Binomial-Option-Pricing-students

BinomialOptionPricingandTheBlack–ScholesmodelAbinomialmodel•HowaEuropeancalloptioncanbevaluedusingasimplebinomialmodel.•Supposeastock’spriceiscurrently$60.•Itisknownthatthepriceattheendofthemonthwillbeeither$66or$54.•Acalloptiononthestockhasanexercisepriceof$63andaone-monthmaturity.•Ifthestockpriceis$66afteronemonth,thevalueoftheoptionis$3($66–$63).•Ifthestockpriceis$54,theoption’svalueis0(sincetheexercisepriceisgreaterthan$54).BinomialModelofStockPriceMovementsAbinomialmodel•ForaportfolioconsistingofalongpositioninXsharesofthestockandashortpositioninacalloption,thevalueoftheportfoliois–66X–3ifthestockpricegoesto$66–$54Xotherwise.•FindXthatmake66X–3=54X.•WhenX=0.25,theportfolioisriskless.•Thatis,theportfolio’svalueisunaffectedbythechangeinthestockpriceovertheonemonth.Thecalloption•Cisthevalueofthecalloptionatthebeginningofthemonth.•Iftherisk-freerateofinterestis1/2percentpermonth,thenthevalueoftheoptionatthebeginningofthemonthmustbe$1.567.TheBlack–Scholesmodel•Extendingbeyondthismodel,theBlack–Scholesmodeluseshistoricalstockpricedatatodeterminetheexactvalueofacalloption.•TheBlack–Scholesoptionpricingformulaassumesthefollowing:•Capitalmarketsarefrictionless(i.e.,therearenotransactioncostsortaxesandallinformationissimultaneouslyandfreelyavailabletoallinvestors).•Thevariabilityintheunderlyingasset’sreturnisconstant.•Theprobabilitydistributionoftheunderlyingasset’spriceislognormal.•Therisk-freerateisconstantandknownovertime.•Nodividendsarepaidontheunderlyingasset.•Noearlyexerciseisallowedontheoption.TheBlack–ScholesmodelUsingtheBlack–ScholesFormula•Supposeyouownacalloptiononastockforwhichthefollowingapplies:•Underlyingstock’sprice=$60•Exercisepriceontheoption=$58•Annualrisk-freerate=5percent•Timetoexpirationontheoption=3months•Standarddeviationoftheunderlyingstock’sreturn=.12•Tocalculatethevalueoftheoption,wefirstcalculated1andd2asfollows:Thecalloption’spriceValuesoftheCumulativeNormalDistribution.7881.7891ValuesoftheCumulativeNormalDistributionThecalloption’sprice•Next,thesevaluesarepluggedintotheBlack–Scholesformulatogetthecalloption’spriceasfollows:Theputoption’sprice•SupposeyouownaputoptiononthestockdescribedinExample10–5.•Theputoptionhasanexercisepriceof$65.Thevaluesofd1=-.8034andofd2=-.7434.•ThevaluesofN(d1)andN(d2)arefoundfromTable10–A1,whichshowsthecumulativenormaldistribution.ValuesoftheCumulativeNormalDistributionValuesoftheCumulativeNormalDistributionTheputoption’sprice•Next,thesevaluesarepluggedintotheBlack–Scholesformulatogettheputoption’spriceasfollows: