occurs !ecause the initial gold price of B(20 is !elow the extraction cost of B(&0"

This example is not easy, neither in concepts nor in implementation" owever, theextra work involved in mastering this example is worth it !ecause the example illustratesthe type of modeling that actually occurs in corporate +nance departments in the real world"

%urthermore, the example illustrates the !ene+ts of the !inomial approach" e merelycalculate the cash Nows associated with each of a num!er of simulations, discountthe cash Nows from each simulation, and average present values across the simulations"6ecause the 6lack47choles model is not amena!le to simulations, it cannot !e used for thistype of pro!lem" #n addition, there are a num!er of other situations where the !inomialmodel is more appropriate than is the 6lack47choles model" %or example, it is well knownthat the 6lack47choles model cannot properly handle options with dividend payments prior

to the expiration date" This model also does not ade9uately handle the valuation of anAmerican put" 6y contrast, the !inomial model can easily handle !oth of these situations"

Thus, any student of corporate +nance should !e well versed in !oth models" The6lack47choles model should !e used whenever appropriate !ecause it is simpler to usethan is the !inomial model" owever, for the more complex situations where the 6lack47choles model !reaks down, the !inomial model !ecomes a necessary tool"


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Suar and Conc!usions

eal options, which are pervasive in !usiness, are not captured !y net present valueanalysis" )hapter

$ valued real options via decision trees" Uiven the work on options in the previous chapter,we are now a!le to value real options according to the 6lack47choles model and the!inomial model"

#n this chapter, we descri!ed and valued four di=erent types ofoptions.

/" Executive stock options, which are technically not real options"

2" The em!edded option in a start1up company"

(" The option in simple !usiness contracts"

3" The option to shut down and reopen a proJect"

e tried to keep the presentation simple and straightforward from a mathematical pointof view" The !inomial approach to option pricing in )hapter 22 was extended to manyperiods" This adJustment !rings us closer to the real world !ecause the assumption of onlytwo prices at the end of an interval is more plausi!le when the interval is short"

Concept ?uestions

/" Ep!oee Stock Options hy do companies issue options to executives ifthey cost the company more than they are worth to the executive- hy not Justgive cash and split the di=erence- ouldn:t that make !oth the company and theexecutive !etter off-

2" Rea! Options hat are the two options that many !usinesses have-

(" Pro0ect Ana!sis hy does a strict @I? calculation typically understate the

value of a company or proJect-

3" Rea! Options nsurance as an Option #nsurance, whether purchased !y a corporation or anindividual, is in essence an option" hat type of option is an insurance policy-

/0" Rea! Options ow would the analysis of real options change if a company hascompetitors-

?uestions and Pro!es connect@


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7AS>C 4?uestions 14;5

/" Ep!oee Stock Options Uary evin is the chief executive o;cer ofGountain!rook Trading )ompany" The !oard of directors has Just granted Gr" evin(0,000 at1the1money European call options on the company:s stock, which iscurrently trading at B&& per share" The stock pays no dividends" The options willexpire in +ve years, and the standard deviation of the returns on the stock is 3&percent" Treasury !ills that mature in +ve years currently yield a continuouslycompounded interest rate of percent"

/" arus has Just !een named the new chiefexecutive o;cer of 6lu6ell %itness )enters, #nc" #n addition to an annual salary ofB($&,000, his three1year contract states that his compensation will include /&,000

at1the1money European call options on the company:s stock that expire in threeyears" The current stock price is B(3 per share, and the standard deviation of thereturns on the +rm:s stock is $3 percent" The company does not pay a dividend"Treasury !ills that mature in three years yield a continuously compounded interest rateof& percent" Assume that Gr" a>arus:s annual salary payments occur at the end of theyear and that these cash Nows should !e discounted at a rate of ' percent" TER)E9>ATE 4?/EST>OS es inmanufacturing steel, and this particlar warehouse is the only facility in the area thatsuits the +rm:s operations" The current price of steel is B(0 per ton" #f the price ofsteel falls over the next six months, the company will purchase 300 tons of steel andproduce 3&,000 steel rods" Each steel rod will cost B/ to manufacture, and thecompany plans to sell the rods for B23 each" #t will take only a matter of days toproduce and sell the steel rods" #f the price of steel rises or remains the same, it will


22/23

not !e pro+ta!le to undertake the proJect, and the company will allow the lease toexpire without producing any steel rods" Treasury !ills that mature in six monthsyield a continuously compounded interest rate of 3"& percent, and the standarddeviation of the returns on steel is 3& percent"


23/23

reluctant to use the historical returns to estimate the standard deviation of the stock:sreturn" owever, you have estimated that the average annual standard deviation forrestaurant company stocks is a!out && percent" 6ecause Exotic )uisines is a newerrestaurant chain, you decide to use a 0 percent standard deviation in your calculations"The company is relatively young, and you expect that all earnings will !e reinvested !acknto the company for the near future" Therefore, you expect no dividends will !e paid forat least the next /0 years" A three1year Treasury note currently has a yield of ("8 percent,and a /01year Treasury note has a yield of3"3 percent"

/" Qou:re trying to value your options" hat minimum value would you assign-hat is the maximum value you would assign-

2" 7uppose that in three years the company:s stock is trading at B0" At that time,should you keep the options or exercise them immediately- hat are some of theimportant determinants in making such a decision-

(" Qour options, like most employee stock options, are not transfera!le or trada!le"Roes this have a signi+cant e=ect on the value of the options- hy-

3" hy do you suppose employee stock options usually have a vesting provision-hy must they !e exercised shortly after you depart the company even after they vest-

&" A controversial practice with employee stock options is repricing" hat

happens is that a company experiences a stock price decrease, which leavesemployee stock options far out of the money or underwater":: #n such cases, manycompanies have repriced:: or restruck:: the options, meaning that the companyleaves the original terms of the option intact !ut lowers the strike price" Iroponents ofrepricing argue that !ecause the option is very unlikely to end in the money !ecause ofthe stock price decline, the motivational force is lost" *pponents argue thatrepricing is in essence a reward for failure" ow do you evaluate this argument- owdoes the possi!ility of repricing a=ect the value of an employee stock option at the timeit is granted-

" As we have seen, much of the volatility in a company:s stock price is due tosystematic or marketwide risks" 7uch risks are !eyond the control of a company and its

employees" hat are the implications for employee stock options- #n light of youranswer, can you recommend an improvement over traditional employee stockoptions-

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