RestrictionsonOptionsPrices(衍生金融工具-人民

Lecture#6:BasicNoArbitrageRestrictionsonOptionsPricesSomeOptionContractsinHongKongHISOptionsOptionsCoveredWarrantsContractsizePricequoteStrikepriceTicksize50indexunits1indexunit50indexpoints(index2000)100indexpts(2000index8000)200indexpts(index8000)400shares1share$2($20shareprice$50)$5($50shareprice$200)$10(200shareprice300)Nopresetformula1/10sharesNopresetformulaHISOptionsOptionsCoveredWarrantsMaturitycycleMaturitydayStyle&SettlementExchangeTradingMethodTradingHoursCurrent&nextmonthplustwomoreinthematuritycycleSecondlastbusinessdayEuropeancashsettlementHKFuturesExchangeOpenoutcry10:00am-12:30pm2:30pm-4:00pmCurrent&nextmonthplustwomoreinthematuritycycleSecondlastbusinessdayAmericanPhysicaldeliveryHKFuturesExchangeMarketmakers10:00am-12:30pm2:30pm-4:00pmEuroorAmericanHKFuturesExchangeMarketmakers10:00am-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:VariableEuropeanCallEuropeanPutAmericanCallAmericanPutStockpriceStrikepriceTimetoexpirationVolatilityRisk-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.,forT2T1,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.IfcS-KB,wehaveanarbitrage.ThisimpliesthatAmericancallsonnon-dividend-payingstockswillneverbeexercisedearlier.(Intuition?)Anarbitrage:TransactionPayoff(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=4.0Sc-p+KBWhatshouldyoudoifthesewerethetrueprices?TransactionInitial(t)Cash-flowFinal(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-CallParityrelatedtothevalueofaforwardc