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1、CHAPTER62utilitytheoryInthisChapter...•utilitydefinitionsandaxioms•riskaversion•certaintyequivalentwealth•maximizationofexpectedutility•ordinalandcardinalutility•vonNeumann–Morgensternutilityfunctions62.1INTRODUCTIONMostofderivativestheoryisabouthedginga
2、ndtheeliminationofrisk.Foravarietyofreasons,someofwhichwehavediscussed,perfecthedgingisnotalwayspossible.Inthatcasewemustdecidehowto‘value’theunhedgedresidualrisk.Onewayistosimplyignoreit,andjustconcernoneselfwithrealexpectationsofoutcomes.Anotheristoex
3、amineboththeaverageoutcomeandthestandarddeviation.We’veseenbothoftheseapproachesinpreviouschapters.Anotherapproachistoconsidertheutilityofeachoutcome,thatisweassociateanumbermeasuringthe‘happiness’(inaloosesense)thateachoutcomegivesus.Although$10billion
4、istwiceasmuchas$5billion,wewouldn’tnecessarilybetwiceashappywiththeformer;we’dbehappier,probably,butsincebothnumbersaresostaggeringlylargeandunimaginable,thedifferenceinourhappinesswouldnotbethatgreat.We’llseehowthisfitsintoquantitativefinancetheoryinlate
5、rchapters.Inthischapterwesettheframework.62.2RANKINGEVENTSAsageneralrulepeoplehave(atleastroughly)arankingofpreferencesforvarious‘commodi-ties.’AnE-typeJaguarispreferabletoanyBMW,forexample.Sometimesitisimpossibletodecidebetweentwocommodities,peanutbutt
6、ercheesecakeversus(myown)pumpkinpie.Thecommoditiesdon’thavetobephysicalquantities,theycouldbeevents,oroutcomes.1006PartFiveadvancedtopicsDefinitionLet’ssupposewehavecommoditiesoreventsA,B,C,....WewriteABifAisstrictlypreferredtoB.Notethatthisisapersonalc
7、hoice.1Ifwecan’tdecidebetweenAandBthenwewriteA∼B.WecanalsowriteABifAisweaklypreferredto,orisatleastasgoodas,B.Wemusthavesomeaxioms,toensurethattherelationshipbetweeneventsmakessense.Theusualaxiomsareasfollows.Axiom1:CompletenessGiventwoeventsAandB,oneo
8、fthefollowingthreemusthold.EitherA∼BorABorBA.Axiom2:ReflexivityA∼A.Axiom3:TransitivityIfABandBCthenAC.Theseaxiomsaresufficienttogiveusarankingbetweenvariousevents.Buttheydonotensuretheexistenceofautilityfunction.Forthisweneeda
