期權(quán)、期貨及其他衍生產(chǎn)品 課件 hull-OFOD11-ppt-21

Options,Futures,andOtherDerivativesEleventhEditionChapter21BasicNumericalProceduresCopyright?2022,2018,2012PearsonEducation,Inc.AllRightsReservedApproachestoDerivativesValuationTreesMonteCarlosimulationFinitedifferencemethodsBinomialTreesBinomialtreesarefrequentlyusedtoapproximatethemovementsinthepriceofastockorotherasset.Ineachsmallintervaloftime,thestockpriceisassumedtomoveupbyaproportionalamountuortomovedownbyaproportionalamountd.MovementsinTimedeltat

(Figure21.1)TreeParametersforAssetPayingaDividendYieldofqParametersp,u,anddarechosensothatthetreegivescorrectvaluesforthemeanandvarianceofthestockpricechangesinarisk-neutralworld.Mean:Variance:Afurtherconditionoftenimposedis:TreeParametersforAssetPayingaDividendYieldofq

(Equations21.4to21.7)Whenissmall,asolutiontotheequationsisTheCompleteTree(Figure21.2)BackwardsInductionWeknowthevalueoftheoptionatthefinalnodes.Weworkbackthroughthetreeusingrisk-neutralvaluationtocalculatethevalueoftheoptionateachnode,testingforearlyexercisewhenappropriate.Example21.1:PutOptionInthiscase,Example(continued;Figure21.3)CalculationofDeltaDeltaiscalculatedfromthenodesattimeCalculationofGammaGammaiscalculatedfromthenodesattimeCalculationofThetaThetaiscalculatedfromthecentralnodesattimes0andCalculationofVegaWecanproceedasfollows:Constructanewtreewithavolatilityof41%insteadof40%.Valueofoptionis4.62.VegaisTreesforOptionsonIndices,CurrenciesandFuturesContractsAswithBlack–Scholes–Merton:Foroptionsonstockindices,qequalsthedividendyieldontheindex.Foroptionsonaforeigncurrency,qequalstheforeignrisk-freerate.Foroptionsonfuturescontracts,BinomialTreeforStockPayingKnownDividendsProcedure:Constructatreeforthestockpricelessthepresentvalueofthedividends.Createanewtreebyaddingthepresentvalueofthedividendsateachnode.ThisensuresthatthetreerecombinesandmakesassumptionssimilartothosewhentheBlack–Scholes–MertonmodelisusedforEuropeanoptions.ControlVariateTechniqueValueAmericanoption,ValueEuropeanoptionusingsametree,ValueEuropeanoptionusingBlack–Scholes–Merton,OptionpriceAlternativeBinomialTreeInsteadofsettingwecanseteachofthe2probabilitiesto0.5andTrinomialTreeTimeDependentParametersinaBinomialTreeMakingrorqafunctionoftimedoesnotaffectthegeometryofthetree.Theprobabilitiesonthetreebecomefunctionsoftime.Wecanmakesafunctionoftimebymakingthelengthsofthetimestepsinverselyproportionaltothevariancerate.MonteCarloSimulationand

pi

(Figure21.13)Howcouldyoucalculatebyrandomlysamplingpointsinthesquare?MonteCarloSimulationandOptionsWhenusedtovalueEuropeanstockoptions,MonteCarlosimulationinvolvesthefollowingsteps:Simulateapathforthestockpriceinariskneutralworld.Calculatethepayofffromthestockoption.Repeatsteps1and2manytimestogetmanysamplepayoffs.Calculatemeanpayoff.Discountmeanpayoffatriskfreeratetogetanestimateofthevalueoftheoption.SamplingStockPriceMovementsInariskneutralworld,theprocessforastockpriceiswhereistherisk-neutralreturnWecansimulateapathbychoosingtimestepsoflengthandusingthediscreteversionofthiswhereisarandomsamplefromAMoreAccurateApproach

(Equation21.17)UseThediscreteversionofthisisandbecausewearenowdealingwithageneralizedWienerprocessExtensionsWhenaderivativedependsonseveralunderlyingvariables,wecansimulatepathsforeachoftheminarisk-neutralworldtocalculatethevaluesforthederivative.SamplingfromNormalDistribution

givesarandomsamplefromToObtainTwoCorrelatedNormalSamplesObtainindependentnormalsamplesandsetUseaprocedureknownasCholesky’sdecompositionwhensamplesarerequiredfrommorethantwonormalvariables.StandardErrorsinMonteCarloSimulationThestandarderroroftheestimateoftheoptionpriceisthestandarddeviationofthediscountedpayoffsgivenbythesimulationtrialsdividedbythesquarerootofthenumberofobservations.ApplicationofMonteCarloSimulationMonteCarlosimulationcandealwithpathdependentoptions,optionsdependentonseveralunderlyingstatevariables,andoptionswithcomplexpayoffs.ItcannoteasilydealwithAmerican-styleoptions.DeterminingGreekLettersForMakeasmallchangetoassetprice.Carryoutthesimulationagainusingthesamerandomnumberstreams.Estimateasthechangeintheoptionpricedividedbythechangeintheassetprice.ProceedinasimilarmannerforotherGreekletters.VarianceReductionTechniquesAntitheticvariabletechniqueControlvariatetechniqueImportancesamplingStratifiedsamplingMomentmatchingUsingquasi-randomsequencesSamplingThroughtheTreeInsteadofsamplingfromthestochasticprocess,wecansamplepathsrandomlythroughabinomialortrinomialtreetovalueaderivative.Ateachnodethatisreached,wesamplearandomnumberbetween0and1.Ifitisbetween0andp,wetaketheupbranch;ifitisbetweenpand1,wetakethedownbranch.FiniteDifferenceMethodsFinitedifferencemethodsaimtorepresentthedifferentialequationintheformofadifferenceequation.Weformagridbyconsideringequallyspacedtimevaluesandstockpricevalues.DefineasthevalueoffattimewhenthestockpriceisTheGrid(Figure21.5)FiniteDifferenceMethods(Equations21.24and21.26)ImplicitFiniteDifferenceMethodSettoobtainforeachnodeanequationoftheformoftheform:ExplicitFiniteDifferenceMethodIfandareassumedtobethesameatthe(i+1,j)pointastheyareatthe(i,j)point,weobtaintheexplicitfinitedifferencemethodThisinvolvessolvingequationsoftheform:ImplicitversusExplicitFiniteDifferenceMethodTheexplicitfinitedifferencemethodisequivalenttothetrinomialtreeapproach.Theimplicitfinitedifferencemethodisequivalenttoamultinomialtreeapproach.ImplicitversusExplicitFiniteDifferenceMethods(Figure21.16)OtherPointsonFiniteDifferenceMethodsItisbettertohavelnSratherthanSastheunderlyingvariable.Improvements