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1、第19卷第1期經(jīng)濟數(shù)學(xué)Vol.19No.12002年3月MATHEMATICSINECONOMICSMar.2002THEVALUATIONOFAMERICANPUTOPTIONSWITHSTOCHASTICVOLATILITYMeiZhengyangLiChulin(DepartmentofMathematics,HuazhongUniversityofScienceandTechnologyWuhan,Hubei430074,China)AbstractInthispaper,byusingfinitedifferencemethod,setupthevaluationof
2、AmericanputoptionmodelwithstochasticvolatilityoffiniteMarkovchains,obtainedthevaluesofAmericanputoption,thatimprovedtheresultsofconstantvolatilitymodelandbinomialprobabilitytreemodel.Theillustrationshowsthatthevaluecalculatedbyourmodelandalgorithmapproachestherealpriceclosermore.KeywordsMarko
3、vprocess,Americanoption,Finitedifference,Riskneutral1.IntroductionInallkindsofoptions,thestudyingofEuropeanoptionisthebest,buttheoptionstradedintheCBOT,PHLX,AMEX,NYSYandothersareAmericanoptionnotEuropeanoption.ValuationofAmericanoptionisverycomplexbecauseithasthefeaturethatexerciseispermitted
4、atanytimepriortoexpiry.HullandWhite[4]andHull[7]derivedanumericmethodtovaluetheAmericanputwhichwiththeconstantvolatility.Ritchey[5]foundthatthepriceofEuropeanoptionwithstochasticvolatilityisthecombinationofthesolutionofBlack-Scholesequation,hesetupaRitcheymodeldescribedthestochasticvolatility
5、withmulti-periodbinomialprobabilitytree,andobtainedsomeconclusions.Buttheelementsofstochasticvolatilityinhismodelwereincreasedaccordingexponentforamulti-periodoption,itwasverydifficultforpositivistsresearch.Consideringtheparametersofstochasticvolatilitymustbeestimated,wereplacethemulti-perio
6、dbinomialprobabilitytreewithfiniteMarkovchain,sothattheaccurateestimateofconstantvolatilityisrelaxedtoerrorestimateofstochasticvolatility.WeobtainanalgorithmtovalueAmericanoption.Forshowingthealgorithmsadvantage,wecalculateagaintheexampleofHull[7],theresultsclosedtotherealpricemorethanHull[7
7、].Finallywediscussedthestabilityofouralgorithm.Asweknown,thevalueofAmericancall,onanondividendpayingasset,hasthesamevalueascorrespondingEuropeancall.ButAmericanput,becauseofthefeatureofexerciseatanytimepriortoexpiry,hasbiggervaluethancorrespo
