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1、Chapter21 BasicNumericalProceduresOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20141ApproachestoDerivativesValuationTreesMonteCarlosimulationFinitedifferencemethodsOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20142BinomialTreesBinomialtrees
2、arefrequentlyusedtoapproximatethemovementsinthepriceofastockorotherassetIneachsmallintervaloftimethestockpriceisassumedtomoveupbyaproportionalamountuortomovedownbyaproportionalamountdOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20143MovementsinTimeDt(Figure21.1,
3、page451)Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20144SuSdSp1–pTreeParametersforassetpayingadividendyieldofqParametersp,u,anddarechosensothatthetreegivescorrectvaluesforthemean&varianceofthestockpricechangesinarisk-neutralworldMean:e(r−q)Dt=pu+(1–p)dVariance
4、:s2Dt=pu2+(1–p)d2–e2(r−q)DtAfurtherconditionoftenimposedisu=1/dOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20145TreeParametersforassetpayingadividendyieldofq(continued)WhenDtissmallasolutiontotheequationsisOptions,Futures,andOtherDerivatives,9thEdition,Copyrigh
5、t©JohnC.Hull20146TheCompleteTree(Figure21.2,page453)Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20147S0u4S0u2S0d2S0d4S0S0uS0dS0S0S0u2S0d2S0u3S0uS0dS0d3BackwardsInductionWeknowthevalueoftheoptionatthefinalnodesWeworkbackthroughthetreeusingrisk-neutralvaluationto
6、calculatethevalueoftheoptionateachnode,testingforearlyexercisewhenappropriateOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20148Example:PutOption(Example21.1,page453-455)S0=50;K=50;r=10%;s=40%;T=5months=0.4167;Dt=1month=0.0833InthiscaseOptions,Futures,andOtherDer
7、ivatives,9thEdition,Copyright©JohnC.Hull20149Example(continued;Figure21.3,page454)Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201410CalculationofDeltaDeltaiscalculatedfromthenodesattimeDtOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201411C
8、alculationof
