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1、Chapter10Schr•odingerSemigroupsEstimatesandImpliedVolatilityWingsAbstractWestudythesmall/largestrikebehavioroftheimpliedvolatil-ityforlocalandstochasticvolatilitymodels.Thederivationusestwo-sidedestimatesforSchr•odingerequationsonRiemannianmanifoldswithscalarpo-tentialsbelongingtodi
2、 erentclasses.10.1IntroductionSinceBlack-Scholes,severalalternativemodelshaveemergedsuchaslocalvolatilitymodels(LVMs)andstochasticvolatilitymodels(SVMs)whichhavebeenreviewedinchapters5and6.Althoughallthesemodelsareableto treasonablythemarketimpliedvolatilitiesacrossa(liquid)rangeofs
3、trikesandmaturities,theoverallpropertiesofeachmodelarequitedi erent.Notabledi erencescomefromthedynamicsoftheimpliedvolatility.Moreprecisely,althoughmatchingtheinitialmarketimpliedvolatilitysurface,twomodels,belongingtodi erentmodelclasses,cangivedi erentpriceswhenpricingexoticoptio
4、nssuchasforward-startingoptions.Similarly,thedi erent(illiquid)large-strikebehaviors,notquotedonthemarket,producedi erentpricesforvolatilityderivatives,suchasvarianceswapswhichstronglydependontheimpliedvolatilitywings.Inordertochoosethebestmodelforpricingandcapturingtherisk,itisther
5、eforeimportanttounderstandthegeneralpropertiesofthesemodels.Belowisalistofcommonpropertiesthatcancharacterizeamodel:Isthemodelexplodingina nitetime?Astudyforsomeparticularstochasticvolatilitymodelswasachievedin[47,29]andreviewedinchapter6usingtheFellercriteria.Whatistheshort-timeb
6、ehavioroftheimpliedvolatility?Canweusethisshort-timeasymptoticstocalibratethemodelifnoanalyticalso-lutionsareavailable?Ageneralshort-timeasymptoticsforLVMsandSVMsatthe rst-orderinthematurityisproposedinchapters5and6289©2009byTaylor&FrancisGroup,LLC290Analysis,Geometry,andModelinginF
7、inanceusingtheheatkernelexpansiononaRiemannmanifoldendowedwithanAbelianconnection.Isitsolvable?Inparticular,canweobtainanalyticalpricesforliquidoptionsandcalibrateecientlythemodel?Anextensiveclassi cationofsolvableLVMsandSVMsisachievedinchapter9.Whatisthelarge-strikebehaviorofthe
8、impliedvolatility?A
