CorporateFinanceFifthEditionChapter21OptionValuationCopyright?2020,2017,2014PearsonEducation,Inc.

AllRightsReservedChapterOutline21.1

TheBinomialOptionPricingModel21.2

TheBlack-ScholesOptionPricingModel21.3Risk-NeutralProbabilities21.4RiskandReturnofanOption21.5CorporateApplicationsofOptionPricingLearningObjectives(1of4)IllustratetheuseoftheBinomialOptionPricingModeltovalueanoption.DefinethereplicatingportfoliofortheBinomialOptionPricingModel.UsetheLawofOnePricetoexplainhowtheBinomialOptionPricingModelprovidesthecorrectvalueundertheassumptionsmadebythemodel.LearningObjectives

(2of4)UsetheBlack-ScholesOptionPricingformulatocalculatethevalueofacalloptiononanon-dividend-payingstock.UsetheBlack-ScholesOptionPricingformulatocalculatethepriceofaEuropeanputoptiononanon-dividendpayingstock.ComputethevalueofaEuropeanoptiononadividend-payingstock.LearningObjectives(3of4)DefinetheBlack-Scholesreplicatingportfolioforacalloptiononanon-dividendpayingstockoraEuropeanputoptiononanon-dividendpayingstock.Discusswhatismeantbyrisk-neutralprobabilities,andshowhowtheseprobabilitiescanbeusedtopriceanyotherassetforwhichthepayoffsineachstateareknown.LearningObjectives(4of4)Defineandcalculatetherisk-neutralprobabilitythatthestockpricewillincreaseinabinomialtree.Calculateandinterpretthebetaofanoption.UsetheBlack-Scholesformulatounlevertheequitybetaofafirmandfindthebetaofdebt.21.1TheBinomialOptionPricingModelBinomialOptionPricingModelAtechniqueforpricingoptionsbasedontheassumptionthateachperiod,thestock’sreturncantakeononlytwovaluesBinomialTreeAtimelinewithtwobranchesateverydaterepresentingthepossibleeventsthatcouldhappenatthosetimesATwo-StateSingle-PeriodModel(1of9)ReplicatingPortfolioAportfolioconsistingofastockandarisk-freebondthathasthesamevalueandpayoffsinoneperiodasanoptionwrittenonthesamestockTheLawofOnePriceimpliesthatthecurrentvalueofthecallandthereplicatingportfoliomustbeequal.ATwo-StateSingle-PeriodModel(2of9)AssumeAEuropeancalloptionexpiresinoneperiodandhasanexercisepriceof$50.Thestockpricetodayisequalto$50andthestockpaysnodividends.Inoneperiod,thestockpricewilleitherriseby$10orfallby$10.Theone-periodrisk-freerateis6%.ATwo-StateSingle-PeriodModel(3of9)Thepayoffscanbesummarizedinabinomialtree.ATwo-StateSingle-PeriodModel(4of9)LetΔbethenumberofsharesofstockpurchased,andletBbetheinitialinvestmentinbonds.Tocreateacalloptionusingthestockandthebond,thevalueoftheportfolioconsistingofthestockandbondmustmatchthevalueoftheoptionineverypossiblestate.ATwo-StateSingle-PeriodModel(5of9)Intheupstate,thevalueoftheportfoliomustbe$10.Inthedownstate,thevalueoftheportfoliomustbe$0.ATwo-StateSingle-PeriodModel(6of9)Usingsimultaneousequations,canbesolvedforATwo-StateSingle-PeriodModel(7of9)Aportfoliothatislong0.5shareofstockandshortapproximately$18.87worthofbondswillhaveavalueinoneperiodthatexactlymatchesthevalueofthecall.ATwo-StateSingle-PeriodModel(8of9)BytheLawofOnePrice,thepriceofthecalloptiontodaymustequalthecurrentmarketvalueofthereplicatingportfolio.Thevalueoftheportfoliotodayisthevalueof0.5sharesatthecurrentsharepriceof$50,lesstheamountborrowed.ATwo-StateSingle-PeriodModel(9of9)NotethatbyusingtheLawofOnePrice,weareabletosolveforthepriceoftheoptionwithoutknowingtheprobabilitiesofthestatesinthebinomialtree.Figure21.1ReplicatinganOptionintheBinomialModelTheBinomialPricingFormula(1of3)AssumeSisthecurrentstockprice,andSwilleithergouptonextperiod.Therisk-freeinterestrateis

isthevalueofthecalloptionifthestockgoesup,and

isthevalueofthecalloptionifthestockgoesdown.TheBinomialPricingFormula(2of3)Giventheaboveassumptions,thebinomialtreewouldlooklikethefollowing:Thepayoffsofthereplicatingportfolioscouldbewrittenasfollows:TheBinomialPricingFormula(3of3)Solvingthetworeplicatingportfolioequationsforthetwounknownsyieldsthegeneralformulaforthereplicatingformulainthebinomialmodel.ReplicatingPortfoliointheBinomialModelThevalueoftheoptionisOptionPriceintheBinomialModelTextbookExample21.1(1of3)ValuingaPutOptionProblemSupposeastockiscurrentlytradingfor$60,andinoneperiodwilleithergoupby20%orfallby10%.Iftheone-periodrisk-freerateis3%,whatisthepriceofaEuropeanputoptionthatexpiresinoneperiodandhasanexercisepriceof$60?TextbookExample21.1(2of3)SolutionWebeginbyconstructingabinomialtree:Thus,wecansolveforthevalueoftheputbyusingEq.21.5andEq.21.6with(thevalueoftheputwhenthestockgoesup)andthevalueoftheputwhenthestockgoesdown).Therefore,TextbookExample21.1(3of3)Thisportfolioisshort0.3333sharesofthestock,andhas$23.30investedintherisk-freebond.Let’scheckthatitreplicatestheputifthestockgoesupordown:Thus,thevalueoftheputistheinitialcostofthisportfolio.UsingEq.21.6:AlternativeExample21.1A(1of2)ProblemAssumeCLWstockhasacurrentpriceof$24.Theone-yearrisk-freerateis5%.Attheendofoneyear,CLWwilleitherbetradingat$21or$31.CLWInc.doesnotpaydividends.WhatisthepriceofaEuropeanputoptionthatexpiresinoneyearandhasastrikepriceof$24?AlternativeExample21.1(2of2)SolutionIfthestockpricerisesto$31,theputoptionwillbeworthlessIfthestockpricefallsto$21,theputoptionwillbeworthAMultiperiodModel(1of10)Consideratwo-periodbinomialtreeforthestockprice.AMultiperiodModel(2of10)Tocalculatethevalueofanoptioninamultiperiodbinomialtree,startattheendofthetreeandworkbackward.Attime2,theoptionexpires,soitsvalueisequaltoitsintrinsicvalue.Inthiscase,thecallwillbeworth$10ifthestockpricegoesupto$60andwillbeworthzerootherwise.AMultiperiodModel(3of10)AMultiperiodModel(4of10)Thenextstepistodeterminethevalueoftheoptionineachpossiblestateattime1.Ifthestockpricehasgoneupto$50attime1,thebinomialtreewilllooklikethis:Thevalueoftheoptionwillbe$6.13(justasinthesingleperiodexample).

AMultiperiodModel(5of10)Thenextstepistodeterminethevalueoftheoptionineachpossiblestateattime1.Ifthestockpricehasgonedownto$30attime1,thebinomialtreewilllooklikethis:Thevalueoftheoptionwillbe$0becauseitisworth$0ineitherstate.AMultiperiodModel(6of10)Thefinalstepistodeterminethevalueoftheoptionineachpossiblestateattime0.SolvingforAMultiperiodModel(7of10)Thefinalstepistodeterminethevalueoftheoptionineachpossiblestateattime0.Thus,theinitialoptionvalueisAMultiperiodModel(8of10)DynamicTradingStrategyAreplicationstrategybasedontheideathatanoptionpayoffcanbereplicatedbydynamicallytradinginaportfoliooftheunderlingstockandarisk-freebond.AMultiperiodModel(9of10)DynamicTradingStrategy.Inthetwo-periodexampleonthepreviousslide,thereplicatingportfoliowillneedtobeadjustedattheendofeachperiod.Theportfoliostartsofflong0.3065sharesofstockandborrowing$8.67.Ifthestockpricedropsto$30,thesharesareworth$9.20,andthedebthasgrownto$9.20.Thenetvalueoftheportfolioisworthless,andtheportfolioisliquidated.AMultiperiodModel(10of10)DynamicTradingStrategyIfthestockpricerisesto$50,thenetvalueoftheportfoliorisesto$6.13.ThenewDofthereplicatingportfoliois0.5.Therefore0.1935moresharesmustbepurchased.

Thepurchasewillbefinancedbyadditionalborrowing.

Attheendthetotaldebtwillbe$18.87.

Thismatchesthevaluecalculatedpreviously.TextbookExample21.2(1of5)UsingtheBinomialOptionPricingModeltoValueaPutOptionProblemSupposethecurrentpriceofNarverNetworkSystemsstockis$50pershare.Ineachofthenexttwoyears,thestockpricewilleitherincreaseby20%ordecreaseby10%.The3%one-yearrisk-freerateofinterestwillremainconstant.Calculatethepriceofatwo-yearEuropeanputoptiononNarverNetworkSystemsstockwithastrikeprice$60.TextbookExample21.2(2of5)SolutionHereisthebinomialtreeforthestockprice,togetherwiththefinalpayoffsoftheputoption:Ifthestockgoesupto$60attime1,weareinexactlythesamesituationasinExample21.1.Usingourresultthere,weseethatifthestockisworth$60attime1,thevalueoftheputoptionis$3.30.TextbookExample21.2(3of5)Ifthestockgoesdownto$45attime1,thenattime2theputoptionwillbewortheither$6ifthestockgoesupor$19.50ifthestockgoesdown.UsingEq.21.5:TextbookExample21.2(4of5)Thisportfolioisshortoneshareofthestock,andhas$58.25investedintheriskrisk-freebond.Becausethevalueofthebondwillgrowtoattime2,thevalueofthereplicatingportfoliowillbe$60lessthefinalpriceofthestock,matchingthepayoffoftheputoption.Thus,thevalueoftheputisthecostofthisportfolio.UsingEq.21.6:Nowconsiderthevalueoftheputoptionattime0.Inperiod1,wehavecalculatedthattheputwillbeworth$3.30ifthestockgoesupto$60and$13.25ifthestockfallsto$45.Thebinomialtreeattime0isTextbookExample21.2(5of5)UsingEq.21.5andEq.21.6,thereplicatingportfolioandputvalueattime0areThus,thevalueofaEuropeanputoptionattime0is$8.68.MakingtheModelRealisticAlthoughbinaryupordownmovementsarenotthewaystockpricesbehaveonanannualorevendailybasis,bydecreasingthelengthofeachperiod,andincreasingthenumberofperiodsinthestockpricetree,arealisticmodelforthestockpricecanbeconstructed.Figure21.2ABinomialStockPricePath21.2TheBlack-ScholesOptionPricingModelBlack-ScholesOptionPricingModelAtechniqueforpricingEuropean-styleoptionswhenthestockcanbetradedcontinuously.ItcanbederivedfromtheBinomialOptionPricingModelbyallowingthelengthofeachperiodtoshrinktozeroandlettingthenumberofperiodsgrowinfinitelylarge.TheBlack-ScholesFormula(1of7)Black-ScholesPriceofaCallOptiononaNon-Dividend-PayingStockWhereSisthecurrentpriceofthestock,Kistheexerciseprice,andN(d)isthecumulativenormaldistributionCumulativeNormalDistribution.TheprobabilitythatanoutcomefromastandardnormaldistributionwillbebelowacertainvalueTheBlack-ScholesFormula(2of7)Whereistheannualvolatility,andTisthenumbernumberofyearslefttoexpiration.Figure21.3NormalDistributionTheBlack-ScholesFormula(3of7)Note:OnlyfiveinputsareneededfortheBlack-Scholesformula.StockpriceStrikepriceExercisedateRisk-freerateVolatilityofthestockTextbookExample21.3(1of3)ValuingaCallOptionwiththeBlack-ScholesFormulaProblemJetBlueAirwaysdoesnotpaydividends.UsingthedatainTable21.1,comparethepriceonJuly24,2009,fortheDecember2009AmericancalloptiononJetBluewithastrikepriceof$6tothepricepredictedbytheBlack-Scholesformula.AssumethatthevolatilityofJetBlueis65%peryearandthattherisk-freerateofinterestis1%peryear.Table21.1JetBlueOptionQuotesSource:ChicagoBoardOptionsExchangeatTextbookExample21.3(2of3)SolutionWeuse$5.03(theclosingprice)fortheper-sharepriceofJetBluestock.BecausetheDecembercontractexpiresontheSaturdayfollowingthethirdFridayofDecember(December19),thereare148daysleftuntilexpiration.ThepresentvalueofthestrikepriceisCalculatingd1andd2fromEq.21.8givesTextbookExample21.3(3of3)Substitutingd1andd2

intotheBlack-ScholesformulagivenbyEq.21.7resultsinInTable21.1,thebidandaskpricesforthisoptionare$0.45and$0.55.Figure21.4Black-ScholesValueonJuly24,2009,oftheDecember2009$6CallonJetBlueStockAlternativeExample21.3(1of2)ProblemAssumeC

L

WInc.doesnotpaydividends.ThestandarddeviationofC

L

Wis45%peryear.Therisk-freerateis5%.C

L

Wstockhasacurrentpriceof$24.UsingtheBlack-Scholesformula,whatisthepricefora?yearAmericancalloptiononC

L

Wwithastrikepriceof$30?AlternativeExample21.3(2of2)SolutionTheBlack-ScholesFormula(4of7)EuropeanPutOptionsBlack-ScholesPriceofaEuropeanPutOptiononaNon-Dividend-PayingStockUsingput-callparity,thevalueofaEuropeanputoptionisTextbookExample21.4(1of3)ValuingaPutOptionwiththeBlack-ScholesFormulaProblemUsingtheBlack-ScholesformulaandthedatainTable21.1,computethepriceofaJanuary2010$5putoptionandcompareittothepriceinthemarket.IstheBlack-Scholesformulathecorrectwaytopricetheseoptions?(Asbefore,assumethatthevolatilityofJetBlueis65%peryearandthattherisk-freerateofinterestis1%peryear.)TextbookExample21.4(2of3)SolutionThecontractexpiresonJanuary16,2010,or176daysfromthequotedate.ThepresentvalueofthestrikepriceisCalculatingfromEq.21.8givesTextbookExample21.4(3of3)Substitutingd1andd2intotheBlack-Scholesformulaforaputoption,usingEq.21.9,givesGiventhebidandaskpricesof$0.85and$0.95,respectively,fortheoption,thisestimateiswithinthebid-askspread.ButtheBlack-ScholesformulaforputsisvalidforEuropeanoptions,andthequotesareforAmericanoptions.Hence,inthiscase,theBlack-Scholesoptionpriceisalowerboundontheactualvalueoftheput,asanAmericanputmightbeexercisedearlytobenefitfrominterestonthestrikeprice.However,giventhatinterestonthe$5strikepriceislessthan$0.03,inthiscasetheapproximationisacloseone.Figure21.5Black-ScholesValueonJuly24,2009,oftheJanuary2010$5PutonJetBlueStockAlternativeExample21.4(1of2)ProblemAssumeC

L

WInc.doesnotpaydividends.ThestandarddeviationofC

L

Wis45%peryear.Therisk-freerateis5%.C

L

Wstockhasacurrentpriceof$24.UsingtheBlack-Scholesformula,whatisthepricefora?yearAmericanputoptiononC

L

Wwithastrikepriceof$30?AlternativeExample21.4(2of2)SolutionTheBlack-ScholesFormula(5of7)Dividend-PayingStocksIfP

V(Div)isthepresentvalueofanydividendspaidpriortotheexpirationoftheoption,thenWhereisthepriceofthestockexcludinganydividends.TheBlack-ScholesFormula(6of7)Dividend-PayingStocksBecauseaEuropeancalloptionistherighttobuythestockwithoutthesedividends,itcanbeevaluatedbyusingtheBlack-ScholesformulawithTheBlack-ScholesFormula(7of7)Dividend-PayingStocksAspecialcaseiswhenthestockwillpayadividendthatisproportionaltoitsstockpriceatthetimethedividendispaid.Ifqisthestock’s(compounded)dividendyielduntiltheexpirationdate,thenTextbookExample21.5(1of3)ValuingaDividend-PayingEuropeanCallOptionwiththeBlack-ScholesFormulaProblemWorldWidePlantswillpayanannualdividendyieldof5%onitsstock.Plotthevalueofaone-yearEuropeancalloptionwithastrikepriceof$20onWorldWidePlantsstockasafunctionofthestockprice.AssumethatthevolatilityofWorldWidePlantsstockis20%peryearandthattheone-yearrisk-freerateofinterestis4%.TextbookExample21.5(2of3)SolutionThepriceofthecallisgivenbythestandardBlack-Scholesformula,Eq.21.7,butwiththestockpricereplacedthroughoutwithForexample,withastockpriceof$30,TextbookExample21.5(3of3)Theplotbelowshowsthevalueofthecall(inred)fordifferentlevelsofthestockprice.Whenthestockpriceissufficientlyhigh,thecallisworthlessthanitsintrinsicvalue.AlternativeExample21.5(1of2)ProblemF

P

A,Inc.hasacurrentstockpriceof$42.40pershare.Thecompanywillpayanannualdividendyieldof6%onitsstock.IfF

P

A’sreturnshaveastandarddeviationof25%andtherisk-freerateis4%,whatisthevalueofaone-yearcalloptiononthestockwithastrikepriceof$40?AlternativeExample21.5(2of2)SolutionThepriceofthecallcanbefoundusingtheBlack-Scholesmodelwherethestockpriceisadjustedforthedividendyield:Thus,ImpliedVolatilityOfthefiverequiredinputsintheBlack-Scholesformula,onlysisnotobservabledirectly.PractitionersusetwostrategiestoestimatethevalueofUsehistoricaldata“Backout”theimpliedvolatilityImpliedvolatilityThevolatilityofanasset’sreturnsthatisconsistentwiththequotedpriceofanoptionontheassetTextbookExample21.6(1of3)ComputingtheImpliedVolatilityfromanOptionPriceProblemUsethepriceoftheMarch2010callonJetBluewithastrikepriceof$5inTable21.1tocalculatetheimpliedvolatilityforJetBluefromJuly2009toMarch2010.Assumetherisk-freerateofinterestis1%peryear.TextbookExample21.6(2of3)SolutionThecallexpiresonMarch20,2010,or239daysafterthequotedate.Thestockpriceis$5.03,andSubstitutingthesevaluesintotheBlack-Scholesformula,Eq.21.7,givesTextbookExample21.6(3of3)WecancomputetheBlack-ScholesoptionvalueCfordifferentvolatilitiesusingthisequation.TheoptionvalueCincreaseswithandequals$1.10(averagebidandaskpriceforthecall).When(YoucanfindthisvaluebytrialanderrororbyusingExcel’sSolvertool.)Ifwelookatthebidpriceof$1.05,theimpliedvolatilityisabout64%,andattheaskpriceof$1.15,theimpliedvolatilityisabout70%.Thus,the65%volatilityweusedinExample21.3andExample21.4iswithinthebid-askspreadfortheoption.TheReplicatingPortfolio(1of4)GivenOptionPriceintheBinomialModelThenBlack-ScholesReplicatingPortfolioofaCallOptionTheReplicatingPortfolio(2of4)OptionDeltaThechangeinthepriceofanoptiongivena$1changeinthepriceofthestock.Thenumberofsharesinthereplicatingportfoliofortheoption.Note:Becausethechangeinthecallpriceisalwayslessthanthechangeinthestockprice.TextbookExample21.7(1of3)ComputingtheReplicatingPortfolioProblemPNASystemspaysnodividendsandhasacurrentstockpriceof$10pershare.Ifitsreturnshaveavolatilityof40%andtherisk-freerateis5%,whatportfoliowouldyouholdtodaytoreplicateaone-yearat-the-moneycalloptiononthestock?TextbookExample21.7(2of3)SolutionWecanapplytheBlack-ScholesformulawithS=10,TextbookExample21.7(3of3)FromEq.21.12,thereplicatingportfoliofortheoptionisThatis,weshouldbuy0.626sharesoftheP

N

Astock,andborrow$4.47,foratotalcostofwhichistheBlack-Scholesvalueofthecalloption.Figure21.6ReplicatingPortfoliofortheCallOptioninExample21.7TheReplicatingPortfolio(3of4)Note:Thereplicatingportfolioofacalloptionalwaysconsistsofalongpositioninthestockandashortpositioninthebond.Thereplicatingportfolioisaleveragedpositioninthestock.Aleveragedpositioninastockisriskierthanthestockitself,whichimpliesthatcalloptionsonapositivebetastockaremoreriskythantheunderlyingstockandthereforehavehigherreturnsandhigherbetas.TheReplicatingPortfolio(4of4)Thereplicatingportfolioofaputoptioniscalculatedasfollows:Black-ScholesReplicatingPortfolioofaPutOptionNote:

Thereplicatingportfolioofaputoptionalwaysconsistsofalongpositioninthebondandashortpositioninthestock,implyingthatputoptionsonapositivebetastockwillhaveanegativebeta.21.3Risk-NeutralProbabilitiesIfallmarketparticipantsareriskneutral,thenallfinancialassets(includingoptions)havethesamecostofcapital,therisk-freerateofinterest.ARisk-NeutralTwo-StateModel(1of4)Assumeaworldconsistingofonlyrisk-neutralinvestors,andconsidertheoriginaltwo-stateexample.Thestockpricetodayisequalto$50.Inoneperioditwilleithergoupby$10orgodownby$10.Theone-periodrisk-freerateofinterestis6%.ARisk-NeutralTwo-StateModel(2of4)Ifistheprobabilitythatthestockpricewillincrease,thenistheprobabilitythatitwillgodown.Thevalueofthestocktodaymustequalthepresentvalueoftheexpectedpricenextperioddiscountedattherisk-freerate.SolvingforARisk-NeutralTwo-StateModel(3of4)Thecalloptionhadanexercisepriceof$50,soitwillbewortheither$10ornothingatexpiration.Thepresentvalueoftheexpectedpayoutsisasfollows:ARisk-NeutralTwo-StateModel(4of4)ThisispreciselythevaluecalculatedusingtheBinomialOptionPricingModelwhereitwasnotassumedthatinvestorswereriskneutral.BecausenoassumptionontheriskpreferencesofinvestorsisnecessarytocalculatetheoptionpriceusingeithertheBinomialModelortheBlack-Scholesformula,themodelsworkforanysetofpreferences,includingrisk-neutralinvestors.ImplicationsoftheRisk-NeutralWorld(1of6)TheBinomialandBlack-Scholesmodelsgivethesameoptionpricenomatterwhattheactualriskpreferencesandexpectedstockreturnsare.Intherealworld,investorsareriskaverseandrequireapositiveriskpremiumtocompensateforrisk,whileinahypotheticalrisk-neutralworld,investorsdonotrequirecompensationforrisk.ImplicationsoftheRisk-NeutralWorld(2of6)Inotherwords,risnottheactualprobabilityofthestockpriceincreasing.Rather,itrepresentshowtheactualprobabilitywouldhavetobeadjustedtokeepthestockpricethesameinarisk-neutralworld.ImplicationsoftheRisk-NeutralWorld(3of6)Risk-NeutralProbabilitiesTheprobabilityoffuturestatesthatareconsistentwithcurrentpricesofsecuritiesassumingallinvestorsareriskneutral.Probabilitesarerisk-neutralprobabilitiesAlsoknownasState-ContingentPrices,StatePrices,orMartingaleprices.ImplicationsoftheRisk-NeutralWorld(4of6)Assumefromthepreviousexamplethatthecurrentpriceof$50hasatrueprobabilityof75%ofincreasingto$60andatrueprobabilityof25%ofdecreasingto$40.ImplicationsoftheRisk-NeutralWorld(5of6)Thisstock’strueexpectedreturnisthereforeGiventherisk-freeinterestrateof6%,thisstockhasa4%riskpremium.Butascalculatedearlier,therisk-neutralprobabilitythatthestockwillincreaseis65%,whichislessthanthetrueprobability.Thus,theexpectedreturnofthestockintherisk-neutralworldis6%.ImplicationsoftheRisk-NeutralWorld(6of6)Toensurethatallassetsintherisk-neutralworldhaveanexpectedreturnequaltotherisk-freerate,relativetothetrueprobabilities,therisk-neutralprobabilitiesoverweightthebadstatesandunderweightthegoodstates.Risk-NeutralProbabilitiesandOptionPricing(1of4)Consideragainthegeneralbinomialstockpricetree.Computetherisk-neutralprobabilitythatmakesthestock’sexpectedreturnequaltotherisk-freeinterestrate:Risk-NeutralProbabilitiesandOptionPricing(2of4)Solvingfortherisk-neutralprobabilityryields:Thevalueoftheoptioncanbecalculatedbycomputingitsexpectedpayoffusingtherisk-neutralprobabilities,anddiscounttheexpectedpayoffattherisk-freeinterestrate.TextbookExample21.8(1of4)OptionPricingwithRisk-NeutralProbabilitiesProblemUsingNarverNetworkSystemsstockfromExample21.2,imagineallinvestorsareriskneutralandcalculatetheprobabilityofeverystateinthenexttwoyears.Usetheseprobabilitiestocalculatethepriceofatwo-yearcalloptiononNarverNetworkSystemsstockwithastrikeprice$60.Then,priceatwo-yearEuropeanputoptionwiththesamestrikeprice.TextbookExample21.8(2of4)SolutionThebinomialtreeinthethree-stateexampleisFirst,weuseEq.21.16tocomputetherisk-neutralprobabilitythatthestockpricewillincrease.Attime0,wehaveTextbookExample21.8(3of4)Becausethestockhasthesamereturns(up20%ordown10%)ateachdate,wecancheckthattheriskneutralprobabilityisthesameateachdateaswell.Considerthecalloptionwithastrikepriceof$60.Thiscallpays$12ifthestockgoesuptwice,andzerootherwise.Therisk-neutralprobabilitythatthestockwillgouptwiceissothecalloptionhasanexpectedpayoffofWecomputethecurrentpriceofthecalloptionbydiscountingthisexpectedpayoffattherisk-freerate:TextbookExample21.8(4of4)Next,considertheEuropeanputoptionwithastrikepriceof$60.Theputendsupinthemoneyifthestockgoesdowntwice,ifitgoesupandthendown,orifitgoesdownandthenup.Becausetherisk-neutralprobabilityofadropinthestockpriceistheexpectedpayoffoftheputoptionisThevalueoftheputtodayisthereforewhichisthepricewecalculatedinExample21.2.Risk-NeutralProbabilitiesandOptionPricing(3of4)DerivativeSecurityAsecuritywhosecashflowsdependsolelyonthepricesofothermarketedassetsTheprobabilitiesintherisk-neutralworldcanbeusedtopriceanyderivativesecurity.Risk-NeutralProbabilitiesandOptionPricing(4of4)MonteCarloSimulationAcommontechniqueforpricingderivativeassetsinwhichtheexpectedpayoffofthederivativesecurityisestimatedbycalculatingitsaveragepayoffaftersimulatingmanyrandompathsfortheunderlyingstock.Intherandomization,therisk-neutralprobabilitiesareusedandsotheaveragepayoffcanbediscountedattherisk-freeratetoestimatethederivativesecurity’svalue.21.4RiskandReturnofanOption(1of2)Tomeasuretheriskofanoption,wemustcomputetheoptionbeta.Thiscanbeaccomplishedbycomputingthebetaofthereplicatingportfolio.Whereisthestock’sbetaandisthebond’sbeta.Becausethebondisriskless,andtheoptionbetaisBetaofanOptionTextbookExample