HullFund8eCh09ProblemSolutions.docx
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HullFund8eCh09ProblemSolutions.docx
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HullFund8eCh09ProblemSolutions
CHAPTER9
MechanicsofOptionsMarkets
PracticeQuestions
Problem9.8.
Acorporatetreasurerisdesigningahedgingprograminvolvingforeigncurrencyoptions.Whataretheprosandconsofusing(a)theNASDAQOMXand(b)theover-the-countermarketfortrading?
TheNASDAQOMXoffersoptionswithstandardstrikepricesandtimestomaturity.Optionsintheover-the-countermarkethavetheadvantagethattheycanbetailoredtomeetthepreciseneedsofthetreasurer.Theirdisadvantageisthattheyexposethetreasurertosomecreditrisk.Exchangesorganizetheirtradingsothatthereisvirtuallynocreditrisk.
Problem9.9.
SupposethataEuropeancalloptiontobuyasharefor$100.00costs$5.00andishelduntilmaturity.Underwhatcircumstanceswilltheholderoftheoptionmakeaprofit?
Underwhatcircumstanceswilltheoptionbeexercised?
Drawadiagramillustratinghowtheprofitfromalongpositionintheoptiondependsonthestockpriceatmaturityoftheoption.
Ignoringthetimevalueofmoney,theholderoftheoptionwillmakeaprofitifthestockpriceatmaturityoftheoptionisgreaterthan$105.Thisisbecausethepayofftotheholderoftheoptionis,inthesecircumstances,greaterthanthe$5paidfortheoption.Theoptionwillbeexercisedifthestockpriceatmaturityisgreaterthan$100.Notethatifthestockpriceisbetween$100and$105theoptionisexercised,buttheholderoftheoptiontakesalossoverall.TheprofitfromalongpositionisasshowninFigureS9.1.
FigureS9.1ProfitfromlongpositioninProblem9.9
Problem9.10.
SupposethataEuropeanputoptiontosellasharefor$60costs$8andishelduntilmaturity.Underwhatcircumstanceswilltheselleroftheoption(thepartywiththeshortposition)makeaprofit?
Underwhatcircumstanceswilltheoptionbeexercised?
Drawadiagramillustratinghowtheprofitfromashortpositionintheoptiondependsonthestockpriceatmaturityoftheoption.
Ignoringthetimevalueofmoney,theselleroftheoptionwillmakeaprofitifthestockpriceatmaturityisgreaterthan$52.00.Thisisbecausethecosttotheselleroftheoptionisinthesecircumstanceslessthanthepricereceivedfortheoption.Theoptionwillbeexercisedifthestockpriceatmaturityislessthan$60.00.Notethatifthestockpriceisbetween$52.00and$60.00theselleroftheoptionmakesaprofiteventhoughtheoptionisexercised.TheprofitfromtheshortpositionisasshowninFigureS9.2.
FigureS9.2ProfitfromshortpositioninProblem9.10
Problem9.11.
Describetheterminalvalueofthefollowingportfolio:
anewlyentered-intolongforwardcontractonanassetandalongpositioninaEuropeanputoptionontheassetwiththesamematurityastheforwardcontractandastrikepricethatisequaltotheforwardpriceoftheassetatthetimetheportfolioissetup.ShowthattheEuropeanputoptionhasthesamevalueasaEuropeancalloptionwiththesamestrikepriceandmaturity.
Theterminalvalueofthelongforwardcontractis:
where
isthepriceoftheassetatmaturityand
istheforwardpriceoftheassetatthetimetheportfolioissetup.(Thedeliverypriceintheforwardcontractisalso
.)
Theterminalvalueoftheputoptionis:
Theterminalvalueoftheportfolioistherefore
ThisisthesameastheterminalvalueofaEuropeancalloptionwiththesamematurityastheforwardcontractandanexercisepriceequalto
.ThisresultisillustratedintheFigureS9.3.
FigureS9.3ProfitfromportfolioinProblem9.11
Wehaveshownthattheforwardcontractplustheputisworththesameasacallwiththesamestrikepriceandtimetomaturityastheput.Theforwardcontractisworthzeroatthetimetheportfolioissetup.Itfollowsthattheputisworththesameasthecallatthetimetheportfolioissetup.
Problem9.12.
Atraderbuysacalloptionwithastrikepriceof$45andaputoptionwithastrikepriceof$40.Bothoptionshavethesamematurity.Thecallcosts$3andtheputcosts$4.Drawadiagramshowingthevariationofthetrader’sprofitwiththeassetprice.
FigureS9.4showsthevariationofthetrader’spositionwiththeassetprice.Wecandividethealternativeassetpricesintothreeranges:
a)Whentheassetpricelessthan$40,theputoptionprovidesapayoffof
andthecalloptionprovidesnopayoff.Theoptionscost$7andsothetotalprofitis
.
b)Whentheassetpriceisbetween$40and$45,neitheroptionprovidesapayoff.Thereisanetlossof$7.
c)Whentheassetpricegreaterthan$45,thecalloptionprovidesapayoffof
andtheputoptionprovidesnopayoff.Takingintoaccountthe$7costoftheoptions,thetotalprofitis
.
Thetradermakesaprofit(ignoringthetimevalueofmoney)ifthestockpriceislessthan$33orgreaterthan$52.ThistypeoftradingstrategyisknownasastrangleandisdiscussedinChapter11.
FigureS9.4ProfitfromtradingstrategyinProblem9.12
Problem9.13.
ExplainwhyanAmericanoptionisalwaysworthatleastasmuchasaEuropeanoptiononthesameassetwiththesamestrikepriceandexercisedate.
TheholderofanAmericanoptionhasallthesamerightsastheholderofaEuropeanoptionandmore.Itmustthereforebeworthatleastasmuch.Ifitwerenot,anarbitrageurcouldshorttheEuropeanoptionandtakealongpositionintheAmericanoption.
Problem9.14.
ExplainwhyanAmericanoptionisalwaysworthatleastasmuchasitsintrinsicvalue.
TheholderofanAmericanoptionhastherighttoexerciseitimmediately.TheAmericanoptionmustthereforebeworthatleastasmuchasitsintrinsicvalue.Ifitwerenotanarbitrageurcouldlockinasureprofitbybuyingtheoptionandexercisingitimmediately.
Problem9.15.
Explaincarefullythedifferencebetweenwritingaputoptionandbuyingacalloption.
Writingaputgivesapayoffof
.Buyingacallgivesapayoffof
.Inbothcasesthepotentialpayoffis
.Thedifferenceisthatforawrittenputthecounterpartychooseswhetheryougetthepayoff(andwillallowyoutogetitonlywhenitisnegativetoyou).Foralongcallyoudecidewhetheryougetthepayoff(andyouchoosetogetitwhenitispositivetoyou.)
Problem9.16.
Thetreasurerofacorporationistryingtochoosebetweenoptionsandforwardcontractstohedgethecorporation’sforeignexchangerisk.Discusstheadvantagesanddisadvantagesofeach.
Forwardcontractslockintheexchangeratethatwillapplytoaparticulartransactioninthefuture.Optionsprovideinsurancethattheexchangeratewillnotbeworsethansomelevel.Theadvantageofaforwardcontractisthatuncertaintyiseliminatedasfaraspossible.Thedisadvantageisthattheoutcomewithhedgingcanbesignificantlyworsethantheoutcomewithnohedging.Thisdisadvantageisnotasmarkedwithoptions.However,unlikeforwardcontracts,optionsinvolveanup-frontcost.
Problem9.17.
Consideranexchange-tradedcalloptioncontracttobuy500shareswithastrikepriceof$40andmaturityinfourmonths.Explainhowthetermsoftheoptioncontractchangewhenthereis
a)A10%stockdividend
b)A10%cashdividend
c)A4-for-1stocksplit
a)Theoptioncontractbecomesonetobuy
shareswithanexerciseprice
.
b)Thereisnoeffect.Thetermsofanoptionscontractarenotnormallyadjustedforcashdividends.
c)Theoptioncontractbecomesonetobuy
shareswithanexercisepriceof40/4=$10.
Problem9.18.
“Ifmostofthecalloptionsonastockareinthemoney,itislikelythatthestockpricehasrisenrapidlyinthelastfewmonths.”Discussthisstatement.
Theexchangehascertainrulesgoverningwhentradinginanewoptionisinitiated.Thesemeanthattheoptionisclose-to-the-moneywhenitisfirsttraded.Ifallcalloptionsareinthemoney,itisthereforelikelythatthestockpricehasrisensincetradingintheoptionbegan.
Problem9.19.
Whatistheeffectofanunexpectedcashdividendon(a)acalloptionpriceand(b)aputoptionprice?
Anunexpectedcashdividendwouldreducethestockpriceontheex-dividenddate.Thisstockpricereductionwouldnotbeanticipatedbyoptionholders.Asaresulttherewouldbeareductioninthevalueofacalloptionandanincreasethevalueofaputoption.(Notethatthetermsofanoptionareadjustedforcashdividendsonlyinexceptionalcircumstances.)
Problem9.20.
OptionsonGeneralMotorsstockareonaMarch,June,September,andDecembercycle.Whatoptionstradeon(a)March1,(b)June30,and(c)August5?
a)March,April,JuneandSeptember
b)July,August,September,December
c)August,September,December,March.
Longerdatedoptionsmayalsotrade.
Problem9.21.
Explainwhythemarketmaker’sbid-offerspreadrepresentsarealcosttooptionsinvestors.
A“fair”pricefortheoptioncanreasonablybeassumedtobehalfwaybetweenthebidandtheofferpricequotedbyamarketmaker.Aninvestortypicallybuysatthemarketmaker’sofferandsellsatthemarketmaker’sbid.Eachtimeheorshedoesthisthereisahiddencostequaltohalfthebid-offerspread.
Problem9.22.
AUnitedStatesinvestorwritesfivenakedcalloptioncontracts.Theoptionpriceis$3.50,thestrikepriceis$60.00,andthestockpriceis$57.00.Whatistheinitialmarginrequirement?
Thetwocalculationsarenecessarytodeterminetheinitialmargin.Thefirstgives
Thesecondgives
Theinitialmarginisthegreaterofthese,or$5,950.Partofthiscanbeprovidedbytheinitialamountof500×3.5=$1,750receivedfortheoptions.
FurtherQuestions
Problem9.23.
Calculatetheintrinsicvalueandtimevaluefromthemid-marke
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