InvestmentsLecture15Pricing:OptionsOptionPricingModelsTwoapproaches:TrackingPortfolio.Solveexplicitlyforreplicatingportfolio.Risk-neutralprobabilities.Adjustprobabilitiessothatexpectedcashflowsgiveoptionprices.Twosettings:Binomial(discretetime)Black-Scholes(continuoustime)3TheBinomialValuationModel:

TheTrackingPortfolioApproach

Considerthefollowingcalloption: S0=$100, K=$125, rf=8Supposeatyear-endstockcantakeonlytwovalues: ST=$200or$50Thenthepayoffforcalloptionis: $75whenstockgoesupand$0whenstockgoesdown.200 7550 0Stock Call100TrackingPortfolioComparetothepayofffromaportfolioof: 1shareofstock+borrowing$46.30at8%Payoffsfromthispositionare: $150whenstockgoesupand$0whenstockgoesdown Cashoutlaytoestablishthisposition=$100-$46.30=$53.70

Payoffofleveredportfolioistwicetheoptionpayoff,therefore: C0=(1/2)$53.70=$26.85Considerthefollowingposition(samedataasbefore) 1shareofstock+2writtencalls Netpayoffs: Stockvalue $50 $200 -Callobligation -0 -$150 Netpayoff $50 $50 PVof$50=$50/1.08=$46.30Hence: S0-2C0=46.30; S0=100andC0=$26.856Hedgeratio(H):Inthiscase:Therefore: H=0.5shares $100 $25

-Callobligation $75 $0

$25 $25PVof$25at8%=$25/1.08=$23.15CalculatingtheHedgeRatio7Valueofhedgedposition=PVofcertainpayoff 0.5S0-C0=$23.15 $50-C0=$23.15 C0=$26.85Hsharesand1callwrittenalwaysresultsinaperfectlyhedgedportfolioYieldssurepayoffattimeTofGeneralbinomialmodelpricingformula:GeneralPrinciple8Example:ArbitrageProfitsGivenpreviousdataexceptC0=$30Initialcashflows: Write1option $30 Purchase0.5shares -$50 Borrow$20at8% $20 (Repayin1year)Cashflowin1yearateachstockprice: Writtenoption $0 -$75 PurchaseHshares $25 $100 Borrow$20at8%

Repayin1year-$21.60-$21.60 Payoffs $3.40 $3.40Foranetinitialinvestmentof$09Basicapproach:IfoptionoverpricedpurchaseHshares,write1optionandborrowthedifferencetomake0netinvestment.IfoptionunderpricedsellHsharesshort,purchase1optionandlendthedifferencetomake0netinvestment.10Step2:Usethatprobabilitytoevaluateoptionspayoffanduserftodiscount

Supposestockpriceis$75andyouwanttovalueanoptionwithK=$90Supposethatrf=2.5%andthatstockpricecanbe$60or$100in1yearStep1:Computetheprobability“asif”investorswereindifferentwithrespecttoriskp=0.421875TheBinomialValuationModel:

TheRisk-NeutralProbabilityApproachRisk-NeutralValuationTovalueaderivativesecurityusingthebinomialmodelandtherisk-neutralpricingapproach:Calculatetherisk-neutralprobabilities:Usetheseprobabilitiestocalculatetheexpectedpayoutfromtheinstrument,andDiscounttheexpectpayoutusingtherisk-freeinterestrate.BinomialOptionPricingBinomialoptionpricingenablesustodeterminethepriceofanoption,giventhecharacteristicsofthestockorotherunderlyingasset.Thebinomialoptionpricingmodelassumesthatthepriceoftheunderlyingassetfollowsabinomialdistribution—thatis,theassetpriceineachperiodcanmoveonlyupordownbyaspecifiedamount.Thebinomialmodelisoftenreferredtoasthe“Cox-Ross-Rubinsteinpricingmodel.”BuildingtheBinomialModelInorderforthesetechniquestobeuseful,thebinomialmodelmustcloselyrepresentstockpricedynamics.Inordertounderstandthemodelandhowtoapplyit,wemustfirstunderstandsomecontinuouscompoundingtechniques.RisklessAssetDynamicsConsiderthebehaviorofthepriceofaninvestmentthatisgrowingatthecontinuallycompoundingraterf:Thisgrowthcanbethoughtofasoccurringinstages:RiskyAssetDynamicsThesamesortofmodelmaybeappliedtoriskyassetpricedynamics:Here,however,thereturnisrandom.Wewillassumeitiscomposedofalotofsmallrandomshocksthatoccuroversmallintervals:RiskyAssetDynamicsOvereachsmallinterval,therandomreturnscanbethoughtofasabranchofabinomialtree:RiskyAssetDynamicsWhydowecareaboutthesedetails?Well,ifeachofthebinomialbranchesisindependentoftheother,stringingthebranchestogetherintoatreegivesrisetostockpriceswhosereturnsarelognormal.Thisfactallowsustomeasurecharacteristicsofthereturns(e.g.thevariance)andthenusethesemeasuredvariablestoconstructabinomialtreethatdescribeshowthestock’spricewillchange.

TherandomwalkmodelTheideathatassetpricesshouldfollowarandomwalkwasarticulatedinSamuelson(1965)Inefficientmarkets,anassetpriceshouldreflectallavailableinformation.Inresponsetonewinformationthepriceisequallylikelytomoveupordown,aswiththecoinflip.Thepriceafteraperiodoftimeistheinitialpriceplusthecumulativeupanddownmovementsduetonewinformation.Modelingstockpricesasarandomwalk.Theabovedescriptionofarandomwalkisnotasatisfactorydescriptionofstockpricemovements.ThereareatleastthreeproblemswiththismodelIfbychancewegetenoughcumulativedownmovements,thestockpricewillbecomenegative.Themagnitudeofthemove($1)shoulddependuponhowquicklythecoinflipsoccurandthelevelofthestockpriceThestock,onaverage,shouldhaveapositivereturn.However,therandomwalkmodeltakenliterallydoesnotpermitthisThebinomialmodelisavariantoftherandomwalkmodelthatsolvesalloftheseproblems.MovingThroughtheTreeSuudSuddSdddSMeasuringVolatilityIfweknowthevolatilityofastock,wecanstarttobuildabinomialtree.Thevolatilityisgivenbythevarianceofthecontinuouslycompoundedreturn.Thesquarerootofthistermisinunitsof%/yr.BuildingtheBinomialTreeInordertohaveourbinomialtreeaccuratelyrepresentthestock’sreturns,wewanttochoosetheupanddownreturns(uandd)sothatthevarianceofeachreturniscorrect.Inaddition,thetreewill“recombine”makingcomputationeasier.So,wechooseExampleThestandarddeviationofreturnsfortheS&P500indexis28.42%peryear.Thecurrentleveloftheindexis881.Builda3-periodbinomialtreethatcanbeusedtopriceoptionswithtwomonthstomaturity.UsethetreetopriceaMay875call.(currentprice=32.9)OptionPriceOtherChoicesfortheReturnsThereareseveralpossibilitiesforuandd:Withmanysteps,thechoiceislargelyirrelevant.Thelastchoicewillgivetheclosestapproximationinacoarsetree.DividendsOptionholdersdonotreceivedividendsontheunderlyingstock.Asaresult,wemustadjusttheoptionpricingformulatoaccountforthis.Severalassumptionscanbemadeabouttheformofdividends.Fornow,assumedividendsarepaidtomaintainaconstantdividendyield(Div/Price=constant).EffectofDividendsonBinomialModelWecanaccountfordividendsbychangingtherisk-neutralprobabilities:Noticethatnochangesneedtobemadetothebinomialtree!Wejustreinterpretthestockpricesasex-dividendprices.ExampleThestandarddeviationofreturnsfortheS&P500indexis28.42%peryear.Thecurrentleveloftheindexis881andthecurrentdividendyieldis1%peryear.Builda3-periodbinomialtreethatcanbeusedtopriceoptionswithtwomonthstomaturity.UsethetreetopriceaJune875call.(currentprice=32.90)OptionPricePutOptionWecomputeputoptionpricesusingthesamestockpricetreeandinthesamewayascalloptionprices.TheonlydifferencewithaEuropeanputoptionoccursatexpiration.–Insteadofcomputingthepriceasmax(0,S–K),weusemax(0,K–S).AmericanOptionThevalueoftheoptionifitisleft“alive”(i.e.,unexercised)isgivenbythevalueofholdingitforanotherperiodThevalueoftheoptionifitisexercisedisgivenbymax(0,S–K)ifitisacallandmax(0,K–S)ifitisaput.ForanAmericancall,thevalueoftheoptionatanodeisgivenbyAmericanOptionsThevaluationofAmericanoptionsproceedsasfollows:Ateachnode,wecheckforearlyexercise.Ifthevalueoftheoptionisgreaterwhenexercised,weassignthatvaluetothenode.Otherwise,weassignthevalueoftheoptionunexercised.Weworkbackwardthroughthethreeasusual.

PropertiesofoptionpricesAmericanvs.European–SinceanAmericanoptioncanbeexercisedatanytime,whereasaEuropeanoptioncanonlybeexercisedatexpiration,anAmericanoptionmustalwaysbeatleastasvaluableasanotherwiseidenticalEuropeanoptionPropertiesofoptionprices(cont.)Callpricecannot:benegativeexceedstockpricebelessthanpriceimpliedbyput-callparityusingzeroforputprice:Putpricecannotbemorethanthestrikepricebelessthanpriceimpliedbyput-callparityusingzeroforputpricePropertiesEarlyexerciseofAmericanoptionsAnon-dividendpayingAmericancalloptionshouldnotbeexercisedearlyThatmeans,onewouldlosemoneybeexercisingearlyinsteadofsellingtheoptionIftherearedividends,itmaybeoptimaltoexerciseearlyI