Chp9OptionPricingWhenUnderlyingStockReturnsar

Chp.9OptionPricingWhenUnderlyingStockReturnsareDiscontinuous•Inthischapter,anoptionpricingformulaisderivedforthemoregeneralcasewheretheunderlyingstockreturnsaregeneratedbyamixtureofbothcontinuousandjumpprocesses.9.1Introduction•ThecriticalassumptionsintheBlack-Scholesderivationisthattradingtakesplacecontinuouslyintimeandthatthepricedynamicsofthestockhaveacontinuoussamplepathwithprovabilityone.•Whatwillhappenedifthereisajump?9.2TheStock-PriceandOption-PriceDynamics•Thetotalchangeofstockpriceisdividedintotwoparts:–normalvibrations(振动),modeledbyastandardgeometricBrownianmotion;–abnormalvibrations,usuallyduetofirmspecificinformation,modeledjumpprocess;•sothestockpricesamplepath:Wienerprocess+Poisson-drivenprocess•ThePoisson-drivenprocessisdescribedasfollows:theeventdoesnotoccurinthetimeinterval,1theeventoccursonceinthetimeinterval,theeventoccrusmorethanonceinthetimeinterval,probtthhohprobtthhohprobtthoh•Y:therandomvariabledescriptionofthedrawingfromadistributiontodeterminetheimpactoftheinformationonthestockprice.Then,neglectingthecontinuouspart•SthStY•Thenthestock-pricereturnscanbedescribedasifthePoissoneventdoesnotoccur1ifthePoissoneventoccursdSkdtdZdqSkdtdZkdtdZY•Ifareconstants,then,,,k2exp2StktZtYnS1njjYnY•AccordingtoIto’slemmathefunctionofstockpriceandtime221,,,,2,,,SSStWSWSFStkSFStFFSYtFStFStFStSFSt,/,WYFSYtFSt•（followingthethreeassetsmodel）•Consideraportfoliostrategywhichholdsthestock,theoption,andtherisklessasset,ifPisthevalueofthereturndynamicsontheportfoliocanwrittenasppppdPkdtdZdqP•2221111221121,,11,,11,pwwpwpwrrwwrwr•1.no-jumps=standardB-Swwrr22102SSStSFrSFrFF0,0dq•2.jumpscannotbehedge**********ifthePoissoneventdoesnotoccur1ifthePoissoneventoccursppppppppdPkdtdqPkdtkdtY•becauseoftheconcaveofoptionpricetostockprice•isalwayspositivewillnotbe0。*2***21,,,1,,11,SpwFSYtFStFStSYSYFStFSYtwwYFSt,,,SFSYtFStFStSYS*1pY•Economicimplications：•（1）followingB-Shedging:longstockandshortoption,平时收益高于预期，跳跃时损失很大；reverseB-Shedging:shortstockandlongoption,平时收益低于预期，跳跃时收益很大；•（2）无跳跃时期权卖方获利，有跳跃时买方获利。*20w*20w9.3AnOptionPricingFormula•Pricingtechnique1:•Ifoneknewtherequiredexpectedreturnontheoption221,,,20,00,max0,SSSSFkSFgSFFSYFSFFSE,wgS•Pricingtechnique2:AssumedthattheCAPMwasavaliddescriptionofequilibriumsecurityreturns.•Stock-pricedynamicsweredescribedtwocomponents:–Continuouspart---newinformation;–Jumppart---importantnewinformation,usuallyfirm(orevenindustry)specificsuchasdiscoveryofanimportantnewoilorthelossofacourtsuit,“nonsystematic”risk.•跳跃部分属于非系统风险，不产生风险溢价，根据CAPM*prwwrr221,,2SSSSFrkSFFrFFSYFS•Eventhoughthejumpsrepresent“pure”nonsystematicrisk,thejumpcomponentdoesaffecttheequilibriumoptionprice.Thatis,onecannot“actasif”thejumpcomponentwasnotthereandcomputethecorrectoptionprice(nonsystematicriskhassnonezeroprice?)•DefineWtobetheB-Soptionpricingformulafortheno-jumpcase.21221/2,;,,exp1exp22yWSErSdErdsyds•DefineXntherandomvariabletohavethesamedistributionastheproductofni.i.d.(identicallydistributedtoY)randomvariables20exp,exp,;,,!nnnnFSWSXkErn•Thereisnotaclosed-formsolution,butitdoesadmittoreasonablecomputationalapproximation•Therearetwospecialcaseswherecanbevastlysimplified.•Example1:Thereisapositiveprobabilityofimmediateruin,i.e.ifthePoissoneventoccurs,thenthestockpricegoesto0.thatisY=0withprobabilityone.22,expexp,;,,,;,,FSWSErWSEr•inednticalwiththestandardB-Ssolutionbutwithalarger“interestrate,”.AswasshowninMerton(1973a,Ch.8),theoptionpriceisanincreasingfunctionoftheinterestrate,andthereforeanoptiononastockthathasapositiveprobabilityofcompleteruinismorevaluablethananoptiononastockthatdosenot.•Example2:Yhasalog-normaldistribution•Define•then2,,;,,nnnfSWSEvr0exp,(,)!nnnFSfSn•Clearly,isthevalueoftheoption,conditionalonknowingthatexactlynPoissonjumpswilloccurduringthelifeoftheoption.Theactualvalueoftheoption,,isjusttheweightedsumofeachofthesepriceswhereeachweightedequalstheprobability.(,)nfS•(9.16)wasdeducedfromthetwinassumptionsthatCAMPisvalidandthejumpcomponentofasecurity’sreturnisuncorrelatedwiththemarket.Onecanhardlyclaimstrongempiricalevidencetosupporttheseassumptions.•Anothertechniquetoderive(9.16):theRossmodelforsecuritypricing.•有m个跳跃过程互相独立的股票，对应有m个共同形式（股票＋期权）的套利组合•*****1,2,jjjjjjdPkdtdqjmP•用m个套利组合与无风险资产组成一个组合：***111,,HHHHmmmHjjHHjjjHjjjjjdHkdtdqHxrrkxkdqxdq•随着m的增加，组合接近无风险（推导），所以有•所以有，此时成立。•两种推导过程基于同样的原理：跳跃过程可以分散。*110mjjjurm*jr•为消除系统风险必须卖出的股票数量。1/NWSd*0221/222exp!log//2/2nnnNdnnSErndnn9.4APossible