Chp9OptionPricingWhenUnderlyingStockReturnsar.ppt

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 Y therandomvariabledescriptionofthedrawingfromadistributiontodeterminetheimpactoftheinformationonthestockprice Then neglectingthecontinuouspart Thenthestock pricereturnscanbedescribedas Ifareconstants then AccordingtoIto slemmathefunctionofstockpriceandtime followingthethreeassetsmodel Consideraportfoliostrategywhichholdsthestock theoption andtherisklessasset ifPisthevalueofthereturndynamicsontheportfoliocanwrittenas 1 no jumps standardB S 2 jumpscannotbehedge becauseoftheconcaveofoptionpricetostockpriceisalwayspositivewillnotbe0 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 9 3AnOptionPricingFormula Pricingtechnique1 Ifoneknewtherequiredexpectedreturnontheoption 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 Eventhoughthejumpsrepresent pure nonsystematicrisk thejumpcomponentdoesaffecttheequilibriumoptionprice Thatis onecannot actasif thejumpcomponentwasnotthereandcomputethecorrectoptionprice nonsystematicriskhassnonezeroprice DefineWtobetheB Soptionpricingformulafortheno jumpcase DefineXntherandomvariabletohavethesamedistributionastheproductofni i d identicallydistributedtoY randomvariables Thereisnotaclosed formsolution butitdoesadmittoreasonablecomputationalapproximationTherearetwospecialcaseswherecanbevastlysimplified Example1 Thereisapositiveprobabilityofimmediateruin i e ifthePoissoneventoccurs thenthestockpricegoesto0 thatisY 0withprobabilityone inednticalwiththestandardB Ssolutionbutwithalarger interestrate AswasshowninMerton 1973a Ch 8 theoptionpriceisanincreasingfunctionoftheinterestrate andthereforeanoptiononastockthathasapositiveprobabilityofcompleteruinismorevaluablethananoptiononastockthatdosenot Example2 Yhasalog normaldistributionDefinethen Clearly isthevalueoftheoption conditionalonknowingthatexactlynPoissonjumpswilloccurduringthelifeoftheoption Theactualvalueoftheoption isjusttheweightedsumofeachofthesepriceswhereeachweightedequalstheprobability 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 用m个套利组合与无风险资产组成一个组合 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 为消除系统风险必须卖出的股票数量 9 4APossibleAnswertoanEmpiricalPuzzle 如果投资者用B S公式来为期权股价 它与理论价格的差距在哪 AssumeYislog normalTheinvestor sestimateisthetrueunconditionalvariance真是的应该是一个条件方差 考虑了跳跃过程 Theproblembecomes iftheinvestorusesashisestima