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1、SIAMJ.MATH.ANAL.c2008SocietyforIndustrialandAppliedMathematicsVol.40,No.1,pp.306–326AVARIATIONALINEQUALITYARISINGFROMEUROPEAN∗INSTALLMENTCALLOPTIONSPRICINGFAHUAIYI†,ZHOUYANG†,ANDXIAOHUAWANG‡Abstract.InthispaperweconsideraparabolicvariationalinequalityarisingfromEuropeancontinuousinstallmentcallopti
2、onspricingandprovetheexistenceanduniquenessofthesolutionto∞theproblem.Moreover,weobtainCregularityandtheboundsofthefreeboundary,aswellasthelimitofthefreeboundaryasτ=T−t→+∞.Eventuallyweshowitsnumericalresultbythebinomialmethod.Keywords.freeboundary,variationalinequality,optionpricing,Europeaninstallm
3、entcalloptionsAMSsubjectclassification.35R35DOI.10.1137/0606703531.Introduction.InthispaperweconsideraparabolicvariationalinequalityarisingfromthemodelofEuropeancontinuousinstallmentcalloptionspricing.Moreprecisely,wewillfindC(S,t)satisfying⎧2⎪⎪∂C+σS2∂C+(r−q)S∂C−rC=L∗⎪⎪t2SSS⎪⎪⎨ifC>0and(S,t)∈(0,+∞)×(0,
4、T],2(1.1)∂C+σS2∂C+(r−q)S∂C−rC≤L∗⎪⎪t2SSS⎪⎪⎪⎪ifC=0and(S,t)∈(0,+∞)×(0,T],⎩+C(S,T)=(S−K),S∈[0,+∞),whereσ,r,L∗,andKarepositiveconstantsandqisanonnegativeconstant.Intheappendixwepresentthefinancialandstochasticbackgroundofthisprob-lem.IfL∗=0inproblem(1.1),itisstandardEuropeancalloption,whichhasanexplicitan
5、alyticformulaofsolutionanddoesnothavefreeboundaryatall(see[11]).WewillfindthatthecaseL∗>0ismorecomplicatedthanthecaseL∗=0.Therearesomepapersinthefieldofinstalloptions,suchas[7],[8],[1],[6],inwhichtheauthorsdevelopedthemodelsandnumericalanalysis.Particularly,Alobaidi,Mallier,andDeakinshowedthebehavioro
6、fthefreeboundarySs(τ)in(1.1)closetoexpirebyLaplacetransformsin[1],thatis,1/2+(1.2)Ss(τ)∼Kexp{−σ(−τlnτ)(1+o(1))}asτ=T−t→0.∗ReceivedbytheeditorsSeptember20,2006;acceptedforpublication(inrevisedform)September17,2007;publishedelectronicallyApril23,2008.TheprojectsupportedbyNationalNaturalSci-enceFoundat
7、ionofChina(10671075),NationalNaturalScienceFoundationofGuangDongprovince(5005930),andUniversitySpecialResearchFundforPh.D.Program(20060574002).http://www.siam.org/journals/sima/40-1/67035.html†Schoolo
