Derivative Markets 2nd Edition- Robert L.Mcdonald

Chapter11BinomialOptionPricing:IICopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-2UnderstandingEarlyExercise•Optionsmayberationallyexercisedpriortoexpiration•Byexercising,theoptionholderreceivesthestockandthusreceivesdividendspaysthestrikepricepriortoexpiration(thishasaninterestcost)losestheinsuranceimplicitinthecallagainstthepossibilitythatthestockpricewillbelessthanthestrikepriceatexpirationCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-3SrKUnderstandingEarlyExercise•Ifvolatilityiszero,thevalueofinsuranceiszero.Then,itisoptimaltodeferexerciseaslongasinterestsavingsonthestrikeexceeddividendslost•Therefore,itisoptimaltoexercisewhenInthespecialcasewhenr=and=0,anyin-the-moneyoptionshouldbeexercisedimmediately•Whenvolatilityispositive,theimplicitinsurancehasvalue,andthevaluevarieswithtimetoexpirationrKSCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-4UnderstandingEarlyExercise(Cont’d)•Thefollowinggraphdisplaystheprice,abovewhichearlyexerciseisoptimalfora5-yearcalloptionwithK=$100,r=5%,and=5%Copyright©2006PearsonAddison-Wesley.Allrightsreserved.11-5UnderstandingEarlyExercise(Cont’d)•Thefollowinggraphdisplaystheprice,abovewhichearlyexerciseisoptimalfora5-yearputoptionwithK=$100,r=5%,and=5%Copyright©2006PearsonAddison-Wesley.Allrightsreserved.11-6UnderstandingRisk-NeutralPricing•Arisk-neutralinvestorisindifferentbetweenasurethingandariskybetwithanexpectedpayoffequaltothevalueofthesurething•p*istherisk-neutralprobabilitythatthestockpricewillgoupCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-7UnderstandingRisk-NeutralPricing(Cont’d)•Theoptionpricingformulacanbesaidtopriceoptionsasifinvestorsarerisk-neutralNotethatwearenotassumingthatinvestorsareactuallyrisk-neutral,andthatriskyassetsareactuallyexpectedtoearntherisk-freerateofreturnCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-8PricinganOptionUsingRealProbabilities•Isoptionpricingconsistentwithstandarddiscountedcashflowcalculations?Yes.However,discountedcashflowisnotusedinpracticetopriceoptionsThisisbecauseitisnecessarytocomputetheoptionpriceinordertocomputethecorrectdiscountrateCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-9pedudhPricinganOptionUsingRealProbabilities(cont’d)•Supposethatthecontinuouslycompoundedexpectedreturnonthestockisandthatthestockdoesnotpaydividends•Ifpisthetrueprobabilityofthestockgoingup,pmustbeconsistentwithu,d,and(11.3)•Solvingforpgivesus(11.4)puS(1p)dSehSCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-10pCpCedudCueudCudhuhd()1PricinganOptionUsingRealProbabilities(cont’d)•Usingp,theactualexpectedpayofftotheoptiononeperiodhenceis(11.5)•Atwhatratedowediscountthisexpectedpayoff?Itisnotcorrecttodiscounttheoptionattheexpectedreturnonthestock,,becausetheoptionisequivalenttoaleveragedinvestmentinthestockandhenceisriskierthanthestockCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-11eSSBeBSBehhrhPricinganOptionUsingRealProbabilities(cont’d)•Denotetheappropriateper-perioddiscountratefortheoptionas•SinceanoptionisequivalenttoholdingaportfolioconsistingofsharesofstockandBbonds,theexpectedreturnonthisportfoliois(11.6)Copyright©2006PearsonAddison-Wesley.Allrightsreserved.11-12CeedudCueudChhuhdPricinganOptionUsingRealProbabilities(cont’d)•Wecannowcomputetheoptionpriceastheexpectedoptionpayoff,discountedattheappropriatediscountrate,givenbyequation(11.6).Thisgives(11.7)Copyright©2006PearsonAddison-Wesley.Allrightsreserved.11-13PricinganOptionUsingRealProbabilities(cont’d)•Itturnsoutthatthisgivesusthesameoptionpriceasperformingtherisk-neutralcalculationNotethatitdoesnotmatterwhetherwehavethe“correct”valueoftostartwithAnyconsistentpairofandwillgivethesameoptionpriceRisk-neutralpricingisvaluablebecausesetting=rresultsinthesimplestpricingprocedure.Copyright©2006PearsonAddison-Wesley.Allrightsreserved.11-14TheBinomialTreeandLognormality•Theusefulnessofthebinomialpricingmodelhingesonthebinomialtreeprovidingareasonablerepresentationofthestockpricedistribution•ThebinomialtreeapproximatesalognormaldistributionCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-15ZYniin1TheRandomWalkModel•Itisoftensaidthatstockpricesfollowarandomwalk•ImaginethatweflipacoinrepeatedlyLettherandomvariableYdenotetheoutcomeoftheflipIfthecoinlandsdisplayingahead,Y=1;otherwise,Y=–1Iftheprobabilityofaheadis50%,wesaythecoinisfairAfternflips,withtheithflipdenotedYi,thecumulativetotal,Zn,is(11.8)•Itturnsoutthatthemoretimesweflip,onaveragethefartherwewillmovefromwherewestartedCopyright©2006PearsonAddison-Wesley.Allrightsreserved.11-16TheRandomWalkModel(cont’d)•WecanrepresenttheprocessfollowedbyZnintermofthechangeinZnZn–Zn-1=YnorHeads:Zn–Zn-1=+1Tails:Zn–Zn-1=–1Copyright©2006PearsonAddison-Wesley.Allrightsreserved.11-17TheRandomWalkModel(cont’d)•Arandomwalk,wherewithheads,thechangeinZis1,andwithtails,thechangeinZis–1Copyright©2006PearsonAddison-Wesley.Allrightsreserved.11-18TheRandomWalkModel(cont’d)•Theideathatassetpricesshouldfollowarando