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1、GaugetheoryofFinance?DidierSornette1,21DepartmentofEarthandSpaceScienceandInstituteofGeophysicsandPlanetaryPhysicsUniversityofCalifornia,LosAngeles,California900952LaboratoiredePhysiquedelaMati`ereCondens´ee,CNRSUMR6622Universit´edesSciences,B.P.70,ParcValrose,06108Nic
2、eCedex2,FranceAbstract:Someproblemswiththerecentstimulatingproposalofa“GaugeTheoryofFinance”areoutlined.First,thederivationofthelog-normaldistributionisshownequivalentbothininformationandmathematicalcontenttothesimplerandwell-knownderivation,datingbackfromBachelierandS
3、amuelson.Similarly,there-derivationofBlack-Scholesequationisshownequivalenttothestandardonebecausethelimitofnouncertaintyisequivalenttothestandardrisk-freereplicationargument.Bothre-derivationsofthelog-normalityandBlack-Scholesresultdonotprovideatestofthetheorybecausei
4、tisdegenerateinthelimitswheretheseresultsapply.Third,thechoiceoftheexponentialformalaBoltzmann,oftheweightofagivenmarketconfiguration,isakeypostulatethatrequiresjustification.Inaddition,the“GaugeTheoryofFinance”seemstoleadto“virtual”arbitrageopportunitiesforpureMarkovran
5、domwalkmarketwhenthereshouldbenone.Theseremarksareofferedinthehopetoimprovetheformulationofthe“GaugeTheoryofFinance”intoacoherentandusefulframework.Ilinskiandcollaborators[1,2]haverecentlyproposedanintringuinganalogybetweenquantumelectrodynamics(QED),gaugefieldtheoryandfi
6、nancialmarkets(seealso[3]forasimplifiedintro-duction).Theirmainideastartsfromtherecognitionthatthequestforarbitrage1opportunitiesbymarketplayersisoneofthepossiblereasonableunderlyingexplanationforanomalousbehaviorssuchasdeviationsfromlog-normalityofpricevariations.Theyt
7、henproceedtodevelopageomet-ricaltheoryinwhichanarbitrageopportunityisequivalenttoalocalcurvatureoftheplaquettescorrespondingtothedifferentfinancialintruments.Thisisalsoveryintuitivesinceaparameter-freedefinitionofa(non-zero)curvatureinvolvesthenon-closureofaparalleltransp
8、ortalongaclosedpath,whichisquitesimilartoaninvestmentfromitsinceptiontoclosure.Inotherwords,itisnothingbutthestandard
