ResricionsonOionsrices衍生金融工具人民

Lecture#6:BasicNoArbitrageRestrictionsonOptionsPricesSomeOptionContractsinHongKongHISOptions

OptionsCoveredWarrantsContractsizePricequoteStrikepriceTicksize

50indexunits1indexunit50indexpoints(index<2000)100indexpts(2000index8000)200indexpts(index>8000)400shares1share$2($20shareprice<$50)$5($50shareprice<$200)$10(200shareprice<300)Nopresetformula1/10sharesNopresetformulaHISOptions

OptionsCoveredWarrantsMaturitycycle

Maturityday

Style&SettlementExchangeTradingMethodTradingHoursCurrent&nextmonthplustwomoreinthematuritycycleSecondlastbusinessdayEuropeancashsettlementHKFuturesExchangeOpenoutcry

10:00am-12:30pm2:30pm-4:00pmCurrent&nextmonthplustwomoreinthematuritycycleSecondlastbusinessdayAmericanPhysicaldeliveryHKFuturesExchangeMarketmakers

10:00am-12:30pm2:30pm-4:00pm

EuroorAmericanHKFuturesExchangeMarketmakers

10:00am-12:30pm2:30pm-4:00pm·

IntroductionExactpricingformulasforoptionsaremoredifficulttoderivethanformulasforforwardsandfutures.Toarriveatapricingformulaforstockoptions,whichwewilldoinafewlectures,weneedtomakeassumptionsonthedynamicbehaviorofthepricesoftheunderlyingstock.Inwhatfollowswillderivesomegeneralrestrictionsonstockoptionpricewithoutassumingadynamicmodelforstockpricemovement.Themainpurposeofdoingthatistoimproveourunderstandingofoptioncontracts.Outline:A.

NotationB.

BasicintuitionC.

BasicarbitragerelationsD.

ArbitragebondsonpricesandPut-CallparityE.

EffectsondividendsonarbitragerestrictionsF.

Conclusions·

NotationCurrentdateMaturityorexpirationdatePriceoftheunderlyingassetCurrentpriceofa$1face-valuebondthatmaturesatTExercise(strike)priceValueofaEuropeancallValueofanAmericancallValueofaEuropeanputValueofaAmericanputTTS(t)B(t,T)=e-r(T-t)

K(orX)c(S,K,t,T)C(S,K,t,T)p(S,K,t,T)P(S,K,t,T)·

BasicIntuitionEffectonthepriceofastockoptionofincreasingonevariablewhilekeepingallothersfixed:Variable

EuropeanCall

EuropeanPut

AmericanCall

AmericanPut

StockpriceStrikepriceTimetoexpirationVolatilityRisk-freerateDividends·

Basicarbitragerelations:Note:Thefollowingrestrictionsholdregardlessofwhethertheunderlyingstockpaysdividendsornot.A.

AcallisneverworthmorethanthestockandaputisneverworthmorethanexercisepriceC(S,K,t,T)S(t)c(S,K,t,T)S(t)P(S,K,t,T)Kp(S,K,t,T)KB.

Europeanputsareneverworthmorethanthepresentvalueoftheexerciseprice.p(S,K,t,T)KB(t,T)<K.Intuitively,thishastoholdsincethtime-TpayofftoEuropeanputholderisbounded(fromabove)byK.C.

Optionsneverhasanegativevalue:C(S,K,t,T)0c(S,K,t,T)0P(S,K,t,T)0p(S,K,t,T)0D.

AmericanoptionsareatleastasvaluableasEuropeanoptions:C(S,K,t,T)c(S,K,t,T)P(S,K,t,T)p(S,K,t,T)E.

Americanoptionswithmoretimetomaturityareatleastasvaluable;i.e.,forT2>T1,C(S,K,t,T2)C(S,K,t,T1)P(S,K,t,T2)P(S,K,t,T1)Note:ThisdoesnotalwaysholdforEuropeanoptions.(Why?)F.AnAmericanoptionisworthatleastitsexercisevalue(whatyouwouldgetifyouexercisetoday).C(S,K,t,T)max[0,S(t)-K]P(S,K,t,T)max[0,K-S(t)]Example:Dowehaveanarbitrageopportunityif,forIntelstockwithS(t)=$100,acalloptionwithK=$90and6-monthtomaturityistradingat$9?Note:ThisruledoesnotalwaysholdforEuropeanoptions.(Why?)·MoreArbitrageBoundsforOptionsonNon-Dividend-PayingStocks:Example:Sameasonthepreviouspage.AssumeS(t)=$100,andthepriceofanIntelcallwithK=$90and6-monthtomaturityis$11.AssumethatIntelwillnotpayanydividendwithinthenext6-monthandassumethattheriskfreeinterestrate(a.c.c.)is10%.Isthereanarbitrage?A.Forastockdoesnotpaydividends:c(S,K,t,T)max[0,S(t)-KB(t,T)]C(S,K,t,T)max[0,S(t)-KB(t,T)]Proof:Toprovethisweonlyneedtoshow(why?)c(S,K,t,T)S(t)-KB(t,T)Weshowthisbycontradiction.Ifc<S-KB,wehaveanarbitrage.ThisimpliesthatAmericancallsonnon-dividend-payingstockswillneverbeexercisedearlier.(Intuition?)Anarbitrage:Transaction

Payoff(att)

Payoff(atT)

-cSt-KBMax[0,S(T)-K]-S(T)KS-KB-cMax[0,S(T)-K]-[S(T)-K]B.ForEuropeanputsonnon-dividend-payingstocks,asimilararbitrageargumentshowsthat:Intuition?)p(S,K,t,T)max[0,KB(t,T)-S]C.CombiningtheserulesimpliesthatthevalueofaEuropeancallonanon-dividend-payingstockmustlieintheregion:max[0,S(t)-KB(t,T)]c(S,K,t,T)S(t).0KB(t,T)S(t)D.CombiningtherulesforEuropeanputs,weseethatthevalueofaEuropeanputonanon-dividend-payingstockmustlieintheregion:max[0,KB(t,T)-S(t)]p(S,K,t,T)KB(t,T)K-B(t,T)S(t)E.IsitpossibletoearlyexerciseAmericanPutsonnon-dividend-payingstocks?Intuitions?Example:S(t)=$1,K=$25,T-t=6-month,r=9.5%(a.c.c)·Put-CallParityforNon-dividend-payingstocksA.ForEuropeanoptions:S(t)=c(S,K,t,T)-p(S,K,t,T)+KB(t,T)Intuition:acertainportfolioofbondsandoptionshasthesamepayoffatmaturityasashareofstock,soitmusthavethesamepriceasashareofstock.Example:K=50,S=50,r=0,T-t=1month,c=4.5,p=4.0Sc-p+KBWhatshouldyoudoifthesewerethetrueprices?Transaction

Initial(t)Cash-flow

Final(T)Cash-flowS(T)<50S(T)>50

-$50$4.5-$4.0$50S(T)0$50-S(T)-$50S(T)-[S(T)-$50]0-$50$0.500B.StaticReplicationwithPut-CallParityWecanmakesyntheticstock,call,put,andbondusingthePut-CallParity.ForEuropeanoptionsonanon-dividend-payingstock,wehave:Syntheticstock:S=c-p+PV(K)Syntheticcall:c=S+p-PV(K)Syntheticput:p=c-S+PV(K)Syntheticbond:PV(K)=S-c+pQuestion:HowisthePut-CallParityrelatedtothevalueofaforwardcontractonastock(whosedeliverypriceisequaltothestrikepriceK)?C.Put-CallParityforAmericanOptionsP(S,K,t,T)+S(t)-KBC(S,K,t,T)C(S,K,t,T)P(S,K,t,T)+S(t)-K(Why?)·EffectsofDividendsontheArbitrageRestrictions:Note:Weassumethatthestockwillpayaknowndividend(oraknowndividendyieldinsomecases)beforetheoptionmaturityandextendourpreviousarbitragerestrictions.A.Boundsforoptionsondividendpayingstocks:C(S,K,t,T)c(S,K,t,T)S(t)-PV(D)-KBP(S,K,t,T)p(S,K,t,T)KB-S(t)+PV(D)Intuition:1.Thevalueofa(EuropeanorAmerican)call(orput)ishigherthanthevalueofalong(orshort)positionina"Europeanforward"withstrikeKandmaturityT.Thatis,thevalueofacall(orput)ishigherthanthePVofthecashflowtotheholderofthecall(orput)whoalwaysexerciseitatmaturity.2.Thecashflowofalongpositionina"Europeanforward"withKandTisK-S(T)attimeTandhasaPVofKB-S(t)+PV(D).3.Ifeitherofthesetworulesisviolated,onecanconstructanarbitragebybuyingtheoptionandshortingtherighthandsideoftheinequality.B.EarlyexercisedecisionsofAmericanoptionsAmericancalls1.Givenpositiveinterestrates,itisneveroptimaltoexerciseanAmericancalloptionbetweenex-dividenddatesorpriortomaturity.|||Today(t)Ex-dividendMaturityDate()ofOption(T)Considertwostrategies:i.Exercisetheoptionnow,valueisS(t)-Kii.Waittilljustbeforetheex-dividenddateandexerciseforsure,eveniftheoptionisout-of-the-money.ThevalueofthestrategyatisSc()-K,whereSc()isthecum-stockpricejustbeforethestockgoesdividend.HencethevalueofthisstrategytodayisS(t)-KB(t,).Whatcanyousay?2.Theoptionwillbeexercisedjustpriortotheex-dividenddateif,andonlyif,theexercisevalueexceedsthenoexercisevalue,thatisSe()+d-K>C(Se(),K,,T),whereSe()isex-dividend-daystockprice.AmericanPuts1.AsitispointedoutearlieritmaybeoptimaltoearlyexerciseAmericanputoption,eveniftheunderlyingstockpaysnodividends.2.DividendswilltendtodelayearlyexerciseofanAmericanputoption.ItneverpaystoearlyexerciseanAmericanputoptionjustpriortoanex-dividenddate.Considertwostrategies:i.Exercisetheputoptionjustbeforetheex-divideddate.Thevalueofheoptionis:K-(Se+d)ii.Exercisetheputoptionjustaftertheex-dividenddate.ThevalueisK-Se.C.Put-CallParityforEuropeancallsonstockwithknowndividendS(t)=c(S,K,t,T)-p(S,K,t,T)+KB(t,T)+PV(D)D.What'sthePut-CallParityforEuropeancallsonstockswithacontinuousdividendyieldq?E.Put-CallParityforAmericancallsondividendpayingstocks:S-PV(D)-KC-PS-KBToprovethis,weshowthatneitheroftheinequalitycanbeviolated,byconsideringtwocases:Ifthesecondinequalityisviolated;i.e.,C-P>S-KB,thenwecanhavethefollowingarbitrage:Question:Ifthewrittencallisexercisedagainstyouearlywhatshouldbethevalueofyourportfolio?Ifthefirstinequalityisviolated;i.e.,ifC-P<S-PV(D)-KTransaction

Init