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MathematicsandFinancialEconomics
/s11579-021-00299-w
Supermartingaledeflatorsintheabsenceofanuméraire
PhilippHarms1·ChongLiu2·ArielNeufeld3
Received:3October2020/Accepted:3May2021
©TheAuthor(s),underexclusivelicencetoSpringer-VerlagGmbHGermany,partofSpringerNature2021
Abstract
InthispaperwestudyarbitragetheoryofÞnancialmarketsintheabsenceofanumraire
,weprovideageneralizationofthe
classicalequivalencebetweennounboundedproÞtswithboundedriskandtheexistenceof
asupermartingaledeß,weintroduceanewapproachbased
ondisintegrationoftheunderlyingprobabilityspaceintospaceswherethemarketcrashes
atdeterministictimes.
KeywordsSupermartingaledeßator·Absenceofanumraire·NUPBR·Fundamental
theoremofassetpricing·ArbitrageoftheÞrstkind
MathematicsSubjectClassification60G48·91B70·91G99
JELclassificationC02
1Introduction
OverviewAnearlyuniversalassumptioninthearbitragetheoryofÞnancialmarketsisthe
existenceofanumraire,.,,
thisassumptionunderliesthecelebratedfundamentaltheoremsofDelbaen,Schachermayer,
Kabanov,andKardaras[3,6,9].Inpractice,however,itisnotalwaysreasonabletomakethis
,itmaywellhappeninthepresenceofcreditorsystemicriskthatall
assetsunderconsiderationdefaultinÞ,whenacountrydefaultsonits
debtandissuesanewcurrency,thisdevaluesnotonlythedomesticbondmarketbutalso
BArielNeufeld
ariel.******@
PhilippHarms
philipp.******@-
ChongLiu
chong.******@
1AbteilungfrMathematischeStochastik,Albert-LudwigUniversittFreiburg,Freiburg,Germany
2MathematicalInstitute,UniversityofOxford,Oxford,UK
3DivisionofMathematicalSciences,NTUSingapore,Singapore,Singapore
123:.
MathematicsandFinancialEconomics
itsnum
,asimilarsituationoccursinÞnancialmodelswheretheassetswithin
adefaultablemarketsegmentarequotedintermsofanindexormarketaverage,ascanbe
,thepurpose
ofthisworkistostudyarbitragetheoryofÞnancialmarketsintheabsenceofanumraire..
Ourmainresultisageneralizationoftheclassicalequivalencebetweennounboundedprof-
itswithboundedrisk(NUPBR)andtheexistenceofasupermartingaledeßator[10].NUPBR
isapivotalnotioninarbitragetheoryandaminimalrequirementforreasonableÞnancialmod-
els[2,3,7,9,10].ItalsoplaysafundamentalroleindeÞningpath-wisestochasticintegralsin
model-freeÞnance[11].Toputitintocontext,atleastformarketswithnumraire,Delbaen
andSchachermayerÕsconditionofnofreelunchwithvanishingrisk(NFLVR)isequivalent
toNUPBRtogetherwithno-arbitrage(NA)[2].However,therearemanyreasonablemodels
suchasthethree--
over,NUPBRisallthatisneededforensuringthatexpectedutilitymaximizationiswell
deÞned,andthemaximizerispreciselythedesiredsupermartingaledeßator[7].Weshow
thatasimilarresultholdsformarketswithoutnum,indiscretetime,NUPBR
remainsequivalenttotheexistenceofasupermartingaledeß,this
-
outthisindependenceassumption,theequivalenceholdssubjecttoadditionalboundedness
conditionsonthemarket,whichcanberephrasedequivalentlyasboundednessconditionson
thedeß.
ArbitragetheorywithoutnuméraireTheconstructionofthesupermartingaledeßatorin
[10]viamaximizationoftheexpectedlog-utilitypresupposestheexistenceofanumraire
toensurethatthemaximizationproblemiswell-deÞ
oftheproofbutturnsouttobeafundamentalproblem,whichrequiresseveraladaptations
ofclassicaldeÞnitionsandarguments,asoutlinednext.
First,Þnedasboundedness
raire,thisimplies
boundednessinprobabilityofthepayoffsatallintermediatetimest<T,see[10].How-
ever,intheabsenceofanumraire,thepayoffsatintermediatetimesmaybeunbounded
inprobability,,andthisrulesouttheexistenceofastrictlypositive
supermartingaledeßator.
Second,thenotionofforkconvexity(alsoknownasswitchingproperty)istooweak.
AccordingtotheclassicaldeÞnition,forkconvexityallowsanagenttoswitchfromany
,marketswithoutnumrairemaynot
Þ
correctmodiÞcationistoallowtheagenttoswitchtoanewassetcontingentonthenewasset
beingpositiveatthegiventimeandstateofnature,asspelledoutinDeÞ.
Third,thefollowingargument,whichiscrucialfortheconstructionofadeßatorin[10,
],breaksdown:iftheterminalpayoffXTofanassetXisoptimalwithintheset
ofallterminalpayoffs,thenthepayoffXtisoptimalwithinthesetofallpayoffsattimet,
foranyintermediatetimet<,thisclearlydoesnotholdonmarketswhere
,someargumentsin[10]concerning
theregularizationofgeneralizedsupermartingalesbreakdownbecausetheyalsorelyonthe
existenceofanumraire.
ThetimewherethemarketcrashesMethodologically,thisworkreliesheavilyonan
analysisoftheÞrsttimeτwhereallassetsinthemarketvanishor,moresuccinctly,thetime
123:.
MathematicsandFinancialEconomics
,onemaypartitionthescenariospaceinto
disjointsubsetst,thereexistsaprocess
whichisstrictlypositiveuptotimetandthereforecanserveasanum,one
obtainsundertheclassicalconditionsof[10]asupermartingaledeßatorZtoneachspacet
ßatorsZtontcanthen
bepastedintoaglobaldeßatorZon.
Thissketchcanbeturnedratherdirectlyintoarigorousproofifτhascountablesupport;
,theconditionalprobabilities(providedtheyexist)maybe
singularwithrespecttoP,andconsequentlyNUPBRondoesnotentailNUPBRon
overcometheseissues,wediscretizetimeintoaÞnitedyadicgridof2nintervalsandapply
ßator
→∞whilepreservingthestrictpositivityisthemost
importantanddifÞßators
or,equivalently,goodupperboundsontheassets.
PreviousliteraturePreviously,arbitragetheoryformarketswithoutnumraireshasbeen
studiedonlyinÞnitediscretetimebyTehranchi[12].However,recentlyarelatedpreprintof
Blint[1]oncontinuous-timemarkets,basedonresearchindependentofours,hasappeared.
Thephilosophyin[1],aswellasinthepresentpaper,isasfollows:oneÞrstlocalizesthe
market,thenconstructslocaldeßatorsoneachlocalizedpiece,andÞnallypastesthemtogether
togetaglobaldeß[1]isperformed
intime,.,theterminaldateTisapproximatedfrombelowbyasequenceofstopping
times(Tn)n≥1suchthatoneachtimehorizon[0,Tn)themarketcontainsanum
contrast,themethodologyofthispaperistolocalizethemarketinÒspaceÓinthesensethat
thesamplespaceispartitionedintodifferentpartssuchthatthereexistsanumraireunder
[1]
worksverywellincontinuoustimebutfailsiftheunderlyingprocessesarenon-adaptedtothe
givenÞltration;ontheotherhand,ourtechniquescanhandlenon-adaptedness(inparticular
forÞnitediscretetimemarkets)butneedmoretechnicalassumptionstoworkincontinuous
,webelievethatbothapproachescanprovidealternativeandcomplementary
,wealsogiveanexplicitformulaforconstructing
adeßator(,),whiletheresultsin[1]areelegantbutrather
abstract.
thesetup,notations,ÞrstmainresultinÞnite
[12]alsoprovedasimilarresult,butourapproachis
continuoustimeandÞndanequivalenceconditionfortheexistenceofasupermartingale
deßator.
2Setupandmainresults
WeÞxaÞnitetimehorizonT∈(0,∞)andaÞlteredprobabilityspace(,F,F,P).More-
overweletI⊆[0,T]beeitherI:={0,1,...,T}forthediscrete-timesettingorI:=[0,T]
forthecontinuous-,wewillusethefollowingnotation.
•IfnotspeciÞeddifferently,everypropertyofarandomvariableorastochastic
processsuchas,.,strictpositivity,cdlgpaths,ÉisunderstoodtoholdP-..
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MathematicsandFinancialEconomics
•Wemeanbyastochasticprocess(Xt)t∈IsimplyacollectionofF-measurablerandom
variables.
•ForanymeasureQon(,F)wedenotebyL0(Q)thesetofall(equivalenceclasses
of)randomvariables,whichweendowwiththemetricwhichinducesconvergenceinQ-
,wedenotebyL0(Q)⊆L0(Q)thesetofnonnegativerandomvariables
+
andbyL0(Q)⊆L0(Q)thesetofstrictlypositiverandomvariablesXinthesensethat
+++
Q[X>0]=1.
•WecallasetC⊆L0(Q)tobeQ-boundedorboundedinL0(Q)ifitisboundedin
probabilitywithrespecttoQ,namely
limsupQ|X|≥M=0.
M→∞X∈C
•Following[14]wesaythatasetC⊆L0(Q)isQ-convexcompactorconvexlycompactif
+
itisconvex,closed,andQ-bounded.
•Following[10],wesaythatastochasticprocess(Xt)t∈[0,T]deÞnedon[0,T]isQ-cdlg
ifthemapping[0,T]t→Xt∈L0(Q)isright-continuousandhasleft-limits.
-timesetting,wecallacollectionofnonnegativeprocesses,
denotedbyX,awealthprocesssetormarketon{0,1,...,T}ifitsatisÞesthefollowing
twoconditions:
(i)EachX∈XsatisÞesX0=1,
(ii)foreachX∈XwehavethatXvanishesonthestochasticinterval[[τX,T]],where
τX:=inf{t∈{0,1,...,T}|Xt=0}withtheconventioninf∅:=∞.
Inthecontinuous-timesetting,wecallacollectionofnonnegativeprocessesXawealth
processsetormarketon[0,T]ifitsatisÞesthefollowingtwoconditions:
(i)EachX∈XhascdlgpathsandsatisÞesX0=1,
(ii)foreachX∈XwehavethatXvanishesonthestochasticinterval[[τX,T]],where
τX:=inf{t∈[0,T]|Xt=0orXt−=0}withtheconventioninf∅:=∞forall
X∈X.
Furthermore,awealthprocesssetXiscalledF-adaptedifeachX∈XisanF-adapted
process.
∈XanumraireforthemarketXifXnumisstrictly
t
positiveforalltimet.
Ourgoalofthispaperistoanalyzemarketswhichdonotnecessarilycontainanumraire,
bothinthecasewherethemarketXisF-adaptedornot.
Inthespiritof[10,13],weintroduceanotionofgeneralizedforkconvexityforwealth
processsets.
ÞnedonIsatisÞesthegeneralizedfork
convexityifthefollowingtwoconditionshold:
(i)Xisconvex,.,λX1+(1−λ)X2∈Xforanyλ∈[0,1],X1,X2inX,
(ii)foranyX1,X2,X3inX,s∈I,andA∈Fs,theprocessdeÞnedby
2
1Xt11
Xt:=Xt1{t<s}+1AX2Xs1{Xs2>0}+Xt1{Xs2=0}
s
3
+1cXtX113+X1131,t∈I,()
AXs3s{Xs>0}t{Xs=0}{t≥s}
belongstoX.
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MathematicsandFinancialEconomics
Inwords,thegeneralizedforkconvexitymeansthattheagentonthismarketwillswitchto
anotherportfolioattimetonlywhenthewealthprocessassociatedtothenewportfoliohas
apositivevalueatthisinstant,otherwiseshewillkeepheroriginalposition.
usualoneintroducedbyŽitkovi«c[13]andalsousedinKaradaras[10],evenifthemarketX
possessesanum,inthenotionofŽitkovi«c[13],theswitchedportfolios
X2andX3in(),weanalyzemarketswhich
maynotcontainanumraire,webelievethatourslightgeneralizationofforkconvexityis
,weobservethatinthepresence
ofanumraire,thepropertyforamarkettosatisfyNUPBR,meaningthattheÞnalvalue
setCT:={XT:X∈X}isP-bounded,doesn