TopicWeightingsinCFALevel77StudySession54

Forward、Futures&

RiskRiskfreeDefinition:Aderivativeisafinancialinstrument(contract)thatderivesitsperformancefromtheperformanceofanunderlyingasset.BuyorSellBuyorSellBuyorSellsometimeintheSwapSwapAforwardcontractisanprivateagreementthatobligatesonepartytobuyandtheotherpartytosellaspecificquantityofanunderlyingasset,atasetprice,atafuturedateIfthefuturericeoftheunderlinassetsincreasethebuerhasaainandthesellerhasaloss.AFuturescontractisaforwardcontractthatisstandardizedandexchange-AreBackedbyaRequireadaily ndASwapcontractisaseriesofforwardcontractsExchangecashflowsonperiodsettlementSwapSwapAnoption AnoptiontobuyanassetataparticularpriceistermedacallBuyerofaRighttoSellerofaObligationtoAnotiontosellanassetat riceistermed utoBuyerofaRighttoSellerofaObligationtoForwardcommitment:isanagreementbetweentwopartiesinwhichoneparty,thebuyer,agreestobuyfromtheotherparty,theseller,anunderlyingassetatafuturedateatapriceestablishedatthestartforward,futuresandswapcontractsContingentclaim:isderivativeinwhichthepayoffsoccurifaspecificeventhappensoptioncontracts面，與清算所交易（AClearinghouseB）Backedbya(defaultrisk)TradeintheaphysicalnottradeinorganizedForwardContingent

WhichofthefollowingisthebestexampleofaAglobalequitymutualAcontracttopurchaseAppleComputeratafixedWhichofthefollowingstatementsaboutderivativesisnot arecreatedinthesotTheyareusedinthepracticeofriskTheytaketheirvaluesfromthevalueofsomethingThebuyerofacalloptionhasrighttobuytheunderlyingassetinthefutureundercertainobligationtoselltheunderlyingassetinthefutureundercertainrightrighttosellthe assetinthefutureundercertainAprivateagreementbetweentwopartiestoexchangeaseriesoffuturecashflowswithatleastoneofthetwoseriesofcashflowsdeterminedbyalateroutcome,isbestcharacterizedasa(n):LowcapitalEase TooriskyHighRisk-freearbitrageandno-arbitrageArbitrageinvolvesearningovertherisk-freeratewithnoriskorearninganimmediategainwithnofutureliabilitiesArbitrageopportunities:arbitrageoccurswhenequivalenassetsorcombinationsofassetssellfortwodifferentpricesLawofoneprice:twosecuritiesorportfolios flowsinthefuture,regardlessoffutureevents,shouldhavethesameRisk-freearbitrageandno-arbitrageruleThewayofarbitrage:sellhigh,buyIfaportfolioconsistingofAandBhasacertainpayoff,theportfolioshouldyieldtherisk-freeriskTheroleofarbitrageistoeliminatemispricingandleadtothemarketefficiency.Thatiswhyarbitragealsoplaysaroleinpricing.Whetherthesetworulesbelowcanrestrictthepricediscoverfunctionofthemarket?Restrictsellshort limittheamountof Correctanswer:RiskfreeDefinition:Aforwardcontractisabilateralcontractthatobligatesone tobu andtheother tosellasecific ofanunderlyingasset,atasetprice,onaspecificdateintheLongandshortforwardLong:buyShort:sellNopaymentswillbemadeattheinceptionofaforwardcontract.SobothpartiesofaforwardcontractisexposedtopotentialdefaultriskPurposesoftradingforward期保值賺錢。存在defaultrisk。CharacteristicsofForwardcontractsEachpartyareexposedtodefaultrisk(orcounterpartySettlingaforwardcontractatPhysicalsettlementdeliveranactualasset，存在儲(chǔ)存成本，多Cashsettlementthepartythathasapositionwithnegativevalueisobligatedtopaythatamounttotheotherparty，多用在金融遠(yuǎn)期SettlingaforwardcontractpriortoEnteringintoanoppositeforwardcontract:withanexpirationdateequaltothetimeremainingontheoriginalcontract Offsettingwiththeoriginalparty:canavoidcreditWhichisthemostcommonwaytoterminateaforwardcontractpriortoexpiration?CashEnterintoanoppositeDeliverstheactualCorrectanswer:Howtoeliminatetheriskonaforwardcontract:enteraoppositetradewithsamecounterpartyatsameenteraoppositetradewithdifferentcounterpartyforanyenteraoppositetradewithsamecounterpartyforanyCorrectanswer:LIBOR,Euribor,andEurodollartimeLondonInterbankOfferRateUSDinterestQuotedasanannualizedratesbasedona360-dayaAdd-onSingleEuriborisasimilarrateforborrowingandlendinginAforwardrateagreement(FRA)isaforwardcontractonaninterestrate(LIBOR)FRA定義：AnFRAcanbeviewedasaforwardcontracttoborrow/lendmoneyatacertainrateatsomefuturedate.Thelongposition:isthepartythatwouldborrowtheTheshortposition:isthepartythatwouldlendthe LIBOR,Euribor,andFRAs（續(xù)交割：settleincash,butnoactualloanismadeatthesettlementIfthereferencerateattheexpirationdat isabovethespecifiedcontractrate,thelongwillreceivecashpaymentfromtheIfthereferencerateattheexirationdateisbelowthecontractratetheshortwillreceivecashpaymentfromthelongnotionalnotionalprincipalfloatingrateatsettlementforwardratedays3601+floatingrateatsettlementdays360Real SyntheticTheunderlyingassetofFRAAfuturescontractisanagreementthatobligatesonepartytobuyandtheotherpartytosellaspecificquantityofanunderlyingasset,atasetprice,atafuturedate.CanbeeitherdeliverableorcashsettlementDeliverablecontractsobligatethelongtobuyandtheshorttosellacertainquantityofanassetforacertainpriceonaspecifiedfutureCashsettlementcontractsaresettledbypayingthecontractvalueincashontheexpirationdate.ArepricedtohavezerovalueatthetimeaninvestorentersintotheLittleornoD ltrik SettlementatDailysettlement(marktoNomargindepositFuturescontractsspecifythequalityandquantityofgoodsthatcanbedelivered,thedeliverytimeandthemannerofdelivery.Eachexchangehasaclearinghousethatguaranteesthattradersinthefuturesmarketwillhonortheirobligations.Aclearinghouseactsasthecounterpartytoeachparticipant.Theclearinghouseisthebuyertoeverysellerandthesellertoeverybuyer.ThereisnoneedtoworryaboutthecounterpartydefaultClearinghouseallowseithersideofthetradetoreversepositionsatafuturedate.FuturescontractDailyPriceMarkingtoInitialmargin:Thefirstdepositiscalledtheinitialmargin.Initialmarginmustbepostedbeforeanytradingtakesplace;Maintenancemargin:Ifthemarginbalanceintheaccountfallsbelowthemaintenancemargin,getamargincall

thetraderVariationmargin:usedtobringthemarginbalancebackuptotheinitialmarginlevel.Initialmarin=$5/contractmaintenancemarin=$2/contract 200010220--3--0406503602 Do“margin”inthestockmarketand“margin”inthefuturesmarket,respectively,meanthataninvestorhasreceivedaloanthatreducestheamountofhisownmoneyrequiredtocompletethetransaction?ABCAfuturestradermustkeepthemoneyinthemarginaccountaboveFuturescontract風(fēng)險(xiǎn)控制方法（續(xù)Pricelimitsareexchanged-imposedlimitsonhowmuchthecontractpricecanchangefromthepreviousday’ssettlementprice;Limitmove:Iftraderswishtotradeatpricesoutsidetheselimit---notradeswilltakeplace.---thesettl cewillbereportedupperorlowerpricelimitsLockedlimit:iftradescannottakeplacebecauseofalimitmove,eitherupordown,thepriceissaidtobelockedlimit,sincenotradescantakeplaceandtradersare“l(fā)ocked”intotheirexistingpositions.方法三：Markingtomarket：ThemarginrequirementofafuturescontractislowbecauseattheendofeverydaythereisadailysettlementprocesscalledmarkingtomarketWhichofthefollowingstatementsaboutfuturescontractsisThefuturesclearinghouseallowstraderstoreversetheirpositionswithouthavingtocontracttheothersideoftheinitialtrade.Tosafeguardtheclearinghouse,theexchangerequirestraderstopostmarginandsettletheiraccountsonaweeklybasis.Offsettingtradesratherthanexchangesforphysicalsareusedtoclosemostfuturescontracts.Whichofthe'followingoccursin ettlementoffuturesInitialmargindepositsarerefundedtothetwoGainsandlossesare LossesarechargedtoonepartyandgainscreditedtotheCharacteristicsofSwapSwapcontract:Aswapcontractobligatestwoparti ngeaseriesofcashflowsonperiodicsettlementdatesoveracertaintimeNopaymentrequiredbyeitherpartyatinitiationexcepttheprincipalvaluesexchangedincurrencyswaps.NottradedinanyorganizedsecondaryDefaultriskisacriticalaspectoftheThreeThreetypesofswapcontracts-InterestRateTheplainvanillainterestrateswapinvolvestradingfixedinterestratepaymentsforfloating-ratepayment(payingfixedandreceivingfloating).Counterparties:ThepartiesinvolvedinanyswapagreementarecalledthecounterpartiesPay-fixedside:Thecounterpartythatwantsvariable-rateinterestagreestopayfixed-rateinterest.Pay-floatingside:Thecounterpartythatreceivesthefixedpaymentandagreestopayvariable-rateinterest.AAACorp:wantstoborrowBBBCorp:wantstoborrowAAABBBBasicOption定義：Aoptiongivesitsownertheright,butnottheobligation,tobuyorsellanunderlyingassetonorbeforeafuturedate(theexpirationdate)atapredeterminedprice(theexercisepriceorstrikeprice)Calloption：Longcall&ShortPutoption：Longput&shortThesellerorshortpositioninanoptionscontractissometimesreferredtoasthewriteroftheoption期權(quán)價(jià)格：optionpremiumpaidbythebuyerof執(zhí)行價(jià)格：Strikeprice(X)representstheexercisepricespecifiedinthecontract.Creditdefaultswaps(CDS)isessentiallyaninsurancecontractforthereference,thereferenceobligationisthefixedincomesecurityonwhichtheswapiswritten-usuallyabondbutpotentiallyalsoaloan.Protectionbuyerreceivesapaymentfromtheprotectionsellerifdefaultoccursonthereferenceentity.Theprotectionbuyerpaysthesellerapremium.ThedefaultswappremiumisalsoreferredtoastheCDSspread.Creditspreadoption:acalloptionthatisbasedonabond’syieldspreadrelativetoabenchmark.Ifthebond’screditqualitydecreases,itsyieldspreadwillThebondholderwillcollectapayoffontheInthemoney:ImmediateexercisewouldgenerateapositiveAtthemoney:ImmediateexercisewouldgeneratenoOutofthemoney:ImmediateexercisewouldgeneratenoCallPutS=SCallPutS=S=＜> K K

XX X IntrinsicValue:theamountthatitisinthemoney,andzero utotion:P=max0X-TimeThedifferencebetweenhepriceofanoption(calleditspremium)anditsintrinsicvalueisduetoitstimevalue vaPriceoftheoptionismorevolatilethanpricesofunderlyingWhichofthefollowingstatementsaboutcalloptionsatexpirationisTheprofitpotentialtothebuyeroftheoptionisThecallbuyer'smaximumlossisthecalloption'sAllAlloftheanswersareWhichofthebelowpositionsisthemostrisky,inthesenseofhavingthelargestpotentiallosses?AlongpositionincallAshortpositioninputAshort(written)positionincallConsideraputoptiononDeter,Inc.,withanexercisepriceof$45.ThecurrentstockpriceofDeteris$52.Whatistheintrinsicvalueoftheputoption,andistheputoptionat‐the‐moneyorout‐of‐the‐money?Intrinsic Correctanswer:WhichstatementaboutoptionvaluationisPriortomaturity,out‐of‐the‐moneyoptionshavenoThevalueofanoptionisitstimevalueplusitsintrinsicThebuyerofacalloptioncontractcanneverlosemorethantheinitialCorrectanswer:PutcallPutcall

TcX/1RfT

Sf或cK1RfPositionsre

SCondition scp

Condition pcX/1RfTCondition cpSX/1RfTCondition

pcSX/1RffCondition cpX/1RTfAfiduciarycallisaportfolioconsistingAlongpositioninaEuropeancalloptionwithanexercisepriceofXthatmaturitiesinTyearsonastock.Alongpositioninapure‐discountrisklessbondthatpaysXinTThecostafiduciarycallisthecostofthecall(C0)plusthecostofthebond(thepresentvalueofX).The aofftoaiduciar callwillbeXifthecallisout‐of‐the‐moneyandSTifthecallisin‐the‐money,asshowninthefollowing:STST(Callisout‐ofor (CallisLongcallLongbond STXXXAprotectiveputisaportfolioconsistingofAlongpositioninaEuropeanputoptionwithanexercisepriceofXthatmaturitiesinTyearsonastock.AlongpositionintheunderlyingThecostofaprotectiveputisthecostoftheput(P0)plusthecostofthestock(S0).ThepayofftoaprotectiveputisXiftheputisin‐the‐moneyandSTiftheputisout‐of‐the‐money.asshowninthefollowing:ST<(putisin-the-ST≥(putisout-oforat-the-LongputX-0LongstockXAswithallarbitragetrades,youwantto“buylowandsellhigh.”ifput-callaritdoesn’thold(ifthecostofafiduciarycalldoesnotequalthecostofaprotectiveput),thenyoubuy(golongin)theunderpricedpositionandsell(goshort)intheoverpricedposition.90-dayEuropeancallandputoptionswithastrikepriceof$45ispriced$7.50and$3.70.Theunderlyingispricedat$48andmakesnocashpaymentsduringhlifeofhoptions.Therisk-freerateis5%.Calculatetheno-arbitragepriceofthecalloption,andillustratehowtoearnanarbitrageprofit.C0=P0+S0?X/(1+Rf)T=$3.70+$48?$45/1.0590/365=Sincethecallisweshouldsellthecallfor$7.50andbuythesyntheticcallforTobuythesyntheticcall,buytheputfor$3.70,buytheunderlyingfor$48,andissue(sellshort)a90-dayzero-couponbondwithafacevalueof$45.Thetransactionwillgenerateanarbitrageprofitof$0.26Lowerbound.Theoretically,nooptionwillsellforle thaniintrinsicvalueandnooptioncantakeonanegativevalue.Upperboundforcalloptions.ThemaximumvalueofeitheranAmericanoraEuropeancalloptionatanytimetisthetime-tsharepriceoftheunderlyingstock.UpperboundforputThepriceforanAmericanputoptioncannotbemorethanitsstrikeprice.Themaximumvalueisthepresentvalueoftheexercisepricediscountedattherisk-freerate.MinimumandMaximumOptionValues（公式MinvalueandMaxvalueofoptionswithoutMinMaxMax[0, Max[0, Max[0,X/(1+R)T－t－S X/(1+RfPt≥Max[0,XAmriancall whentheunderlyingmakesnocashpayments,noreasontoexercisethecallearly,C whentheunderlyingmakescashpaymentsduringthelifeoftheoption, exercisecanha enC0>=c0P0>p0,nearlyalwaysaslongasthereisapossibilityofbankruptcy,P0always>(consideranAmericanputonabankruptcompany,stock→0,cannotgoanylower,thenputoptionholdermayexerciseit)AEuropeanstockindexcalloptionhasastrikepriceof$1160andatimetoexpirationof0.25years.Givenarisk-freerateof4%,iftheunderlyingindexistradingat$1,200andhasamultiplierof1,thenthelowerboundfortheoptionpriceisclosestto: ThelowerboundonaEuropeancalliseitherzeroortheunderlyingpriceminusthe resentvalueoftheexercise ricewhicheveris reater.$1200-($1160/1.04^0.25)=$51.32.R58：R58：BasicsofDerivativePricingandArbitrage,replication,andriskForwardMarketsandPriceandFuturesContracts&forwardSwapMarketsandOptionMarketsandBinomialR58.BasicsofDerivativePricingandThepriceisthepredeterminedpriceinthecontractthatthelongshouldpaytotheshorttobu theunderlin assetatthesettlementdateThecontractvalueiszerotobothpartiesatTheno-arbitrae rincile:thereshouldnotbeariskless rofittobe acombinationofaforwardcontractpositionwithpositioninotherasset.Twoassetsorportfolioswithidenticalfuturecashflows,regardlessoffutureevents,shouldhavesamepriceRiskRisk-neutralinvestorsarewillingtobuyriskyinvestmentsforwhichtheyexpecttoearnonlytherisk-freerate.Theydonotexpecttoearnapremiumforbearingrisk.Theexpectedpayoffofthederivativecanbediscountedattherisk-freerate.Andshouldyieldtherisk-freerateofreturn,ifitgeneratescertainpayoffsR58.PricingandPricingaforwardcontractistheprocessofdeterminingtheno-pricethatwillmakethevalueofthecontractbezerotobothsidesatValuationofaforwardcontractmeansdeterminingthevalueofthecontracttothelong(ortheshort)atsometimeduringthelitIfFP>S0×(1+RfAtAtsettlementBorrowS0attherisk-freeUsethemoneytobuyUnderlyingShortaforwardDelivertheunderlyingtotheGetFPfromtheRepay loanamountProfit=FP-S0×(1+RfReverseCash-and-CarryArbitragewhentheForwardContractisIfFP<AtAtsettlement ShortselltheunderlyingbondtogetS0InvestS0attherisk-freeLongaforwardPaytheshortFPtogettheunderlying CloseouttheshortpositionbydeliveringthebondReceiveinvestmentR58.PricingandbuyaT-billtodayatthespotprice(S0 hrt forwardcontractattheforwardprice(FP)FPS0(1Rf)Forwardvalueoflongpositionatinitiation,duringthecontractlife,andatexpirationForwardContractZero,becausethecontractispricedtoprevent S (1R)T (1R)T R58.ForwardPricingandbuyaT-billtodayatthespotprice(S0 hrt forwardcontractattheforwardprice(FP)FPFPS0(1RfTBondR58.PricingandValuationwithcostandForwardcontractsonadividend-payingfFP(S0PVD0)(1Rf4.4ForwardContractsonCouponFPFP(S0PVC0)(1Rf

(StPVCt)

fR)TfCalculatetheno-arbitrageforwardpricefora100-dayforwardonastockthatiscurrentlypricedat$30.00andisexpectedtopayadividendof$0.40in15days,$0.40in85days,and$0.50in175days.Theannualrisk-freerateis5%,andtheyieldcurveisflat.Ignorethedividendin175daysbecauseitoccursafterthematurityoftheforwardcontract.PVD=

=FP=($30-$0.7946)1.05100/365=After60days,thevalueofthestockinthepreviousexampleis$36.00.Calculatethevalueoftheequityforwardcontractonthestocktothelongposition,assumingtherisk-freerateisstill5%andtheyieldcurveisflat.There'sonlyonedividendremaining(in25days)beforethecontractmatures(in40days)asshownbelow,so:

=

(longposition)=（$36-$0.3987)

=R58.PricingandValuationwithcostandIt’smoreeasytoconceiveofderivativethatwouldproduceidenticalpayoffsthanmanyinvestments.Thepayoffsformostderivativescomedirectlyfromthevalueoftheunderlyingattheexpirationofthederivative.ThevalueofthederivativeatexpirationisThepriceofthederivativeistiedtothepriceoftheThederivativecanbeusedtohedgetheBorrowunlimitedamountsofmoneyatrisk-freeGainsfromanoffsettingpositionmightnotbeOnepositioncannotbeperfecthedgedinReplication:replicatethepayoffsononeassetorportfoliowiththoseofadifferentassetorportfolio.Arisk-freeasset(orportfolio)canbecreatedfromapositionintheunderlyingassetthatishedgedwithapositioninaderivativesecurity.Asset+Derivative=Risk-freeAsset-Risk-freeasset=-Derivative-Risk-freeasset=-WhichofthefollowingdoesnotrepresentabenefitofholdinganAnoptimisticexpectedoutlookfortheDividendsiftheassetisastockorinteresiftheasset WhichWhichofthefollowingbestdescribesanarbitrageopportunity?ItisearnariskpremiumintheshortbuyanassetatlessthanitsfundamentalmakeaprofitatnoriskwithnocapitalAninvestorwhorequiresnopremiumtocompensatefortheassumptionofriskissaidtobewhichofthefollowing?RiskRiskRiskWhichofthefollowingbestdescribesthedifferencebetweenthepriceofaforwardcontractanditsvalue?Theforwardpriceisfixedatthestart,andthevaluestartsatzeroandthenThepricedeterminestheprofittothebuyer,andthevaluedeterminestheprofittotheseller.Theforwardcontractvalueisabenchmarkagainstwhichthepriceiscomparedforthepurposesofdeterminingwhetheratradeisadvisable. Whichofthefollowingbestdescribesthevalueoftheforwardcontractatexpiration?Thevalueisthepriceoftheunderlying:minustheforwarddividedbytheforwardPricesofFuturesvs.ForwardIfthecorrelationbetweentheunderlyingassetvalueandinterestrateis…Prefertogolonginafuturescontract,andthefutureswillbegreaterthanthepriceofanotherwiseforwardHavenoPrefertogolonginaforwardcontract,andtheforwardwill reaterthan riceofanotherwisecomfuturesPricingandvaluationofFuturesThevalueofafuturescontractiszeroatcontractFuturescontractsaremarkedtomarketdaily,thevaluejustaftermarkingtomarketisresettozero.Betweenthetimesatwhichthecontractismarkedtomarket,thevaluecanbedifferentfromzero.V(long)=currentfuturesprice?futurespriceatthelastmark-to-marketAnotherviewoffutures:settlepreviousfutures,andthenopenanothernewfutureswithsamedateofmaturity.R58.SwapPricingandAswapcontractisanagreementbetweentwopartiestoexchangeaseriesoffuturecashflows.Therearethreekindsofswaps:interestrateswaps,currencyswapsandequityswaps.Aplainvanillaswapisaninterestrateswapinwhichonepartypaysafixedrateandtheotherpaysafloatingrate.Thetermsofthelongandshortarenotusedhere,insteadwesaythefixed-ratepayerandfloating-rate(variable-rate)payer.Thepriceisjustthefixedrate(calledtheswaprate)thatmakesthecontractvaluezerotobothpartiesatinitiation.Aftersomedaysthemarketsituationchanges,onepartywillmakemoneyandtheotherlosemoney.Thecontractvalueisnolongerzerotobothparties.EquivalenceofswapstoAninterestrateswapisidenticaltoissuingafixed-ratebondandusingtheproceedstobuyafloating-ratebond.Acurrencyswapisidenticaltoissuingafixed-orfloating-ratebondinonecurrency,convertingtheproceedstoanothercurrency,andusingtheproceedstobuyafloating-orfixed-ratebondinanothercurrency.Anequityswapisidenticaltoissuingafixed-orfloating-ratebondandusingtheproceedstobuyastockoranindex.Aforwardcontractisanagreementtoexchangefuturecashflowsonce,soaswapcanbeviewedasaseriesofforwardcontracts.Aninterestrateswap,currencyswapandequityswapareidenticaltoaofFRA,currencyforwardsandequity ards,respctivelyThereare,however,somedifferencesbetweenswapsandAseriesofFRAswillnotallhavethesameforwardrates,unlesstheyieldcurveisflat.Soweoftenrefertoaswapasaseriesofoff-marketInaddition,ininterestrateswaps,thenextpaymentisknownoneperiodahead.ThisisnotthecaseforanFRAs.R58.SwapPricingand1.Aswapisequivalenttoaseriesforwardcontracts,eachcreatedattheswapEachimp dc obe -market,becauseitiscreatedattheswapprice,nottheappropriateforwardprice,whichwouldbethepricecreatedintheforwardmarket.Thepriceofaswapiszeroatfluctuatesoverthelifeoftheisobtainedthrouh rocessofreThevalueofaswapisequaltothepresentvalueoffixedpaymentsfromthenetcashflowpaymentsfromtheunderlyingatheendoftheAbinomialmodelisbasedontheideathat,overthenextperiod,somevaluewillchanetooneoftwo ossiblevaluesbinomial.Toconstructabinomialmodel,weneedtoknowthebeginningassetvalue,thesizeofthetwopossiblechanges,andtheprobabilitiesofeachofthesechangesoccurring.Westartoffbyhavingonlyonebinomialperiod,whichmeansthattheunderlyingpricemovestotwonewpricesatoptionexpiration.WeletS0bethepriceoftheunderlyingstocknow.Oneperiodlater,thestockpricecanmoveuptoS1ordowntoS1.Wethenidentifyafactor,u,astheupmoveonthestockanddasthedownmove.Thus, =

=S0d.Wefurtherassumethat+=S S1Risk-neutralprobabilityofanupmoveisπuRisk-neutralprobabilityofanupmoveisπu;Risk-neutralprobabilityofdownmoveisπd=1-u1 uThusweget:C+=Max(0,S+?X);C?=Max(0,S? valueofanoption:c

C

DeltaCCS

(1R(sharesperExample:Calculatethevaluetodayofa1-yearcalloptiononthestockwiththestrikepriceof$20.Thepriceofthestockis$20now,andthesizeofanup-move1.25.Therisk-freerateisStep1:Calculatetheu=1.25;d=1/u=0.8;Su=20×1.25=25;C+=Max(0,25?20)=5 C?=Max(0,16?20)=Step2:Calculaterisk-neutralprobabilities,πuandπd=1?πuπu=(1+0.07?0.8)/(1.25?0.8)= πd=1?πu=Step3:Drawtheone-periodbinomialSS+=Su=25;C+=5S?=Sd=16;C?=0yearPricingaputoptionissimilartothatofacall.TheonlydifferenceisthatP+=Max(0,X?S+)andP?=Max(0,X?S?).Example:Usetheinformationinthepreviousexampletocalculatethevaluetodayofaputonthesamestockwiththestrikepriceof$20P+=Max(0,20?25)=0; P?=Max(0,20?16)=4P=(0.6×0+0.4×4)/1.07=1.6/1.07=1.50FactorsaffectthevalueofanSensitivityUnderlyingPositivelyNegativelyPositivelyPositivelyRisk‐freePositivelyNegativelyTimetoPositivelyPositivelyStrikeNegativelyPositivelyCarryingPositivelyNegativelyR59：Riskmanagementapplicationsofoption（OptionMarketsandContracts的延伸）★★Call/put/coveredcallprotectiveputvalue/profit/shapeBasicThekeyhereisyourabilitytointerpretoptionpayoffda calculateprofit/lossdiagramsOptionBuyerofacalloption－longWriter(seller)ofacalloption－shortBuyerofaputoption－longWriter(seller)ofaputoption－shortBuyingaValueatexpirationofbuyingacall:max(0,S-Profitfrombuyingacall:valueatexpirationminusoptionpremium,Breakevenunderlyingpriceatexpiration:exercisepriceplusoptionpremium(X+c)Whensellingacall,theseresultsareValueatexpirationofsellingacall:-max0S-Profitfromsellingacall:optionpremiumminusvalueatexpiration,- Breakevenunderlyingpri