期权期货及其他衍生产品第9版-赫尔Ch

Chapter5

DeterminationofForwardandFuturesPrices1ConsumptionvsInvestmentAssetsInvestmentassetsareassetsheldbysignificantnumbersofpeoplepurelyforinvestmentpurposes(Examples:gold,silver)Consumptionassetsareassetsheldprimarilyforconsumption(Examples:copper,oil)2ShortSelling(106)ShortsellinginvolvessellingsecuritiesyoudonotownYourbrokerborrowsthesecuritiesfromanotherclientandsellstheminthemarketintheusualway3ShortSelling(continued)AtsomestageyoumustbuythesecuritiessotheycanbereplacedintheaccountoftheclientYoumustpaydividendsandotherbenefitstheownerofthesecuritiesreceivesTheremaybeasmallfeeforborrowingthesecurities4ExampleYoushort100shareswhenthepriceis$100andcloseouttheshortpositionthreemonthslaterwhenthepriceis$90Duringthethreemonthsadividendof$3pershareispaidWhatisyourprofit?Whatwouldbeyourlossifyouhadbought100shares?5NotationforValuingFuturesandForwardContracts6S0:SpotpricetodayF0:FuturesorforwardpricetodayT:Timeuntildeliverydater:Risk-freeinterestrateformaturityTAnArbitrageOpportunity?Supposethat:Thespotpriceofanon-dividend-payingstockis$40The3-monthforwardpriceis$43The3-monthUS$interestrateis5%perannumIsthereanarbitrageopportunity?7AnotherArbitrageOpportunity?Supposethat:Thespotpriceofnondividend-payingstockis$40The3-monthforwardpriceisUS$39The1-yearUS$interestrateis5%perannum(continuouslycompounded)Isthereanarbitrageopportunity?8TheForwardPriceIfthespotpriceofaninvestmentassetisS0andthefuturespriceforacontractdeliverableinTyearsisF0,then

F0=S0erT

whereristheT-yearrisk-freerateofinterest. Inourexamples,S0

=40,T=0.25,andr=0.05sothatF0

=40e0.05×0.25=40.509IfShortSalesAreNotPossible.. Formulastillworksforaninvestmentassetbecauseinvestorswhoholdtheassetwillsellitandbuyforwardcontractswhentheforwardpriceistoolow10

WhenanInvestmentAssetProvidesaKnownIncome(page110,equation5.2)

F0

=(S0

–I)erT

whereIisthepresentvalueoftheincomeduringlifeofforwardcontract11WhenanInvestmentAssetProvidesaKnownYield

(Page112,equation5.3)F0=S0e(r–q)T

whereqistheaverageyieldduringthelifeofthecontract(expressedwithcontinuouscompounding)12ValuingaForwardContractAforwardcontractisworthzero(exceptforbid-offerspreadeffects)whenitisfirstnegotiatedLateritmayhaveapositiveornegativevalueSupposethatKisthedeliverypriceandF0istheforwardpriceforacontractthatwouldbenegotiatedtoday13ValuingaForwardContract

(pages112-114)ByconsideringthedifferencebetweenacontractwithdeliverypriceKandacontractwithdeliverypriceF0wecandeducethat:thevalueofalongforwardcontractis (F0–K)e–rT

thevalueofashortforwardcontractis (K–F0

)e–rT14ForwardvsFuturesPricesWhenthematurityandassetpricearethesame,forwardandfuturespricesareusuallyassumedtobeequal.(Eurodollarfuturesareanexception)Intheory,wheninterestratesareuncertain,theyareslightlydifferent:AstrongpositivecorrelationbetweeninterestratesandtheassetpriceimpliesthefuturespriceisslightlyhigherthantheforwardpriceAstrongnegativecorrelationimpliesthereverse15StockIndex(117)CanbeviewedasaninvestmentassetpayingadividendyieldThefuturespriceandspotpricerelationshipistherefore

F0=S0e(r–q)T

whereqistheaveragedividendyieldontheportfoliorepresentedbytheindexduringlifeofcontract16StockIndex(continued)FortheformulatobetrueitisimportantthattheindexrepresentaninvestmentassetInotherwords,changesintheindexmustcorrespondtochangesinthevalueofatradableportfolioTheNikkeiindexviewedasadollarnumberdoesnotrepresentaninvestmentasset(SeeBusinessSnapshot5.3,page116)17IndexArbitrageWhenF0>S0e(r-q)T

anarbitrageurbuysthestocksunderlyingtheindexandsellsfuturesWhenF0<S0e(r-q)T

anarbitrageurbuysfuturesandshortsorsellsthestocksunderlyingtheindex18IndexArbitrage

(continued)IndexarbitrageinvolvessimultaneoustradesinfuturesandmanydifferentstocksVeryoftenacomputerisusedtogeneratethetradesOccasionallysimultaneoustradesarenotpossibleandthetheoreticalno-arbitragerelationshipbetweenF0andS0doesnothold(seeBusinessSnapshot5.4onpage117)19FuturesandForwardsonCurrencies(Page117-120)AforeigncurrencyisanalogoustoasecurityprovidingayieldTheyieldistheforeignrisk-freeinterestrateItfollowsthatifrf

istheforeignrisk-freeinterestrate

20ExplanationoftheRelationshipBetweenSpotandForward(Figure5.1)211000unitsofforeigncurrency(timezero)

1000S0dollarsattimezero1000S0erT

dollarsattimeTConsumptionAssets:StorageisNegativeIncomeF0

S0e(r+u)Twhereuisthestoragecostperunittimeasapercentoftheassetvalue.Alternatively,

F0

(S0+U)erT

whereUisthepresentvalueofthestoragecosts.22TheCostofCarry(Page123)Thecostofcarry,c,isthestoragecostplustheinterestcostslesstheincomeearnedForaninvestmentassetF0=S0ecT

ForaconsumptionassetF0

S0ecTTheconvenienceyieldontheconsumptionasset,