期权、期货及其他衍生产品课件3金融工程学

5 1 DeterminationofForwardandFuturesPrices Chapter5 5 2 ConsumptionvsInvestmentAssets Investmentassetsareassetsheldbysignificantnumbersofpeoplepurelyforinvestmentpurposes Examples stock bond gold silver Consumptionassetsareassetsheldprimarilyforconsumption Examples copper oil 5 3 ShortSelling Page99 101 ShortsellinginvolvessellingsecuritiesyoudonotownYourbrokerborrowsthesecuritiesfromanotherclientandsellstheminthemarketintheusualwayAtsomestageyoumustbuythesecuritiesbacksotheycanbereplacedintheaccountoftheclient 5 4 ShortSelling continued Ifatanytimewhilethecontractisopenthebrokerisnotabletoborrowshares theinvestorisforcedtocloseouttheposition evenifnotreadytodoso calledshort squeezed 挤空 挟仓 Youmustpaydividendsandotherbenefitstheownerofthesecuritiesreceives Example 5 5 Theinvestorisrequiredtomaintainamarginaccountwiththebroker Assumptions Themarketparticipantsaresubjecttonotransactioncostswhentheytrade Themarketparticipantsaresubjectstothesametaxrateonallnettradingprofits Themarketparticipantscanborrowmoneyatthesamerisk freerateofinterestastheycanlendmoney Themarketparticipantstakeadvantageofarbitrageopportunitiesastheyoccur 5 6 5 7 NotationforValuingFuturesandForwardContracts Forwardpriceforaninvestmentasset 5 8 5 9 WhenInterestRatesareMeasuredwithContinuousCompounding F0 S0erT远期价格大于即期价格ifF0 S0erT arbitrageurscanbuytheassetandshortforwardcontractsontheasset ifF0 S0erT theycanshorttheassetandenterintolongforwardcontractsonit Thisequationrelatestheforwardpriceandthespotpriceforanyinvestmentassetthatprovidesnoincomeandhasnostoragecosts WhenanInvestmentAssetProvidesaKnownDollarIncome 5 10 5 11 套利者净收益 910 00 886 60 23 40 5 12 WhenanInvestmentAssetProvidesaKnownDollarIncome page105 equation5 2 F0 S0 I erTwhereIisthepresentvalueoftheincomeduringlifeofforwardcontract Inourexample Sothat 5 13 WhenanInvestmentAssetProvidesaKnownYield Page107 equation5 3 F0 S0e r q Twhereqistheaverageyieldduringthelifeofthecontract expressedwithcontinuouscompounding ExampleConsidera6 monthforwardcontractonanassetthatisexpectedtoprovideincomeequalto2 oftheassetpriceonceduringa6 monthperiod Theriskfreerateofinterestis10 perannum Theassetpriceis 25 Theyieldis4 perannum itfollowsthatq 0 0396 2ln 1 4 2 5 14 5 15 ValuingaForwardContractPage108 SupposethatKisdeliverypriceinaforwardcontractandF0isforwardpricethatwouldapplytothecontracttodayThevalueofalongforwardcontract is F0 K e rTSimilarly thevalueofashortforwardcontractis K F0 e rT 5 16 Example Alongforwardcontractonanot dividendpayingstockwasenteredintosometimeago Itcurrentlyhas6monthstomaturity Therisk freerateofinterestis10 perannum thestockpriceis 25 andthedeliverypriceis 24 5 17 Thevalueofalongforwardcontract 5 18 Noincome income Knownyield 5 19 ForwardvsFuturesPrices Forwardandfuturespricesareusuallyassumedtobethesame Wheninterestratesareuncertaintheyare intheory slightlydifferentAstrongpositivecorrelationbetweeninterestratesandtheassetpriceimpliesthefuturespriceisslightlyhigherthantheforwardpriceAstrongnegativecorrelationimpliesthereverse 5 20 StockIndex Page110 112 CanbeviewedasaninvestmentassetpayingadividendyieldThefuturespriceandspotpricerelationshipisthereforeF0 S0e r q Twhereqistheaveragedividendyieldontheportfoliorepresentedbytheindexduringlifeofcontract 5 21 StockIndex continued FortheformulatobetrueitisimportantthattheindexrepresentaninvestmentassetInotherwords changesintheindexmustcorrespondtochangesinthevalueofatradableportfolioTheNikkeiindexviewedasadollarnumberdoesnotrepresentaninvestmentasset SeeBusinessSnapshot5 3 page111 5 22 IndexArbitrage WhenF0 S0e r q TanarbitrageurbuysthestocksunderlyingtheindexandsellsfuturesWhenF0 S0e r q Tanarbitrageurbuysfuturesandshortsorsellsthestocksunderlyingtheindex 5 23 IndexArbitrage continued IndexarbitrageinvolvessimultaneoustradesinfuturesandmanydifferentstocksVeryoftenacomputerisusedtogeneratethetradesOccasionally e g onBlackMonday simultaneoustradesarenotpossibleandthetheoreticalno arbitragerelationshipbetweenF0andS0doesnothold 5 24 AforeigncurrencyisanalogoustoasecurityprovidingadividendyieldThecontinuousdividendyieldistheforeignrisk freeinterestrateItfollowsthatifrfistheforeignrisk freeinterestrate FuturesandForwardsonCurrencies Page112 115 5 25 WhytheRelationMustBeTrueFigure5 1 page113 Example Supposethatthe2yearinterestratesinAustraliaandtheUnitedStatesare5 and7 respectively andthespotexchangerateis0 6200USDperAUD The2yearforwardexchangerateshouldbe 5 26 Supposethatthe2yearforwardexchangerateislessthanthis say0 6300 Anarbitrageurcan 1 Borrow1000AUDat5 perannumfor2years convertto620USDandinvesttheUSDat7 2 Enterintoaforwardcontracttobuy1105 17AUDfor1105 17 0 63 696 26USD 5 27 Supposethatthe2yearforwardrateis0 6600 Anarbitrageurcan 1 Borrow1000USDat7 perannumfor2years convertto1000 0 6200 1616 90AUD andinvesttheAUDat5 2 Enterintoaforwardcontracttosell1782 53AUDfor1782 53 0 66 1176 47USD 5 28 Aforeigncurrencycanberegardedasaninvestmentassetpayingaknownyield Theyieldistheriskfreerateofinterestintheforeigncurrency 5 29 5 30 FuturesonConsumptionAssets Page117 118 F0 S0e r u Twhereuisthestoragecostperunittimeasapercentoftheassetvaluenetofanyyieldearnedontheasset Alternatively F0 S0 U erTwhereUisthepresentvalueofthestoragecosts netofincome Storagecostscanbetreatedasnegativeincome SupposeF0 S0 U erT aarbitrageurcan 1 BorrowanamountS0 Uattheriskfreerateanduseittopurchaseoneunitofthecommodityandtopaystoragecost 2 Shortaforwardcontractononeunitofthecommodity EquationF0 S0 U erTcannothold SupposeF0 S0 U erT 1 Sellthecommodity savethestoragecosts andinvesttheproceedsattheriskfreeinterestrate 2 Takealongpositioninaforwardcontract 5 31 Individualsandcompanieswhoownaconsumptioncommodityusuallyplantouseitinsomeway Theyarereluctanttosellthecommodityinthespotmarketandbuyforwardforfuturescontracts becauseforwardandfuturescontractscannotbeconsumed wecanassertforaconsumptioncommodityis F0 S0 U erTF0 S0e r u T 5 32 ConvenienceYieldsy 5 33 5 34 TheCostofCarry Page118 119 Thecostofcarry c isthestoragecostplustheinterestcostslesstheincomeearnedc storagecost interestcost incomeearnForaninvestmentassetF0 S0ecTForaconsumptionassetF0 S0ecTTheconvenienceyieldontheconsumptionasset y isdefinedsothatF0 S0e c y T 5 35 FuturesPrices ExpectedFutureSpotPrices Page119 121 Su