Derivatives

CFA二級(jí)培訓(xùn)項(xiàng)目

1-181

Reading37

PricingandValuationofForwardCommitments

2-181

Case:DonaldTroubadour

DonaldTroubadourisaderivativestraderforSouthernShores

Investments.Thefirmseeksarbitrageopportunitiesintheforwardand

futuresmarketsusingthecarryarbitragemodel.

Troubadouridentifiesanarbitrageopportunityrelatingtoafixed-

incomefuturescontractanditsunderlyingbond.Currentdataonthe

futurescontractandunderlyingbondarepresentedinExhibit1.The

currentannualcompoundedrisk-freerateis0.30%.

3-181

Case:DonaldTroubadour

Exhibit1CurrentDataforFuturesandUnderlyingBond

FuturesContract

UnderlyingBond

Quotedfuturesprice

125.00

0.90

Quotedbondprice

112.00

0.08

Accruedinterestsincelast

couponpayment

Conversionfactor

Timeremainingto

contractexpiration

Three

Accruedinterestatfutures

0.20

monthscontractexpiration

Accruedinterestover

lifeoffuturescontract

0.00

Troubadournextgathersinformationonthreeexistingpositions.

4-181

Case:DonaldTroubadour

Position1(Nikkei225FuturesContract):

TroubadourholdsalongpositioninaNikkei225futurescontractthat

hasaremainingmaturityofthreemonths.Thecontinuously

compoundeddividendyieldontheNikkei225StockIndexis1.1%,

andthecurrentstockindexlevelis16,080.Thecontinuously

compoundedannualinterestrateis0.2996%.

Position2(Euro/JGBForwardContract):

Onemonthago,Troubadourpurchasedeuro/yenforwardcontracts

withthreemonthstoexpirationataquotedpriceof100.20(quotedas

apercentageofpar).Thecontractnotionalamountis￥100,000,000.

Thecurrentforwardpriceis100.05.

5-181

Case:DonaldTroubadour

Position3(JPYIUSDCurrencyForwardContract):

Troubadourholdsashortpositioninayen/USdollarforwardcontract

withanotionalvalueof$1,000,000.Atcontractinitiation,theforward

ratewas￥112.10per$1.Theforwardcontractexpiresinthreemonths.

Thecurrentspotexchangerateis￥112.00per$1,andtheannually

compoundedrisk-freeratesare-0.20%fortheyenand0.30%forthe

USdollar.Thecurrentquotedpriceoftheforwardcontractisequalto

theno-arbitrageprice.

TroubadournextconsidersanequityforwardcontractforTexasSteel,

Inc.(TSI).InformationregardingTSIcommonsharesandaTSIequity

forwardcontractispresentedinExhibit2.

6-181

Case:DonaldTroubadour

Exhibit2SelectedInformationforTSI

ThepricepershareofTSI’scommonshareis$250.

Theforwardpricepershareforanine-monthTSIequityforward

contractis$250.562289.

Assumeannualcompounding.

TroubadourtakesashortpositionintheTSIequityforwardcontract.

Hissupervisorasks,"Underwhichscenariowouldourposition

experiencealoss?“Threemonthsaftercontractinitiation,Troubadour

gathersinformationonTSIandtherisk-freerate,whichispresentedin

Exhibit3.

7-181

Case:DonaldTroubadour

Exhibit3SelectedDataonTSIandtheRisk-FreeRate

ThepricepershareofTSI’scommonshareis$245.

Therisk-freerateis0.325%(quotedonanannualcompounding

basis).

TSIrecentlyannounceditsregularsemiannualdividendof$1.50per

sharethatwillbepaidexactlythreemonthsbeforecontract

expiration.

ThemarketpriceoftheTSIequityforwardcontractisequaltotheno

arbitrageforwardprice.

8-181

Case:DonaldTroubadour

BasedonExhibit2andassumingannualcompounding,thearbitrage

profitonthebondfuturescontractisclosestto:

A.0.4158.

B.0.5356.

C.0.6195.

9-181

Case:DonaldTroubadour

Answer:B

Theno-arbitragefuturespriceisequaltothefollowing:

F(T)=FV(T)[B(T+Y)+AI-PVCI]

0

0,T0,T

0

0

F(T)=(1+0.003)0

0

.25(112.00+0.08-0)

F0(T)=(1+0.003)0.25(112.08)=112.1640

Theadjustedpriceofthefuturescontractisequaltotheconversionfactormultiplied

bythequotedfuturesprice:

F(T)=CF(T)QF(T)

F0(T)=(0.90)(125)=112.50

0

0

Addingtheaccruedinterestof0.20inthreemonths(futurescontractexpiration)to

theadjustedpriceofthefuturescontractgivesatotalpriceof112.70.Thisdifference

meansthatthefuturescontractisoverpricedby112.70-112.1640=0.5360.1he

availablearbitrageprofitisthepresentvalueofthisdifference:0.5360/(1.003)0.25=

0.5356.

10-181

Case:DonaldTroubadour

Thecurrentno-arbitragefuturespriceoftheNikkei225futures

contract(Position1)isclosestto:

A.15,951.81.

B.16,047.86.

C.16,112.21.

11-181

Case:DonaldTroubadour

Answer:B

Theno-arbitragefuturespriceis

F(T)=Se

(r-γ)T

c

0

0

F0(T)=16,080e(0.0029960.011)(3/12)=16,047.68

12-181

Case:DonaldTroubadour

ThevalueofPosition2isclosestto:

A.-￥149,925.

B.-￥150,000.

C.-￥150,075.

13-181

Case:DonaldTroubadour

Answer:A

ThevalueofTroubadour'seuro/JGBforwardpositioniscalculatedas

V(T)=PV[F(T)-F(T)]

t,T

Vt(T)=(100.05-100.20)/(1+0.0030)2/12=-0.149925(per￥100par

value)

t

t

0

Therefore,thevalueoftheTroubadour'sforwardpositionis

Vt(T)=[-0.149925/100](￥100,000,000)=-￥149,925

14-181

Case:DonaldTroubadour

ThevalueofPosition3isclosestto:

A.-￥40,020.

B.￥139,913.

C.￥239,963.

15-181

Case:DonaldTroubadour

Answer:C

Thecurrentno-arbitragepriceoftheforwardcontractis

F(￥/$,T)=S(￥/$)FV(1)/FV(1)

￥,t,T$,t,T

t

t

Ft(￥/$,T)=￥112.00(1-0.002)0

.25/(1+0.003)0.25=￥111.8602

Therefore,thevalueofTroubadour'spositioninthe￥/$forward

contract,onaperdollarbasis,is

V(T)=PV[F(￥/$,T)-F(￥/$,T)]=(112.10-111.8602)/(1-0.002).25=

0

t

￥,t,T

0

t

￥

0.239963per$1

Troubadour'spositionisashortpositionof$1,000,000,sotheshort

positionhasapositivevalueof(￥0.239963/$)x$1,000,000￥239,963

becausetheforwardratehasfallensincethecontractinitiation.

16-181

Case:DonaldTroubadour

BasedonExhibit2,Troubadourshouldfindthatanarbitrage

opportunityrelatingtoTSIsharesis

A.notavailable.

B.availablebasedoncarryarbitrage.

C.availablebasedonreversecarryarbitrage.

17-181

Case:DonaldTroubadour

Answer:A

Thecarryarbitragemodelpriceoftheforwardcontractis

FV(S)S(1+r)T=$250(1+0.003)0.75=$250.562289

0

0

ThemarketpriceoftheTSIforwardcontractis$250.562289.Acarryor

reversecarryarbitrageopportunitydoesnotexistbecausethemarket

priceoftheforwardcontractisequaltothecarryarbitragemodelprice.

18-181

Case:DonaldTroubadour

ThemostappropriateresponsetoTroubadour'ssupervisor'squestion

regardingtheTSIforwardcontractis:

A.adecreaseinTSI'sshareprice,allelseequal.

B.anincreaseintherisk-freerate,allelseequal

C.adecreaseinthemarketpriceoftheforwardcontract,allelse

equal.

19-181

Case:DonaldTroubadour

Answer:B

Fromtheperspectiveofthelongposition,theforwardvalueisequalto

thepresentvalueofthedifferenceinforwardprices:

V(T)=PV[F(T)-F(T)]

t,T

t

t

0

whereF(T)=FV(S+θ-γ).

t

t,T

t

t

t

Allelseequal,anincreaseintherisk-freeratebeforecontract

expirationwouldcausetheforwardprice,F(T),toincrease.This

t

increaseintheforwardpricewouldcausethevalueoftheTSIforward

contract,fromtheperspectiveoftheshort,todecrease.Therefore,an

increaseintherisk-freeratewouldleadtoalossontheshortposition

intheTSIforwardcontract.

20-181

Case:DonaldTroubadour

BasedonExhibits2and3,andassumingannualcompounding,theper

sharevalueofTroubadour'sshortpositionintheTSIforwardcontract

threemonthsaftercontractinitiationisclosestto:

A.$1.6549.

B.$5.1561.

C.$6.6549.

21-181

Case:DonaldTroubadour

Answer:C

Theno-arbitragepriceoftheforwardcontract,threemonthsaftercontractinitiation,

is

F(T)=FV0.25,T(S0.25+θ-γ)

.250.250.25

0

F(T)=[$245+0-$1.50/(1+0.00325)

(0.5-0.25)](1+0.00325)

0

.25

(0.75–0.25)=$243.8966

Therefore,fromtheperspectiveofthelong,thevalueoftheTSIforwardcontractis

.75-0.25=

-$6.6549

BecauseTroubadourisshorttheTSIforwardcontract,thevalueofhispositionisa

gainof$6.6549.

V(T)=PV[F(T)-F0(T)]=($243.8966-$250.562289)/(1+0.00325)0

0.25,T0.25

0

.25

22-181

Case:SonalJohnson

SonalJohnsonisariskmanagerforabank.Shemanagesthebank's

risksusingacombinationofswapsandforwardrateagreements(FRAs).

Johnsonpricesathree-yearLibor-basedinterestrateswapwithannual

resetsusingthepresentvaluefactorspresentedinExhibit1.

Exhibit1

PresentValueFactors

Maturity(years)

PresentValueFactors

0.990099

1

2

3

0.977876

0.965136

23-181

Case:SonalJohnson

JohnsonalsousesthepresentvaluefactorsinExhibit1tovaluean

interestrateswapthatthebankenteredintooneyearagoasthe

receive-floatingparty.Selecteddatafortheswaparepresentedin

Exhibit2.Johnsonnotesthatthecurrentequilibriumtwo-yearfixed

swaprateis1.00%.

Exhibit2

SelectedDataonFixedforFloatingInterestRateSwap

Swapnotionalamount

Originalswapterm

$50,000,000

Threeyears,withannualresets

Fixedswaprate(sinceinitiation)3.00%

24-181

Case:SonalJohnson

Oneofthebank'sinvestmentsisexposedtomovementsinthe

Japaneseyen,andJohnsondesirestohedgethecurrencyexposure.

Shepricesaone-yearfixed-for-fixedcurrencyswapinvolvingyenand

USdollars,withaquarterlyreset.Johnsonusestheinterestratedata

presentedinExhibit3topricethecurrencyswap.

Exhibit3

SelectedJapaneseandUSInterestRateData

DaystoMaturity

YenSpotInterestRates

USDollarSpotInterestRates

9

1

2

3

0

0.05%

0.10%

0.15%

0.25%

0.20%

0.40%

0.55%

0.70%

80

70

60

25-181

Case:SonalJohnson

Johnsonnextreviewsanequityswapwithanannualresetthatthe

bankenteredintosixmonthsagoasthereceive-fixed,pay-equityparty.

Selecteddataregardingtheequityswap,whichislinkedtoanequity

index,arepresentedinExhibit4.Atthetimeofinitiation,the

underlyingequityindexwastradingat100.00.

Exhibit4

SelectedDataonEquitySwap

Swapnotionalamount

Originalswapterm

Fixedswaprate

$20,000,000

Fiveyears,withannualresets

2.00%

26-181

Case:SonalJohnson

Theequityindexiscurrentlytradingat103.00,andrelevantUSspot

rates,alongwiththeirassociatedpresentvaluefactors,arepresentedin

Exhibit5.

Exhibit5

SelectedUSSpotRatesandPresentValueFactors

Maturity(years)

SpotRate

0.4%

PresentValueFactors

0.998004

0.5

1.5

2.5

3.5

4.5

1.00%

1.20%

2.00%

2.60%

0.985222

0.970874

0.934579

0.895255

27-181

Case:SonalJohnson

Johnsonreviewsa6x9FRAthatthebankenteredinto90daysagoas

thepay-fixed/receive-floatingparty.SelecteddatafortheFRAare

presentedinExhibit6,andcurrentLibordataarepresentedinExhibit7.

Basedonherinterestrateforecast,Johnsonalsoconsiderswhetherthe

bankshouldenterintonewpositionsin1x4and2x5FRAs.

Exhibit6

FRAterm

FRArate

6x9FRAData

6x9

0.70%

FRAnotionalamount

FRAsettlementterms

US$20,000,000

Advancedset,advancedsettle

28-181

Case:SonalJohnson

Exhibit7

CurrentLibor

3

3

3

3

3

3

3

3

0-dayLibor

0-dayLibor

0-dayLibor

0-dayLibor

0-dayLibor

0-dayLibor

0-dayLibor

0-dayLibor

0.75%

0.82%

0.90%

0.92%

0.94%

0.95%

0.97%

1.00%

29-181

Case:SonalJohnson

Threemonthslater,the6x9FRAinExhibit6reachesexpiration,at

whichtimethethree-monthUSdollarLiboris1.10%andthesix-month

USdollarLiboris1.20%.Johnsondeterminesthattheappropriate

discountratefortheFRAsettlementcashflowsis1.10%.

30-181

Case:SonalJohnson

BasedonExhibit1,Johnsonshouldpricethethree-yearLibor-based

interestrateswapatafixedrateclosestto:

A.0.34%.

B.1.16%.

C.1.19%.

31-181

Case:SonalJohnson

Answer:C

Theswappricingequationis

1

PV(1)

0,t

VFIX

n

n

PV(1)

0

,t

i

i1

Thatis,thefixedswaprateisequalto1minusthefinalpresentvalue

factor(inthiscase,Year3)dividedbythesumofthepresentvalues(in

thiscase,thesumofYears1,2,and3).Thesumofpresentvaluesfor

Years1,2,and3iscalculatedas

n

PV(1)0.990099+0.977876+0.965136=2.933111

0

,t

i

i1

Thus,thefixed-swaprateiscalculatedas

32-181

Case:SonalJohnson

Thus,thefixed-swaprateiscalculatedas

0.965136

1

V

0.01189or1.19%

FIX

2

.933111

33-181

Case:SonalJohnson

Fromthebank'sperspective,usingdatafromExhibit1,thecurrent

valueoftheswapdescribedinExhibit2isclosestto:

A.-$2,951,963.

B.-$1,967,975.

C.-$1,943,000.

34-181

Case:SonalJohnson

Answer:B

Thevalueofaswapfromtheperspectiveofthereceive-fixedpartyis

calculatedas

n'

VNA(FSFS)PV

0

t

t,ti

i1

Theswaphastwoyearsremaininguntilexpiration.Thesumofthe

presentvaluesforYears1and2is

n'

PV0.990099+0.977876=1.967975

t,t

i

i1

35-181

Case:SonalJohnson

Giventhecurrentequilibriumtwo-yearswaprateof1.00%andthe

fixedswaprateatinitiationof3.00%,theswapvalueperdollarnotional

iscalculatedas

V=(0.030.01)1.967975=0.0393595

Thecurrentvalueoftheswap,fromtheperspectiveofthereceive-fixed

party,is$50,000,000X0.0393595=$1,967,975.Fromtheperspective

ofthebank,asthereceive-floatingparty,thevalueoftheswapis-

$

1,967,975.

36-181

Case:SonalJohnson

BasedonExhibit3,Johnsonshoulddeterminethattheannualized

equilibriumfixedswaprateforJapaneseyenisclosestto:

A.0.0624%.

B.0.1375%.

C.0.2496%.

37-181

Case:SonalJohnson

Answer:C

Theequilibriumswapfixedrateforyeniscalculatedas

1

PV0,t4,JPY(1)

rFIX,JPY

4

PV0,t4,JPY(1)

i1

Theyenpresentvaluefactorsarecalculatedas

1

PV(1)

0

,t

i

NADi

NTD

1

r(

)

Spoti

181

Case:SonalJohnson

90-dayPVfactor=1/[1+0.0005(90/360)]=0.999875.

180-dayPVfactor=1/[1+0.0010(180/360)]0.999500.

270-dayPVfactor=11[1+0.0015(270/360)]0.998876.

360-dayPVfactor=11[1+0.0025(360/360)]0.997506.

Sumofpresentvaluefactors=3.995757.

Therefore,theyenperiodicrateiscalculatedas

1

0.997506

rFIX,JPY

0.000624or0.0624%

3

.995757

Theannualizedrateis(360/90)timestheperiodicrateof0.0624%,or

0.2496%.

39-181

Case:SonalJohnson

Fromthebank'sperspective,usingdatafromExhibits4and5,thefair

valueoftheequityswapisclosestto:

A.-$1,139,425.

B.-$781,323.

C.-$181,323.

40-181

Case:SonalJohnson

Answer:B

Thevalueofanequityswapiscalculatedas

St

VFB(C)()NAE

t

t

0

S

t

Theswapwasinitiatedsixmonthsago,sothefirstresethasnotyet

passed;thus,therearefiveremainingcashflowsforthisequityswap.

Thefairvalueoftheswapisdeterminedbycomparingthepresent

valueoftheimpliedfixedratebondwiththereturnontheequityindex.

Thefixedswaprateof2.00%,theswapnotionalamountof$20,000,000,

andthepresentvaluefactorsinExhibit5resultinapresentvalueofthe

impliedfixed-ratebond'scashflowsof$19,818,677:

41-181

Case:SonalJohnson

Date(in

years)

PVFactors

FixedCashFlowPV(fixedcashflow)

0

.5

.5

.5

.5

.5

0.998004or

$400,000

$400,000

$400,000

$400,000

$20,400,000

$399,202

1/[1+0.004(180/360)]

1

2

3

4

0.985222or

1/[1+0.01(540/360)]

$394,089

0.970874or

1/[1+0.012(900/360)]

$388,350

0.934579or

1/[1+0.02(1260/360)]

$373,832

0.895255or

1/[1+0.026(1620/360)]

$18,263,205

Total

$19,818,677

42-181

Case:SonalJohnson

Thevalueoftheequitylegoftheswapiscalculatedas

(103/100)($20,000,000)=$20,600,000.

Therefore,thefairvalueoftheequityswap,fromtheperspectiveofthe

bank(receive-fixed,pay-equityparty)iscalculatedas

Vt=$19,818,677-$20,600,000=-781,323

43-181

Case:SonalJohnson

BasedonExhibit5,thecurrentvalueoftheequityswapdescribedin

Exhibit4wouldbezeroiftheequityindexwascurrentlytradingthe

closestto:

A.97.30.

B.99.09.

C.100.00.

44-181

Case:SonalJohnson

Answer:B

Theequityindexlevelatwhichtheswap'sfairvaluewouldbezerocan

becalculatedbysettingtheswapvaluationformulaequaltozeroand

solvingforSt:

St

0

FB(C)()NAE

t

0

St

Thevalueofthefixedlegoftheswaphasapresentvalueof

$

19,818,677,or99.0934%ofparvalue:

45-181

Case:SonalJohnson

Date(inyears)

PVFactors

FixedCashFlow

$400,000

PV(fixedcashflow)

0.5

1.5

2.5

3.5

0.998004

0.985222

0.970874

0.934579

$399,202

$394,089

$388,350

$373,832

$400,000

$400,000

$400,000

4

.5

0.895255

$20,400,000

$18,263,205

$19,818,677

Total

46-181

Case:SonalJohnson

Treatingtheswapnotionalvalueasparvalueandsubstitutingthe

presentvalueofthefixedlegandS0intotheequationyields

99.0934-

S

100

0

t

100

SolvingforStyields

St=99.0934

47-181

Case:SonalJohnson

Fromthebank'sperspective,basedonExhibits6and7,thevalueofthe

6x9FRA90daysafterinceptionisclosestto:

A.$14,817.

B.$19,647.

C.$29,635.

48-181

Case:SonalJohnson

Answer:A

Thecurrentvalueofthe6x9FRAiscalculatedas

V(0,h,m){[FRA(g,hg,m)FRA(0,h,m)]t}/[1D(hmg)t

]

g

m

g

hmg

The6x9FRAexpiressixmonthsafterinitiation.Thebankenteredinto

theFRA90daysago;thus,theFRAwillexpirein90days.Tovaluethe

FRA,thefirststepistocomputethenewFRArate,whichistherateon

Day90ofanFRAthatexpiresin90daysinwhichtheunderlyingisthe

9

0-dayLibor,orFRA(90,90,90):

V(0,h,m){[FRA(g,hg,m)FRA(0,h,m)]t}/[1D(hmg)t

hmg

]

g

m

g

FRA(g,hg,m){[1Lg(hgm)thgm]/[1L(hg)t]1}/t

0

hg

m

FRA(90,90,90){[1L(1809090)(180/360)]/[1L(18090)(90/360)]1}/(90/360)

9

0

90

FRA(90,90,90){[1L(180)(180/360)]/[1L(90)(90/360)]1}/(90/360)

9

0

90

49-181

Case:SonalJohnson

Exhibit7indicatesthatL90(180)=0.95%andL90(90)=0.90%,so

FRA(90,90,90){[10.0095(180/360)]/[10.009(90/360)]1}/(90/360)

[(1.00475/1.00225)1](4)0.009978,or0.9978%

Therefore,giventheFRArateatinitiationof0.70%andnotional

principalof$20millionfromExhibit1,thecurrentvalueoftheforward

contractiscalculatedas

V(0,h,m)V(0,180,90)

g

90

V(0,180,90)$20,000,000[(0.0099780.007)(90/360)]/[10.0095(180/360)]

90

$14,887.75/1.00475$14,817.37

50-181

Case:SonalJohnson

BasedonExhibit7,theno-arbitragefixedrateonanew1x4FRAis

closestto:

A.0.65%.

B.0.73%.

C.0.98%.

51-181

Case:SonalJohnson

Answer:C

Theno-arbitragefixedrateonthe1x4FRAiscalculatedas

FRA(0,h,m){[1L(hm)t]/[1L(h)t]1}/t

0

hm

0

h

m

Fora1x4FRA,thetworatesneededtocomputetheno-arbitrageFRA

fixedrateareL(30)=0.75%andL(120)=0.92%.Therefore,theno-

arbitragefixedrateonthe1x4FRArateiscalculatedas

FRA(0,30,90){[10.0092(120/360)]/[10.0075(30/360)]1}/(90/360)

[(1.003066/1.000625)1]40.009761,or0.98%rounded

52-181

Case:SonalJohnson

BasedonExhibit7,thefixedrateonanew2x5FRAisclosestto:

A.0.61%.

B.1.02%.

C.1.71%.

53-181

Case:SonalJohnson

Answer:B

Thefixedrateonthe2x5FRAiscalculatedas

FRA(0,h,m){[1L(hm)t]/[1L(h)t]1}/t

0

hm

0

h

m

Fora2x5FRA,thetworatesneededtocomputetheno-arbitrageFRA

fixedrateareL(60)=0.82%andL(l50)=0.94%.Therefore,theno-

arbitragefixedrateonthe2x5FRArateiscalculatedas

FRA(0,60,90){[10.0094(150/360)]/[10.0082(60/360)]1}/(90/360)

[(1.003917/1.001367)1]40.010186,or1.02%rounded

54-181

Case:SonalJohnson

BasedonExhibit6andthethree-monthUSdollarLiboratexpiration,

thepaymentamountthatthebankwillreceivetosettlethe6x9FRAis

closestto:

A.$19,945.

B.$24,925.

C.$39,781.

55-181

Case:SonalJohnson

Answer:A

Givenathree-monthUSdollarLiborof1.10%atexpiration,the

settlementamountforthebankasthereceive-floatingpartyis

calculatedas

Settlementamount(receivefloating)=NA{[L(m)-FRA(0,h,m)]t}/[1+D(m)t]

h

m

h

m

=

=

$20,000,000[(0.011-0.007)(90/360)]/[1+0.011(90/360)]

$20,000/1.00275=$19,945.15

Therefore,thebankwillreceive$19,945(rounded)asthereceive-

floatingparty.

56-181

Reading38

ValuationofContingentClaims

57-181

Case:BrunoSousa

BrunoSousahasbeenhiredrecentlytoworkwithsenioranalystCamila

Rocha.Rochagiveshimthreeoptionvaluationtasks.

AlphaCompany

Sousa'sfirsttaskistoillustratehowtovalueacalloptiononAlpha

Companywithaone-periodbinomialoptionpricingmodel.Itisanon-

dividend-payingstock,andtheinputsareasfollows.

Thecurrentstockpriceis50,andthecalloptionexercisepriceis50.

Inoneperiod,thestockpricewilleitherriseto56ordeclineto46.

Therisk-freerateofreturnis5%perperiod.

Basedonthemodel,RochaasksSousatoestimatethehedgeratio,the

risk-neutralprobabilityofanupmove,andthepriceofthecalloption.In

theillustration,Sousaisalsoaskedtodescriberelatedarbitragepositions

touseifthecalloptionisoverpricedrelativetothemodel.

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Case:BrunoSousa

BetaCompany

Next,Sousausesthetwo-periodbinomialmodeltoestimatethevalueofa

EuropeanstylecalloptiononBetaCompany'scommonshares.Theinputs

areasfollows.

Thecurrentstockpriceis38,andthecalloptionexercisepriceis40.

Theupfactor(u)is1.300,andthedownfactor(d)is0.800.

Therisk-freerateofreturnis3%perperiod.

Sousathenanalyzesaputoptiononthesamestock.Alloftheinputs,

includingtheexerciseprice,arethesameasforthecalloption.He

estimatesthatthevalueofaEuropean-styleputoptionis4.53.Exhibit1

summarizeshisanalysis.SousanextmustdeterminewhetheranAmerican-

styleputoptionwouldhavethesamevalue.

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Case:BrunoSousa

Exhibit1Two-PeriodBinomialEuropean-StylePutOptiononBeta

Company

Item

Underlying

Put

Value

64.22

0

Item

Underlying

Put

Value

49.4

0.2517

Item

Underlying

Put

Value

38

Hedge

Ratio

0.01943

Item

Underlying

Put

Value

39.52

0.48

4.5346

Item

Underlying

Put

Value

30.4

Hedge

Ratio

-

0.4307

Item

Underlying

Put

Value

24.32

15.68

8.4350

Hedge

Ratio

-

1

Time=0

Time=1

Time=2

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Case:BrunoSousa

Sousamakestwostatementswithregardtothevaluationofa

European-styleoptionundertheexpectationsapproach.

Statement1Thecalculationinvolvesdiscountingattherisk-free

rate.

Statement2Thecalculationusesrisk-neutralprobabilitiesinstead

oftrueprobabilities.

RochaasksSousawhetheritiseverprofitabletoexerciseAmerican

optionspriortomaturity.Sousaanswers,“Icanthinkoftwopossible

cases.ThefirstcaseistheearlyexerciseofanAmericancalloptionona

dividend-payingstock.Thesecondcaseistheearlyexerciseofan

Americanputoption.”

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Case:BrunoSousa

InterestRateOption

Thefinaloptionvaluationtaskinvolvesaninterestrateoption.Sousa

mustvalueatwo-year,European-stylecalloptiononaone-yearspot

rate.Thenotionalvalueoftheoptionis1million,andtheexerciserate

is2.75%.Therisk-neutralprobabilityofanupmoveis0.50.Thecurrent

andexpectedone-yearinterestratesareshowninExhibit2,alongwith

thevaluesofaone-yearzero-couponbondof1notionalvalueforeach

interestrate.

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Case:BrunoSousa

Exhibit2Two-YearInterestRateLatticeforanInterestRateOption

Maturity

Value

Rate

1

0.952381

5%

Maturity

Value

Rate

1

0.961538

4%

Maturity

Value

Rate

Maturity

Value

Rate

1

0.970874

3%

1

0.970874

3%

Maturity

Value

Rate

1

0.980392

2%

Maturity

Value

Rate

1

0.990099

1%

Time=0

Time=1

Time=2

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Case:BrunoSousa

RochaasksSousawhythevalueofasimilarin-the-moneyinterestrate

calloptiondecreasesiftheexercisepriceishigher.Sousaprovidestwo

reasons.

Reason1Theexercisevalueofthecalloptionislower.

Reason2Therisk-neutralprobabilitiesarechanged.

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Case:BrunoSousa

TheoptimalhedgeratiofortheAlphaCompanycalloptionusingthe

oneperiodbinomialmodelisclosestto:

A.0.60.

B.0.67.

C.1.67.

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Case:BrunoSousa

Answer:A

Thehedgeratiorequirestheunderlyingstockandcalloptionvaluesfor

theupmoveanddownmove.S+=56,andS-=46.c+=Max(0,S+-X)=

Max(0,56-50)=6,andc-=Max(0,S--X)=Max(0,46-50)=0.Thehedge

ratiois

CC

60

60.60

h

SS

564610

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Case:BrunoSousa

Therisk-neutralprobabilityoftheupmovefortheAlphaCompany

stockisclosestto:

A.0.06.

B.0.40.

C.0.65.

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Case:BrunoSousa

Answer:C

Forthisapproach,therisk-freerateisr=0.05,theupfactorisu=S+IS

=

56/50=1.12,andthedownfactorisd=S-IS=46/50=0.92.The

risk-neutralprobabilityofanupmoveis

[FV(1)d]/(ud)(1rd)/(ud)

(10.050.92)/(1.120.92)=0.13/0.20=0.65

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Case:BrunoSousa

ThevalueoftheAlphaCompanycalloptionisclosestto:

A.3.71.

B.5.71.

C.6.19.

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Case:BrunoSousa

Answer:A

Thecalloptioncanbeestimatedusingtheno-arbitrageapproachor

theexpectationsapproach.Withtheno-arbitrageapproach,thevalue

ofthecalloptionis

chSPV(hSc)

h(cc)/(SS)(60)/(5646)0.60

c(0.6050)(1/1.05)[(-0.6046)0]

c30[(1/1.05)27.6]3026.2863.714

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Case:BrunoSousa

Usingtheexpectationsapproach,therisk-freerateisr=0.05,theup

factorisu=S+/S=56/50=1.12,andthedownfactorisd=S-/S=

4

6/50=0.92.Thevalueofthecalloptionis

cPV[c(1c)]

[FV(1)d]/(ud)(1.050.92)/(1.120.92)0.65

c(1/1.05)[0.65(6)+0.65(0)](1/1.05)(3.9)3.714

Bothapproachesarelogicallyconsistentandyieldidenticalvalues.

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Case:BrunoSousa

FortheAlphaCompanyoption,thepositionstotakeadvantageofthe

arbitrageopportunityaretowritethecalland:

A.shortsharesofAlphastockandlend.

B.buysharesofAlphastockandborrow.

C.shortsharesofAlphastockandborrow.

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Case:BrunoSousa

Answer:B

Youshouldsell(write)theoverpricedcalloptionandthengolong(buy)the

replicatingportfolioforacalloption.Thereplicatingportfolioforacalloptionisto

buyhsharesofthestockandborrowthepresentvalueof(hS--c-).

chSPV(hSc)

h(cc)/(SS)(60)/(5646)0.60

Fortheexampleinthiscase,thevalueofthecalloptionis3.714.Iftheoptionis

overpricedat,say,4.50,youshorttheoptionandhaveacashflowatTime0of+4.50.

Youbuythereplicatingportfolioof0.60sharesat50pershare(givingyouacash

flowof-30)andborrow(1/1.05)x[(0.60x46)-0]=(111.05)x27.6=26.287.Your

cashflowforbuyingthereplicatingportfoliois-30+26.287=-3.713.Yournetcash

flowatTime0is+4.503.713=0.787.YournetcashflowatTime1fo