analysis, geometry, and modeling in finance advanced methods in option pricing（2-2）

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1、Chapter7Multi-AssetEuropeanOptionandFlatGeometryAbstractAstandardmethodtopriceamulti-assetEuropeanoptionin-corporatinganimpliedvolatilityistousealocalvolatilityMonte-Carlocom-putation.Althoughstraightforward,thismethodisquitetime-consuming,particularlywhenthenumberofassets

2、islargeandweevaluatetheGreeks.Applyingourgeometricalframeworktothismulti-dimensionalproblem,weexplainhowtoobtainaccurateapproximationsofmulti-assetEuropeanop-tions.WeusetheheatkernelexpansiontoobtainanasymptoticsolutiontotheKolmogorovequationforan-dimensionallocalvolatilit

3、ymodel.Theresultingmanifoldisthe atEuclideanspaceRn.Wepresenttwoapplications.Therstapplicationwelookatisthederivationofanasymptoticimpliedvolatilityforabasketoption.Inparticular,wetrytoreconstructthebasketimpliedvolatilityfromtheimpliedvolatilityofeachasset.Inthesecondapp

4、lication,weobtainaccurateapproximationforCollateralizedCommodityObligations(CCO),whicharerecentcommodityderivativesthatmimictheCollateralizedDebtObligations(CDO).7.1Localvolatilitymodelsand atgeometryIntheforwardmeasurePT,eachforwardfi(i=1;


;n)isalocalmartingaletandweass

5、umethattheyfollowalocalvolatilitymodeldfi=Ci(t;fi)dW;dWdW=dt(7.1)ttiijijwithadeterministicrateandwiththeinitialconditionfi=fi.Themetrict=00(4.78)att=0underlyingthismodelisdfidfjds2=2ij(7.2)Ci(fi)Cj(fi)187188Analysis,Geometry,andModelinginFinancewherewehavesetCi(fi)Ci(0;

6、fi).ByusingtheCholeskydecomposition,wewritetheinverseofthecorrelationmatrixas1=LyLorincomponentsij=LLkikjByconventionijdenotesthecomponentsoftheinverseofthecorrelationmatrix.[L]ikisann-matrix.Similarlythecorrelationcanbewrittenas
y=L1L1orincomponents=LikLjkijHer

7、eLijarethecomponentsoftheinverseoftheCholeskymatrixL.IfweintroducethenewcoordinatesZfjdxjui(f)=L(7.3)ijjjfjC(x)0weobtainthatthemetric(7.2)(att=0)is at(thefactor2isintroducedforaconveniencepurpose)ds2=2duiduiThegeodesicdistancebetweenthetwopointsfffigandfffigis00iithengiv

8、enbytheEuclideandistanceXnd(u)2=2u:u2(ui)2(7.4)i=1Aftersomealgebraicmanipulations,thecon