Analysis, Geometry, and Modeling in Finance 5

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1、Chapter4DierentialGeometryandHeatKernelExpansionAEMETPHTOMH
EIEIIT1

2、PlatoAbstractInthischapter,wepresentthekeytoolofthisbook:theheatkernelexpansiononaRiemannianmanifold.Intherstsection,inordertointroducethistechniquenaturally,weremindthereaderofthelinkbe-tweenthemulti-dimensionalKolmogoro

3、vequationandthevalueofaEuro-peanoption.Inparticular,anasymptoticimpliedvolatilityintheshort-timelimitwillbeobtainedifwecanndanasymptoticexpansionforthemulti-dimensionalKolmogorovequation.Thisisthepurposeoftheheatkernelexpansion.RewritingtheKolmogorovequationasaheatkernelequationonaRiemannianma

4、nifoldendowedwithanAbelianconnection,wecanapplyHadamard-DeWitt'stheoremgivingtheshort-timeasymptoticsolutiontotheKolmogorovequation.Anextensiontothetime-dependentheatkernelwillalsobepresentedasthiscaseisparticularlyimportantinnanceinordertoincludetermstructures.Inthenextchapters,wewillpresents

5、everalappli-cationsofthistechnique,forexamplethecalibrationoflocalandstochasticvolatilitymodels.4.1Multi-dimensionalKolmogorovequationInthispart,werecallthelinkbetweenthevaluationofamulti-dimensionalEuropeanoptionandthebackward(andforward)Kolmogorovequation.Forthesakeofsimplicity,weassumeazeroi

6、nterestrate.1Letnooneinapttogeometrycomein."InscribedovertheentrancetothePlatoAcademyinAthens.75©2009byTaylor&FrancisGroup,LLC76Analysis,Geometry,andModelinginFinance4.1.1ForwardKolmogorovequationWeassumethatourmarketmodeldependsonnIt^oprocesseswhichcanbetradedassetsorunobservableMarkovprocess

7、essuchasastochasticvolatility.Letusdenotethestochasticprocessesx(xi)and(i).i=1;


;ni=1;


;nTheseprocessesxisatisfythefollowingSDEsinarisk-neutralmeasurePdxi=bi(t;x)dt+i(t;x)dW(4.1)tttidWidWj=ij(t)dtwiththeinitialconditionx0=.Here[ij]i;j=1;


;nisacorrelationmatrix(i.e.,asymmetricnon-dege

8、neratematrix).REMARK4.1Inchapter2,theSDEsweredrivenbyindependentBrow-nianmotions.TheSDEs(4.1)canbeframedinthissettingbyapplyingaCholeskydecomposition:wewritethecorrelationmatrixas=LLyorincomponentsXnij=LikLjkk=1Thenthecorrelated