Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options-中国Betway体育网页登陆

摘要 ThevalueofaEuropeanoptionsatisfiestheBlack-Scholesequationwithappropriatelyspecifiedfinalandboundaryconditions.Wetransformtheproblemtoaninitialboundaryvalueproblemindimensionlessform.Therearetwoparametersinthecoefficientsoftheresultinglinearparabolicpartialdifferentialequation.Forarangeofvaluesoftheseparameters,thesolutionoftheproblemhasaboundaryoraninitiallayer.Theinitialfunctionhasadiscontinuityinthefirst-orderderivative,whichleadstotheappearanceofaninteriorlayer.Weconstructanalyticallytheasymptoticsolutionoftheequationinafinitedomain.Basedontheasymptoticsolutionwecandeterminethesizeoftheartificialboundarysuchthattherequiredsolutioninafinitedomaininxandatthefinaltimeisnotaffectedbytheboundary.Also,westudycomputationallythebehaviourinthemaximumnormoftheerrorsinnumericalsolutionsincasessuchthatoneoftheparametersvariesfromfinite(orprettylarge)tosmallvalues,whiletheotherparameterisfixedandtakeseitherfinite(orprettylarge)orsmallvalues.Crank-Nicolsonexplicitandimplicitschemesusingcenteredorupwindapproximationstothederivativearestudied.Wepresentnumericalcomputations,whichdetermineexperimentallytheparameter-uniformratesofconvergence.Wenotethatthisrateisratherweak,dueprobablytomixedsourcesoferrorsuchasinitialandboundarylayersandthediscontinuityinthederivativeofthesolution.

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Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

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