Lecture_8BinomialOptionPricing(衍生金融工具-人民

Lecture8:BinomialOptionPricingWehavederivedupperandlowerboundsforoptionsbyusingsimplenoarbitragearguments.Althoughtheseboundslimitthepriceoftheoption,thedifferenceoftheupperandlowerboundscanbequitelarge.Forexample,consideraEuropeancalloptionwithstrikepriceof100,maturitydateinsixmonths,andwheretheunderlyingassetpriceis100.Weknowtheoptionpricemustbeintherangeof2.96and100,assumingtheinterestrateof6%.Topriceoptionsmoreprecisely,wemustmakeadditionalassumptionsabouttheprobabilitydistributiondescribingthepossiblepricechangesintheunderlyingasset.Thepurposeofthislectureistostudyamodelofassetprice.Basicassumptions1.Assumethatthestockpricecantakeoneoftwopossiblevaluesattheendofoneperiod.2.Thereexistsarisk-freesecurity3.Therearenoarbitrageopportunities4.ThereisnointerestrateuncertaintyTheimportanceofbinomialpricemodel1.Ityieldsimportantinsightsintothepricingandhedgingallderivatives.2.Thebasiclogicofthisapproachissimilartothelogicofthemajorityderivativesecuritymodelsinusetoday.3.Ifshortrateisaconstant,undersomeconditions,thebinomialmodelofstockpricewillconvergetothestockpricedynamicsusedtodriveBlack-Scholesoptionpricingformula.4.Binomialtreecanbeusedtomodelstockpricedynamicswhenthevolatilityisafunctionofstockprice.5.Inthepractice,abinomialtreemaybeestimatedorconstructedbasedonsimpleoptions(impliedbinomialtree)andthenitisusedtopriceexoticoptions.6.Innumericalanalysis,binomialtreeapproachisoneofthemostimportanttools.Generalassetspricingmethods1.Generalequilibriumapproach:itisusedtopricebasicassets.Youhavetoconsiderinvestors/consumers'utilityfunctions,producers'productionfunctions.Theassetpricesaredeterminedbymarketequilibriumconditionsthroughindividualstomaximizetheirobjectivefunctions.1.No-arbitrageapproach:Itisusedtopricederivatives.Ingeneral,youtakethebasicassets'pricesasgivenandthinkthatthepayoffsfromaderivativecanbeduplicatedbypayoffsofaportfolioofbasicassets.Thusthepriceofderivativeisthepriceoftheportfolioofbasicasset.2.No-arbitrageapproachcommonlyisusedunderassumptionofcompletemarket.Whenmarketisincompleteyoumayhavetouseequilibriumapproachevenforderivatives.Anexampleofthisisthestochasticvolatilityoftheunderlyingasset.NotationsS:currentstockpricer:riskfreerate(a.c.c)u:upwardmovementfactorinassetpriceovertimeintervalt.d:downwardmovementfactorinassetpriceovertimeintervalt.q:probabilityofupwardmovementinassetprice.X:exercisepriceofoptionT:timetomaturityofoption(currenttimeis0)t:lengthoftimeinterval.:volatility=ln(u)/(t)0.5Forconvenience,wesometimesuseu=1/dandimposetherestriction:dertuOne-periodbinomialgeneratingprocessa.StockpricedynamicsSu(22)S(20)Sd(18)a.Optionpricedynamicsfuffdfcanbeacall,aputoranotherderivativesecurity.Thecalloptiondynamics(X=21)cu=max[Su-X,0](1)ccd=max[Sd-X,0](0)Theputoptiondynamicspu=max[X-Su,0](0)ppd=max[X-Sd,0](3)Derivingthebinomialoptionpricingmodela.Consideraportfolioconsistingof:sharesofstockSoneshortcalloptiononstockSb.Weconstructaportfolioinawaytoensurethatitsvalueatthematurityoftheoptionremainsconstantirrespectiveofthestockprice.Su-cu=Sd-cdc.Solvingfor,thehedgeratio,toyield=(cu-cd)/[S(u-d)]Thehedgeportfolio'spayoffatmaturityisknownbeforehand.Therefore,theportfolio'sprice(today)isequaltothepresentvalueofitspayoffdiscountedattheriskfreerateofinterest:S-c=(Su-cu)e-rt.Thisimpliesc=S-(Su-cu)e-rt=[q*cu+(1-q*)cd]e-rt,whereq*=(ert-d)/(u-d)Ex:S=20,u=1.1,d=0.9,r=12%,t=0.25.Wehave22-1=18,=0.25,(Su-cu)=4.5,S-c=4.367,q*=0.6523,c=0.633.Risk-neutralvaluation:a.Theoptionpricedoesnotdependuponq.Itinsteadreliesupontheriskneutralprobability(equivalentmartingalemeasure)q*.b.Underthissetup,thecurrentstockpriceequalstheexpectedfuturevaluediscountedattheriskfreerate,i.e.,S==[q*Su+(1-q*)Sd]e-rt.Thisimpliesthattheoptionpriceisindependentoftheexpectedrateofreturnoftheunderlyingasset.a.Thishasimportantimplication.Considertwoindividuals,oneanoptimisticandtheotherapessimist.Theoptimistic(pessimist)believesthattheprobabilitythestockpricegoingupis90%(10%).Providedthesetwoagreesthatthestockpricetodayis20,andthestockpriceintheupstateis22andthatthestockpriceinthedownstateis18,thentheybothwillagreethatthetradedoption'svaluetodayis0.633.ReplicationapproachIntheaboveapproach,thecombinationofastockandacallreplicatedarisk-freeasset.Themorenaturalwayistothinkthatastockandabond(therisk-freeasset)canreplicatethepayoffsofoptions.Basedontheaboveexample,ifweinvestonedollarinriskfreeassettodaywewillgete12%*0.25threemonthslatter.Supposewebuym0sharesofstockandinvestb0dollarsintherisk-freeasset.ThevalueoftheportfolioisV(0)=m020+b0.Butwhatmustm0andb0betomimicthepayoffsofthecalloption?m022+b0e12%*0.25=1(why?)m018+b0e12%*0.25=0(why?)Canwedesignaportfoliotosatisfytheaboveconditions?Ingeneral,theanswerisyes.(why?)m0=1/(22-18)=0.25=b0=(1-0.25*22)e-12%*0.25=-4.367V(0)=0.25*20-4.367=0.633Whatshouldbethevalueofthetradedcalloption?Supposetradedoptionispricedat0.7whatcanwedo?Supposetradedoptionispricedat0.6whatshouldwedo?Optiondelta(hedgeratio)Theoptiondelta(hedgeratio),,representstheslopeofthecallorputoption