CHAPTER15

TheBlack-Scholes-MertonModel

PracticeQuestions

Problem15.1.

WhatdoestheBlack–Scholes–Mertonstockoptionpricingmodelassumeaboutthe

probabilitydistributionofthestockpriceinoneyear?Whatdoesitassumeaboutthe

probabilitydistributionofthecontinuouslycompoundedrateofreturnonthestockduring

theyear?

TheBlack–Scholes–Mertonoptionpricingmodelassumesthattheprobabilitydistributionof

thestockpricein1year(oratanyotherfuturetime)islognormal.Itassumesthatthe

continuouslycompoundedrateofreturnonthestockduringtheyearisnormallydistributed.

Problem15.2.

Thevolatilityofastockpriceis30%perannum.Whatisthestandarddeviationofthe

percentagepricechangeinonetradingday?

Thestandarddeviationofthepercentagepricechangeintimetistwhere

thevolatility.Inthisproblem03and,assuming252tradingdaysinoneyear,

is

t12520004

t0300040019or1.9%.

sothat

Problem15.3.

Explaintheprincipleofrisk-neutralvaluation.

Thepriceofanoptionorotherderivativewhenexpressedintermsofthepriceofthe

underlyingstockisindependentofriskpreferences.Optionsthereforehavethesamevaluein

arisk-neutralworldastheydointherealworld.Wemaythereforeassumethattheworldis

riskneutralforthepurposesofvaluingoptions.Thissimplifiestheanalysis.Inarisk-neutral

worldallsecuritieshaveanexpectedreturnequaltorisk-freeinterestrate.Also,ina

risk-neutralworld,theappropriatediscountratetouseforexpectedfuturecashflowsisthe

risk-freeinterestrate.

Problem15.4.

Calculatethepriceofathree-monthEuropeanputoptiononanon-dividend-payingstock

withastrikepriceof$50whenthecurrentstockpriceis$50,therisk-freeinterestrateis

10%perannum,andthevolatilityis30%perannum.

InthiscaseS50,K50,

r01,03,T025,and

0

dln(5050)(010092)02502417

03025

1

dd0302500917

2

1

TheEuropeanputpriceis

50N(00917)e0102550N(02417)

5004634e010255004045237

or$2.37.

Problem15.5.

WhatdifferencedoesitmaketoyourcalculationsinProblem15.4ifadividendof$1.50is

expectedintwomonths?

Inthiscasewemustsubtractthepresentvalueofthedividendfromthestockpricebefore

usingBlack–Scholes-Merton.Hencetheappropriatevalueof

Sis

0

S50150e01667014852

AsbeforeK50,r01,003,andT025.Inthiscase

dln(485250)(010092)02500414

03025

1

dd0302501086

2

1

TheEuropeanputpriceis

50N(01086)e010254852N(00414)

5005432e01025485204835303

or$3.03.

Problem15.6.

Whatisimpliedvolatility?Howcanitbecalculated?

TheimpliedvolatilityisthevolatilitythatmakestheBlack

–Scholes-Mertonpriceofan

optionequaltoitsmarketprice.Theimpliedvolatilityiscalculatedusinganiterative

procedure.Asimpleapproachisthefollowing.Supposewehavetwovolatilitiesonetoohigh

(i.e.,givinganoptionpricegreaterthanthemarketprice)andtheothertoolow(i.e.,giving

anoptionpricelowerthanthemarketprice).Bytestingthevolatilitythatishalfwaybetween

thetwo,wegetanewtoo-highvolatilityoranewtoo-lowvolatility.Ifwesearchinitiallyfor

twovolatilities,onetoohighandtheothertoolowwecanusethisprocedurerepeatedlyto

bisecttherangeandconvergeonthecorrectimpliedvolatility.Othermoresophisticated

approaches(e.g.,involvingtheNewton-Raphsonprocedure)areusedinpractice.

Problem15.7.

Astockpriceiscurrently$40.Assumethattheexpectedreturnfromthestockis15%andits

volatilityis25%.Whatistheprobabilitydistributionfortherateofreturn(withcontinuous

compounding)earnedoveratwo-yearperiod?

Inthiscase015and025.Fromequation(15.7)theprobabilitydistributionfor

therateofreturnoveratwo-yearperiodwithcontinuouscompoundingis:

0.252,0.252

0.15

2

2

i.e.,

(0.11875,0.03125)

Theexpectedvalueofthereturnis11.875%perannumandthestandarddeviationis17.7%

perannum.

Problem15.8.

AstockpricefollowsgeometricBrownianmotionwithanexpectedreturnof16%anda

volatilityof35%.Thecurrentpriceis$38.

a)WhatistheprobabilitythataEuropeancalloptiononthestockwithanexerciseprice

of$40andamaturitydateinsixmonthswillbeexercised?

b)WhatistheprobabilitythataEuropeanputoptiononthestockwiththesameexercise

priceandmaturitywillbeexercised?

a)Therequiredprobabilityistheprobabilityofthestockpricebeingabove$40insix

monthstime.SupposethatthestockpriceinsixmonthsisS

T

0.352

2

lnS~ln380.16

0.5,0.3520.5

T

i.e.,

lnS~3.687,0.2472

T

Sinceln403689,werequiretheprobabilityofln(ST)>3.689.Thisis

36893687

1N

1N(0008)

0247

SinceN(0.008)=0.5032,therequiredprobabilityis0.4968.

b)Inthiscasetherequiredprobabilityistheprobabilityofthestockpricebeinglessthan

$40insixmonthstime.Itis

10496805032

Problem15.9.

Usingthenotationinthechapter,provethata95%confidenceintervalfor

Sisbetween

T

Se

0

and

Se

0

(22)T196

T

(22)T196

T

Fromequation(15.3):

2

2

lnS~lnST,T

2

T

0

95%confidenceintervalsforlnSaretherefore

T

2

lnS()T196T

2

0

and

2

lnS()T196T

2

0

95%confidenceintervalsforSaretherefore

T

e

and

and

e

lnS0(

lnS0(

22)T196

T

T

22)T196

T

i.e.

Se

0

Se

0

(22)T196

(22)T196

T

Problem15.10.

Aportfoliomanagerannouncesthattheaverageofthereturnsrealizedineachofthelast10

yearsis20%perannum.Inwhatrespectisthisstatementmisleading?

ThisproblemrelatestothematerialinSection15.3.Thestatementismisleadinginthata

certainsumofmoney,say$1000,wheninvestedfor10yearsinthefundwouldhaverealized

areturn(withannualcompounding)oflessthan20%perannum.

Theaverageofthereturnsrealizedineachyearisalwaysgreaterthanthereturnperannum

(withannualcompounding)realizedover10years.Thefirstisanarithmeticaverageofthe

returnsineachyear;thesecondisageometricaverageofthesereturns.

Problem15.11.

Assumethatanon-dividend-payingstockhasanexpectedreturnof

andavolatilityof.

Aninnovativefinancialinstitutionhasjustannouncedthatitwilltradeaderivativethatpays

offadollaramountequaltolnSTattimeTwhereSdenotesthevaluesofthestock

T

priceattimeT.

a)Userisk-neutralvaluationtocalculatethepriceofthederivativeattimetintermof

thestockprice,S,attimet

b)Confirmthatyourpricesatisfiesthedifferentialequation(15.16)

a)Attimet,theexpectedvalueoflnSisfromequation(15.3)

T

lnS(2/2)(Tt)

Inarisk-neutralworldtheexpectedvalueof

lnS(r2/2)(Tt)

lnSTistherefore

Usingrisk-neutralvaluationthevalueofthederivativeattimetis

er(Tt)[lnS(r2/2)(Tt)]

b)If

fer(Tt)[lnS(r2/2)(Tt)]

then

ftrer(Tt)

[lnS(r2/2)(Tt)]er(Tt)r/2

2

fer(Tt)

S

S

2fer(Tt)

S2

S2

Theleft-handsideoftheBlack-Scholes-Mertondifferentialequationis

er(Tt)rlnSr(r

/2

2/2)(Tt)(r2/2)r

2

rlnSr(r

2/2)(Tt)

er(Tt)

rf

Hencethedifferentialequationissatisfied.

Problem15.12.

ConsideraderivativethatpaysoffSattimeTwhereSisthestockpriceatthattime.

n

T

T

WhenthestockpaysnodividendsanditspricefollowsgeometricBrownianmotion,itcanbe

shownthatitspriceattimet(tT)hastheform

h(tT)Sn

whereSisthestockpriceattimetandhisafunctiononlyoftandT.

(a)BysubstitutingintotheBlack–Scholes–Mertonpartialdifferentialequationderivean

ordinarydifferentialequationsatisfiedbyh(tT).

(b)Whatistheboundaryconditionforthedifferentialequationforh(tT)?

(c)Showthat

h(tT)e[052n(n1)r(n1)](Tt)

whereristherisk-freeinterestrateand

isthestockpricevolatility.

IfG(St)h(tT)SnthenGthSn,GShnSn1,and2GS2hn(n1)Sn2

t

wherehht.SubstitutingintotheBlack–Scholes–Mertondifferentialequationwe

t

obtain

hrhn12hn(n1)rh

2

t

ThederivativeisworthSnwhentT.Theboundaryconditionforthisdifferential

equationisthereforeh(TT)1

Theequation

h(tT)e[052n(n1)r(n1)](Tt)

satisfiestheboundaryconditionsinceitcollapsestoh1whentT.Itcanalsobeshown

thatitsatisfiesthedifferentialequationin(a).Alternativelywecansolvethedifferential

equationin(a)directly.Thedifferentialequationcanbewritten

h

tr(n1)12n(n1)

h

2

Thesolutiontothisis

or

lnh[r(n1)12n(n1)](Tt)

2

h(tT)e[052n(n1)r(n1)](Tt)

Problem15.13.

WhatisthepriceofaEuropeancalloptiononanon-dividend-payingstockwhenthestock

priceis$52,thestrikepriceis$50,therisk-freeinterestrateis12%perannum,thevolatility

is30%perannum,andthetimetomaturityisthreemonths?

r012,030

andT025.

InthiscaseS52,K50,

0

ln(5250)(0120322)02505365

030025

d

1

dd03002503865

2

1

ThepriceoftheEuropeancallis

52N(05365)50e012025N(03865)

520704250e00306504

506

or$5.06.

Problem15.14.

WhatisthepriceofaEuropeanputoptiononanon-dividend-payingstockwhenthestock

priceis$69,thestrikepriceis$70,therisk-freeinterestrateis5%perannum,thevolatility

is35%perannum,andthetimetomaturityissixmonths?

r005,035

andT05.

InthiscaseS69,K70,

0

ln(6970)(00503522)0501666

03505

d

1

dd0350500809

2

1

ThepriceoftheEuropeanputis

70e00505N(00809)69N(01666)

70e0025053236904338

640

or$6.40.

Problem15.15.

ConsideranAmericancalloptiononastock.Thestockpriceis$70,thetimetomaturityis

eightmonths,therisk-freerateofinterestis10%perannum,theexercisepriceis$65,and

thevolatilityis32%.Adividendof$1isexpectedafterthreemonthsandagainaftersix

months.Showthatitcanneverbeoptimaltoexercisetheoptiononeitherofthetwodividend

dates.UseDerivaGemtocalculatethepriceoftheoption.

UsingthenotationofSection15.12,DD1,K(1er(Tt2))65(1e0101667)107,

1

2

andK(1er(t2t1))65(1e01025)160.Since

DK(1er(Tt2)

)

)

1

and

DK(1er(t2t1)

2

Itisneveroptimaltoexercisethecalloptionearly.DerivaGemshowsthatthevalueofthe

optionis1094.

Problem15.16.

Acalloptiononanon-dividend-payingstockhasamarketpriceof$2.50.Thestockpriceis

$15,theexercisepriceis$13,thetimetomaturityisthreemonths,andtherisk-freeinterest

rateis5%perannum.Whatistheimpliedvolatility?

Inthecasec25,S15,

0

K13,T025,r005.Theimpliedvolatilitymustbe

calculatedusinganiterativeprocedure.

Avolatilityof0.2(or20%perannum)givesc220.Avolatilityof0.3gives

c232.A

volatilityof0.4givesc2507.Avolatilityof0.39gives

c2487.Byinterpolationthe

impliedvolatilityisabout0.396or39.6%perannum.

TheimpliedvolatilitycanalsobecalculatedusingDerivaGem.Selectequityasthe

UnderlyingTypeinthefirstworksheet.SelectBlack-ScholesEuropeanastheOptionType.

Inputstockpriceas15,therisk-freerateas5%,timetoexerciseas0.25,andexercisepriceas

13.Leavethedividendtableblankbecauseweareassumingnodividends.Selectthebutton

correspondingtocall.Selecttheimpliedvolatilitybutton.InputthePriceas2.5inthesecond

halfoftheoptiondatatable.HittheEnterkeyandclickoncalculate.DerivaGemwillshow

thevolatilityoftheoptionas39.64%.

Problem15.17.

Withthenotationusedinthischapter

(a)WhatisN(x)?

(b)ShowthatSN(d)Ker(Tt)N(d),whereSisthestockpriceattimet

1

2

ln(SK)(r22)(Tt)

d

1

Tt

ln(SK)(r22)(Tt)

d

2

Tt

(c)CalculatedSanddS.

1

2

(d)Showthatwhen

cSN(d)Ker(Tt)N(d)

1

2

c

t

rKer(Tt)N(d)SN(d)

2Tt

1

2

wherecisthepriceofacalloptiononanon-dividend-payingstock.

(e)ShowthatcSN(d).

1

(f)ShowthatthecsatisfiestheBlack–Scholes–Mertondifferentialequation.

(g)ShowthatcsatisfiestheboundaryconditionforaEuropeancalloption,i.e.,that

cmax(SK0)asttendstoT.

(a)SinceN(x)isthecumulativeprobabilitythatavariablewithastandardized

normaldistributionwillbelessthanx,N(x)istheprobabilitydensityfunctionfora

standardizednormaldistribution,thatis,

1

N(x)

e

2

x

2

2

N(d)N(d

Tt)

(b)

1

2

2(Tt)

1

2

d

1

2

exp2dTt

2

2

2

1

dTt(Tt)

2

N(d)exp

2

2

2

Because

ln(SK)(r22)(Tt)

d

2

Tt

itfollowsthat

1

Ke

r(Tt)

expdTt

2(Tt)

2

S

2

Asaresult

SN(d)Ke

1

N(d)

2

r(Tt)

whichistherequiredresult.

(c)

lnS(r2)(Tt)

d

1

K

2

Tt

lnSlnK(r2)(Tt)

2

Tt

Hence

d

1

SSTt

1

Similarly

lnSlnK(r2)(Tt)

d

2

2

Tt

and

d2

1

SSTt

Therefore:

dd

SS

1

2

(d)

cSN(d)Ke

N(d)

r(Tt)

1

2

c

d

d

SN(d)1rKe

N(d)Ke

N(d)

2

r(Tt)

r(Tt)

2

t

t

t

1

2

From(b):

Hence

SN(d)Ke

1

N(d)

2

r(Tt)

c

t

dd

12

rKe

N(d)SN(d)

tt

r(Tt

)

2

1

Since

ddTt

1

2

dd

(Tt)

1

2

ttt

2Tt

Hence

c

t

rKer(Tt)N(d)SN(d)

2Tt

1

2

(e)

FromdifferentiatingtheBlack–Scholes–Mertonformulaforacallpricewe

obtain

c

S

d

Ker(Tt)N(d)d

1

N(d)SN(d)

2

dS

S

1

1

2

Fromtheresultsin(b)and(c)itfollowsthat

cN(d)

S

1

(f)Differentiatingtheresultin(e)andusingtheresultin(c),weobtain

2c

d

1

S2N(d)S

1

1

N(d)

1

STt

Fromtheresultsind)ande)

c

t

c1

S2

2crKer(Tt)

N(d)SN(d)

rS

2S2

S2

2Tt

1

2

1

2

1

2S2N(d)

1

rSN(d)

1

STt

r[SN(d)Ker(Tt)N(d)]

1

2

rc

ThisshowsthattheBlack–Scholes–Mertonformulaforacalloptiondoesindeed

satisfytheBlack–Scholes–Mertondifferentialequation

(g)Considerwhathappensintheformulaforcinpart(d)astapproachesT.If

SK,danddtendtoinfinityandN(d)andN(d)tendto1.IfSK,

1

2

1

2

danddtendtozero.Itfollowsthattheformulaforctendstomax(SK0)

.

1

2

Problem15.18.

ShowthattheBlack–Scholes–Mertonformulasforcallandputoptionssatisfyput–callparity.

TheBlack–Scholes–MertonformulaforaEuropeancalloptionis

cSN(d)KerTN(d)

0

1

2

sothat

cKerTSN(d)KerTN(d)KerT

0

1

2

or

cKerTSN(d)KerT[1N(d)]

0

1

2

or

cKerTSN(d)KerTN(d)

0

1

2

TheBlack–Scholes–MertonformulaforaEuropeanputoptionis

pKerTN(d)SN(d)

2

0

1

sothat

pSKerTN(d)SN(d)S

0

0

2

0

1

or

pSKerTN(d)S[1N(d)]

0

2

0

1

or

pSKerTN(d)SN(d)

0

2

0

1

Thisshowsthattheput–callparityresult

cKerTpS

0

holds.

Problem15.19.

Astockpriceiscurrently$50andtherisk-freeinterestrateis5%.UsetheDerivaGem

softwaretotranslatethefollowingtableofEuropeancalloptionsonthestockintoatableof

impliedvolatilities,assumingnodividends.Aretheoptionpricesconsistentwiththe

assumptionsunderlyingBlack–Scholes–Merton?

StockPrice

Maturity=3months

Maturity=6months

Maturity=12months

45

50

55

7.00

3.50

1.60

8.30

5.20

2.90

10.50

7.50

5.10

UsingDerivaGemweobtainthefollowingtableofimpliedvolatilities

StockPrice

Maturity=3months

Maturity=6months

Maturity=12months

45

50

55

37.78

34.15

31.98

34.99

32.78

30.77

34.02

32.03

30.45

Tocalculatefirstnumber,selectequityastheUnderlyingTypeinthefirstworksheet.Select

Black-ScholesEuropeanastheOptionType.Inputstockpriceas50,therisk-freerateas5%,

timetoexerciseas0.25,andexercisepriceas45.Leavethedividendtableblankbecausewe

areassumingnodividends.Selectthebuttoncorrespondingtocall.Selecttheimplied

volatilitybutton.InputthePriceas7.0inthesecondhalfoftheoptiondatatable.Hitthe

Enterkeyandclickoncalculate.DerivaGemwillshowthevolatilityoftheoptionas37.78%.

Changethestrikepriceandtimetoexerciseandrecomputetocalculatetherestofthe

numbersinthetable.

TheoptionpricesarenotexactlyconsistentwithBlack–Scholes–Merton.Iftheywere,the

impliedvolatilitieswouldbeallthesame.Weusuallyfindinpracticethatlowstrikeprice

optionsonastockhavesignificantlyhigherimpliedvolatilitiesthanhighstrikepriceoptions

onthesamestock.ThisphenomenonisdiscussedinChapter20.

Problem15.20.

ExplaincarefullywhyBlack’sapproachtoevaluatinganAmericancalloptionona

dividend-payingstockmaygiveanapproximateanswerevenwhenonlyonedividendis

anticipated.DoestheanswergivenbyBlack’sapproachunderstateoroverstatethetrue

optionvalue?Explainyouranswer.

Black’sapproachineffectassumesthattheholderofoptionmustdecideattimezerowhether

itisaEuropeanoptionmaturingattimet(thefinalex-dividenddate)oraEuropeanoption

n

maturingattimeT.Infacttheholderoftheoptionhasmoreflexibilitythanthis.Theholder

canchoosetoexerciseattimetifthestockpriceatthattimeisabovesomelevelbutnot

n

otherwise.Furthermore,iftheoptionisnotexercisedattimet,itcanstillbeexercisedat

n

timeT.

ItappearsthatBlack’sapproachshouldunderstatethetrueoptionvalue.Thisisbecausethe

holderoftheoptionhasmorealternativestrategiesfordecidingwhentoexercisetheoption

thanthetwostrategiesimplicitlyassumedbytheapproach.Thesealternativestrategiesadd

valuetotheoption.

However,thisisnotthewholestory!ThestandardapproachtovaluingeitheranAmericanor

aEuropeanoptiononastockpayingasingledividendappliesthevolatilitytothestockprice

lessthepresentvalueofthedividend.(TheprocedureforvaluinganAmericanoptionis

explainedinChapter21.)Black’sapproachwhenconsideringexercisejustpriortothe

dividenddateappliesthevolatilitytothestockpriceitself.Black’sapproachtherefore

assumesmorestockpricevariabilitythanthestandardapproachinsomeofitscalculations.

Insomecircumstancesitcangiveahigherpricethanthestandardapproach.

Problem15.21.

ConsideranAmericancalloptiononastock.Thestockpriceis$50,thetimetomaturityis

15months,therisk-freerateofinterestis8%perannum,theexercisepriceis$55,andthe

volatilityis25%.Dividendsof$1.50areexpectedin4monthsand10months.Showthatit

canneverbeoptimaltoexercisetheoptiononeitherofthetwodividenddates.Calculatethe

priceoftheoption.

Withthenotationinthetext

DD150t03333t08333T125r008andK55

1

2

1

2

K1e

55(1e00804167)180

r(Tt)

2

Hence

Also:

DK1e

r(Tt)

2

2

K1e

55(1e00805)216

r(t2t)

1

Hence:

DK1er(t2t1)

1

ItfollowsfromtheconditionsestablishedinSection15.12thattheoptionshouldneverbe

exercisedearly.

Thepresentvalueofthedividendsis

15e0333300815e083330082864

TheoptioncanbevaluedusingtheEuropeanpricingformulawith:

S50286447136K55025r008T125

0

ln(4713655)(00802522)12500545

d

025125

1

dd02512503340

2

1

N(d)04783N(d)03692

1

2

andthecallpriceis

or$4.17.

471360478355e00812503692417

Problem15.22.

ShowthattheprobabilitythataEuropeancalloptionwillbeexercisedinarisk-neutral

worldis,withthenotationintroducedinthischapter,

N(d).Whatisanexpressionforthe

2

valueofaderivativethatpaysoff$100ifthepriceofastockattime

TisgreaterthanK?

Theprobabilitythatthecalloptionwillbeexercisedistheprobabilitythat

SKwhere

T

SisthestockpriceattimeT.Inariskneutralworld

T

lnS~lnS(r2/2)T,2T

T

0

TheprobabilitythatSKisthesameastheprobabilitythat

lnSlnK.Thisis

T

T

22)T

lnKlnS(r

1N

0

T

22)T

ln(SK)(r

N

0

T

N(d)

2

TheexpectedvalueattimeTinariskneutralworldofaderivativesecuritywhichpaysoff

$100whenSKistherefore

T

100N(d)

2

Fromriskneutralvaluationthevalueofthesecurityattimezerois

100erTN(d)

2

Problem15.23.

Usetheresultinequation(15.17)todeterminethevalueofaperpetualAmericanputoption

onanon-dividend-payingstockwithstrikepriceKifitisexercisedwhenthestockprice

equalsHwhereH<K.AssumethatthecurrentstockpriceSisgreaterthanH.Whatisthe

valueofHthatmaximizestheoptionvalue?DeducethevalueofaperpetualAmericanput

optionwithstrikepriceK.

IftheperpetualAmericanputisexercisedwhenS=H,itprovidesapayoffof(K?H).We

obtainitsvalue,bysettingQ=K?Hinequation(15.17),as

S

H

2r/2

2r/2

V(KH)(KH)

HS

Now

dVH

2

KHrH2r/21

S

2r/

2

dH

S

S

2

H

S

2r(KH)

2r/

2

1

H2

d2V2rKH2r/2

2r(KH)2rH

2r/21

1

H2S

2SS

dH2

dV/dHiszerowhen

H2

2

2rK

H2r2

and,forthisvalueofH,d2V/dH2isnegativeindicatingthatitgivesthemaximumvalueofV.

ThevalueoftheperpetualAmericanputismaximizedifitisexercisedwhenSequalsthis

valueofH.HencethevalueoftheperpetualAmericanputis

S2r/2

(KH)

H

whenH=2rK/(2r+2).Thevalueis

2KS(2r

)

2r/2

2

2r

2rK

2

ThisisconsistentwiththemoregeneralresultproducedinChapter26forthecasewherethe

stockprovidesadividendyield.

Problem15.24.

Acompanyhasanissueofexecutivestockoptionsoutstanding.Shoulddilutionbetakeninto

accountwhentheoptionsarevalued?Explainyouanswer.

Theanswerisno.Ifmarketsareefficienttheyhavealreadytakenpotentialdilutioninto

accountindeterminingthestockprice.ThisargumentisexplainedinBusinessSnapshot15.3.

Problem15.25.

Acompany’sstockpriceis$50and10millionsharesareoutstanding.Thecompanyis

consideringgivingitsemployeesthreemillionat-the-moneyfive-yearcalloptions.Option

exerciseswillbehandledbyissuingmoreshares.Thestockpricevolatilityis25%,the

five-yearrisk-freerateis5%andthecompanydoesnotpaydividends.Estimatethecostto

thecompanyoftheemployeestockoptionissue.

TheBlack-Scholes-MertonpriceoftheoptionisgivenbysettingS50,K50,

0

r005,025,andT5.Itis16.252.FromananalysissimilartothatinSection

15.10thecosttothecompanyoftheoptionsis

10

16252125

103

orabout$12.5peroption.Thetotalcostistherefore3milliontimesthisor$37.5million.If

themarketperceivesnobenefitsfromtheoptionsthestockpricewillfallby$3.75.

FurtherQuestions

Problem15.26.

Ifthevolatilityofastockis18%perannum,estimatethestandarddeviationofthe

percentagepricechangein(a)oneday,(b)oneweek,and(c)onemonth.

(a)182521.13%

(b)18522.50%

(c)18125.20%

Problem15.27.

Astockpriceiscurrently$50.Assumethattheexpectedreturnfromthestockis18%per

annumanditsvolatilityis30%perannum.Whatistheprobabilitydistributionforthestock

priceintwoyears?Calculatethemeanandstandarddeviationofthedistribution.Determine

the95%confidenceinterval.

S50,018and030.Theprobabilitydistributionofthestockprice

Inthiscase

0

intwoyears,S,islognormalandis,fromequation(15.3),givenby:

T

lnS~ln500.18

2,0.322

T

i.e.,

lnS~(4.18,0.422)

T

Themeanstockpriceisfromequation(15.4)

50e018250e0367167

andthestandarddeviationisfromequation(15.5)

50e0182e13183

0092

95%confidenceintervalsforlnSare

T

418196042and418196042

i.e.,

335and501

Thesecorrespondto95%confidencelimitsfor

Sof

T

e335ande501

i.e.,

2852and15044

Problem15.28.(Excelfile)

Supposethatobservationsonastockprice(indollars)attheendofeachof15consecutive

weeksareasfollows:

30.2,32.0,31.1,30.1,30.2,30.3,30.6,33.0,

32.9,33.0,33.5,33.5,33.7,33.5,33.2

Estimatethestockpricevolatility.Whatisthestandarderrorofyourestimate?

Thecalculationsareshowninthetablebelow

u009471

i

u2001145

i

andanestimateofstandarddeviationofweeklyreturnsis:

001145009471

2002884

13

1413

Thevolatilityperannumistherefore0028845202079or20.79%.Thestandarderror

ofthisestimateis

0207900393

214

or3.9%perannum.

Week

ClosingStockPrice

($)

PriceRelative

WeeklyReturn

SS

uln(SS)

i

i1

i

i

i1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

30.2

32.0

31.1

30.1

30.2

30.