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1、LECTURESONREALOPTIONSRobertS.PindyckSloanSchoolofManagementMassachusettsInstituteofTechnology1UseofOptionPricingMethods•HardtohandleoptionswithNPVmethods.Needoptionpricingmethods.•Wewillillustratethiswithasimplecase:TheOptiontoInvest.Example:•Youaredecidingwhethertobuildaplantthatwouldpro
2、ducewidgets.Theplantcanbebuiltquickly,andwillcost$1million.Acarefulanalysisshowsthatthepresentvalueofthecashßowsfromtheplant,ifitwereupandrunningtoday,is$1.2million.Shouldyoubuildtheplant?Answer:Notclear.2YouHaveanOptiontoInvest•Issueiswhetheryoushouldexercisethisoption.•Ifyouexercisetheo
3、ption,itwillcostyouI=$1million.YouwillreceiveanassetwhosevaluetodayisV=$1.2million.OfcourseVmightgoupordowninthefuture,asmarketconditionschange.•Comparetocalloptiononastock,wherePispriceofstockandEXisexerciseprice:OptionPayoffCalloptiononstock:Max(P-EX,0)OptiontoinvestinMax(V-I,0)factory3D
4、ynamicsofProjectValue,Vµ=ExpectedreturnonV.Thisexpectedreturnwillbeconsistentwiththeproject’s(nondiversiÞable)risk.δ=Payoutrateonproject.Thisistherateofcashpayout,asfractionofV.Soµ=δ+expectedrateofcapitalgain.1.NoRisk.Thenrateofcapitalgainis:∆V=(µ−δ)∆tVandµ=rf,therisk-freeinterestrate.4Dy
5、namicsofProjectValue,V(Cont’d)2.VisRisky.∆V=(µ−δ)∆t+σetVwhereetisrandom,zero-mean.SoVfollowsaran-domwalk,likethepriceofastock.•IfallriskisdiversiÞable,µ=rf.•IfthereisnondiversiÞablerisk,µ>rf.•Formally,writeprocessforVas:dV=(µ−δ)dt+σdzV√wheredz=etdtistheincrementofaWienerprocess,andetnorma
6、llydistributed,with(dz)2=dt.•SoVfollowsageometricBrownianmotion(GBM).•Note(dV)2=σ2V2dt.5MathematicalBackground•WienerProcess:Ifz(t)isaWienerprocess,anychangeinz,∆z,overtimeinterval∆t,satisÞes1.Therelationshipbetween∆zand∆tisgivenby:√∆z=²t∆t,where²tisanormallydistributedrandomvariablewithz
7、eromeanandastandarddeviationof1.2.Therandomvariable²tisseriallyuncorrelated,i.e.,E[²t²s]=0fort6=s.Thusvaluesof∆zforanytwodifferenttimeintervalsareindependent.•WhatdotheseconditionsimplyforchangeinzoveranintervaloftimeT?—Breakintervalupintonunitsoflength∆teach,withn=T
