Lécuye , Emy; Riedel, F ank; S anca, Lo enzo
Wo king Pape
A bi age p icing in con ex, cash-addi i e ma ke s
Cen e o Ma hema ical Economics Wo king Pape s, No. 694
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Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Lécuye , Emy; Riedel, F ank; S anca, Lo enzo (2024) : A bi age p icing in con ex,
cash-addi i e ma ke s, Cen e o Ma hema ical Economics Wo king Pape s, No. 694, Biele eld
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694
Oc obe 2024
A bi age P icing in Con ex, Cash-Addi i e
Ma ke s
Emy L´ecuye , F ank Riedel, and Lo enzo S anca
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
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A bi age P icing in Con ex, Cash-Addi i e Ma ke s∗
Emy L´ecuye 1, F ank Riedel2, and Lo enzo S anca3
1LERN, Uni e si ´e Rouen No mandie
2Cen e o Ma hema ical Economics, Biele eld Uni e si y and School o Economics, Uni e si y o
Johannesbu g
3Collegio Ca lo Albe o and Uni e si y o Tu in (ESOMAS Depa men )
Oc obe 9, 2024
Abs ac
We conside supe hedging and no-a bi age p icing in ma ke s wi h a con ex and
cash-addi i e s uc u e and de i e an explici unc ional o m o he supe - eplica ion
p ice. Using con ex duali y me hods, we show ha he supe hedging p ice maximizes
he di e ence be ween he expec ed payo and a con idence unc ion ha accoun s o
he eliabili y o he p obabili y used in p icing. We demons a e ha he exis ence
o a s ic ly posi i e p obabili y wi hin he domain o he con idence unc ion, which
maximizes he supe - eplica ion p ice o a speci ic payo and ac s as a lowe bound
o all o he payo s, is necessa y and su icien o p e en a bi age oppo uni ies.
Fu he mo e, we explo e en opy p icing as a no able example o a supe - eplica ion
p icing unc ional and p o ide condi ions on he ma ke s uc u e unde which he
supe - eplica ion p ice akes he o m o en opy p icing. We show ha he con idence
unc ion in en opy p icing can be exp essed using he Kullback-Leible di e gence.
1 In oduc ion
In inancial ma ke s, he o de size can a ec a secu i y’s p ice pe uni , as many empi i-
cal s udies show (Almg en, Thum, Haup mann and Li 2005, Mo o, Vicen e, Moyano, Ge ig,
∗Funded by he Deu sche Fo schungsgemeinscha (DFG, Ge man Resea ch Founda ion) – P ojec -ID
317210226 – SFB 1283. We hank Rouzbeh Jei anzadeh o his wo k as a esea ch assis an .
1
Fa me , Vaglica, Lillo and Man egna 2009, T´o h, Lempe ie e, De emble, La aillade, Kock-
elko en and Bouchaud 2011, Bac y, Iuga, Lasnie and Lehalle 2015, and Donie and Bona
2015). This li e a u e consis en ly demons a e ha he empo a y p ice impac inc eases
app oxima ely wi h he squa e oo o he o de size.
L´ecuye and Ma ins-da Rocha (2021) ex end he undamen al heo em o asse p icing
in con ex ma ke s uc u es o accoun o his phenomenon, showing ha he p ice o a
secu i y is con ex in he quan i y aded. He e, we add wo na u al assump ions o he model
s udied by L´ecuye and Ma ins-da Rocha (2021). We assume he a ailabili y o a iskless
secu i y and ha he p icing unc ional is cash-addi i e. Cash-addi i i y is a well-known
pos ula e in Ma hema ical Finance ha models he absence o ic ions on he ma ke o
iskless secu i ies (F¨ollme , Schied and Lyons 2004).
Ou i s main esul , Theo em 3.2, shows ha he esul ing supe - eplica ion p ice has
an explici unc ional o m ha depends on a con idence unc ion α. I cap u es he highes
de ia ion om an idealized ic ionless ma ke , e lec ing he un a o able impac o ma ke
ic ion on he p icing o a inancial ins umen . We also show ha he supe - eplica ion
p ice is well-de ined and con inuous.
Ou second main esul , Theo em 3.3, shows ha i is necessa y and su icien o he
supe - eplica ion p ice o be suppo ed by a ic ionless no-a bi age p ice o sa is y he no-
a bi age condi ions. When he supe - eplica ion p ice is linea , his esul s in he s anda d
cha ac e iza ion o no-a bi age.
In Sec ion 4, we explo e an impo an example o supe - eplica ion p icing, en opy p ic-
ing, and p o ide condi ions o he ma ke s uc u e such ha he supe - eplica ion p ice has
an en opy p icing o m. The main condi ion is segmen ed-addi i i y, a condi ion ha s a es
ha he alue o a combined po olio equals he sum o he alues o i s indi idual compo-
nen s when hose componen s yield non-ze o payo s in dis inc , non-o e lapping s a es o he
wo ld. In his case, we show ha he con idence unc ion αis gi en by he Kullback-Leible
di e gence. This example p o ides a majo ac able example o a supe - eplica ion p ice.
O e all, ou esul s ex end he undamen al heo em o asse p icing in con ex ma ke
s uc u es, p o ide a amewo k o accoun o ma ke impac , ansac ion cos s, and axes,
and o e insigh s in o he supe - eplica ion p ice and i s ela ion o he adi ional concep
o no-a bi age.
This a icle con ibu es o he g owing body o li e a u e ha ex ends he undamen al
heo em o asse p icing (FTAP) o accoun o ic ions p esen in inancial ma ke s such
as ansac ion cos s, axes, and ma ke impac . The FTAP was i s demons a ed in ic-
2
ionless ma ke s by Ha ison and K eps (1979). Subsequen ly, Jouini and Kallal (1995) and
Lu me (1996) ex ended he FTAP o sublinea p icing ules. When ma ke s a e comple e,
Ce eia-Vioglio, Macche oni and Ma inacci (2015) cha ac e ized a p icing ule sa is ying
cash-addi i i y, mono onici y, and pu -call pa i y. Bu zoni, Riedel and Sone (2021) ex-
ended he FTAP u he by in oducing a mo e gene al se ing ha encompasses ma ke
unce ain y. Unde he assump ion ha he se o ne ades is a con ex cone, hey show
ha ma ke s iabili y is equi alen o he exis ence o a p icing ule aking he o m o a
lowe semi-con inuous sublinea ma ingale expec a ion wi h ull suppo .
The p icing ules conside ed by Jouini and Kallal (1995), Lu me (1996), Ce eia-Vioglio,
Macche oni and Ma inacci (2015), and Bu zoni, Riedel and Sone (2021) a e posi i ely ho-
mogeneous, meaning ha he size o an o de execu ed in he ma ke s does no modi y he
uni a y p ice o he o de , i.e., uni a y p ices a e cons an in quan i y aded. As a esul ,
hese p icing ules canno accoun o ma ke impac , he ac ha uni a y p ices ac ually
inc ease wi h he quan i y aded.
Se e al o he models exis ha can accoun o ma ke impac assuming he p icing
ule is con ex. Fo example, Jouini and Kallal (1999) showed ha a con ex p icing ule
is iable i , and only i , he e is no asymp o ic ee-lunch. L´ecuye and Ma ins-da Rocha
(2021) p oposed a new app oach making assump ions on he p imi i es o he model, he
ma ke s uc u e, which is he ac ual p ice paid o a po olio, and he payo s ecei ed.
They assumed he ma ke s uc u e is con ex and de e mined he ele an concep o no-
a bi age, which hey called obus no-a bi age. They also de i ed he supe - eplica ion
p icing ule om he p imi i es and show ha i is con ex. Mo eo e , i is iable ( ha is i
sa is ies obus no a bi age) i he ma ke s uc u e sa is ies no obus a bi age. Howe e ,
due o he gene ali y o hei model, hey do no p o ide an explici exp ession o he supe -
eplica ion p icing ule making i challenging o use in p ac ice. The e o e, in his a icle we
add ess a special case o his model by assuming ha a iskless asse is a ailable and ha
he ma ke s uc u e is cash-addi i e in addi ion o being con ex.
Sec ion 2.1 in oduces he no a ion and he main ma hema ical objec s used in he pape .
Sec ion 2.2 p esen s he model and 3 con ains he main esul s o he pape . Sec ion 4
conside s he special case o en opic p icing ules. The p oo s a e all in he Appendix.
3
2 Con ex Cash-Addi i e Ma ke s
2.1 P elimina ies
We use he ollowing no a ions and de ini ions. Gi en a non-emp y ini e se K, we call
an elemen X∈RKa ec o and we deno e i by X= (X(k))k∈Ko X= (Xk)k∈K. Fo
e e y X∈RK, we deno e supp X:= {k∈K:X(k)= 0} he suppo o X, i.e. he se
o poin s a which Xis non-ze o. A ec o X∈RKis nonnega i e (s ic ly posi i e) when
X(k)⩾0 ( esp. X(k)>0 ) o all k∈K. The se o nonnega i e (s ic ly posi i e) ec o s
is deno ed by RK
+( esp. RK
++). Gi en X∈RK,∥X∥deno es he s anda d Euclidean no m
o X. A unc ion :Rn→Rmis spaces), U⊆Xis open, and F:U→Y. The Ga eaux
de i a i e o a unc ion :Rn→Rma X∈Rnin he di ec ion Y∈Rmis de ined as
′(X;Y) = lim
h→0
(X+hY )−F(X)
h.
We say ha is Ga eaux di e en iable i ′(X;Y) exis s ini e o e e y Xand Y. A
unc ion :RK→Ris sub-addi i e when (X+Y)⩽ (X) + (Y) o e e y X, Y ∈RK,
and posi i ely homogeneous when (λX) = λ (X) o e e y λ > 0 and X∈RK. Sub-
addi i e and posi i ely homogeneous unc ions a e called sublinea .
We le ∆(K) deno e he se o p obabili y measu es de ined on he p obabili y space
(K, 2K). A p obabili y P∈∆(K) has ull suppo i P(k)>0 o e e y k∈K. Conside ing
wo p obabili ies P, Q ∈∆(K), he no a ion P≪Qmeans ha Pis absolu ely con inuous
wi h espec o Qin he sense ha o e e y A⊆K, we ha e Q(A) = 0 =⇒P(A) = 0. The
expec a ion unde P∈∆(K) o a andom ec o X∈RKis deno ed by EP[X] := P·X.
2.2 The Ma ke
We conside wo poin s in ime: = 0 and = 1. The second poin in ime, = 1, is
cha ac e ized by unce ain y, ep esen ed by a ini e se Ω o possible s a es o na u e. A he
poin in ime 0, agen s can access inancial ma ke s, whe e a se o Jsecu i ies is a ailable
o ading. One o hese secu i ies is a iskless asse ha pays 1 in e e y s a e o na u e.
Agen s can o m po olios o secu i ies, deno ed by θ∈RJ. The i s coo dina e o he
po olio, θ1, ep esen s he quan i y o he iskless asse ha is pu chased (posi i e) o sold
(nega i e), while he emaining coo dina es, θj o j∈J {1}, ep esen he quan i ies o
he o he secu i ies pu chased o sold.
4
We assume ha po olios a e no subjec o any es ic ions, and hei p ices a e ep e-
sen ed by a unc ion p:RJ→R. He e, p(θ) is he cos o ading po olio θa = 0. A
mapping G:RJ→RΩmodels he payo s o po olios, G(θ, ω)∈Rdeno ing he payo o
po olio θin s a e ω. The couple (p, G) is called a ma ke s uc u e.
De ini ion 2.1. The ma ke s uc u e (p, G) is
1. con ex i pis a con ex unc ion sa is ying p(0) = 0 and G(·, ω) is conca e o e e y
ω∈Ω, and G(0) = 0,
2. cash-addi i e i i sa is ies he ollowing p ope ies o all θ∈RJand all k∈R:
p(θ+ke1) = p(θ) + kand G(θ+ke1) = G(θ) + k1Ω,
whe e 1Ω∈RΩis he ec o wi h all coo dina es equal o 1 and e1∈RJis he ec o
whose i s coo dina e is equal o 1 and all o he coo dina es a e equal o 0.
Rema k 2.1.We assume ze o in e es a es. Posi i e in e es s can easily be accommoda ed.
In L´ecuye and Ma ins-da Rocha (2021), he au ho s ex end he undamen al heo em
o asse p icing o con ex ma ke s uc u es. In his pape , we ocus on he special case
whe e he ma ke s uc u e is cash-addi i e in addi ion o being con ex. In he es o he
pape , we hus main ain he ollowing assump ion.
Assump ion 2.1. The ma ke s uc u e (p, G) is con ex and cash-addi i e.
Cash-addi i i y has he impo an consequence ha e e y payo can be supe hedged
because a i ial supe hedge is gi en by a su icien ly la ge quan i y o he iskless secu i y.
Be o e we come o he undamen al heo em, le us discuss some a che ypical ma ke s
ha a e co e ed by ou se up. The ic ionless case, when bo h pand Ga e linea , is
na u ally a pa icula case o ou se up.
Example 2.1 (T ansac ion Cos s).Conside he si ua ion when he asse s a e aded a
bid-ask p ices 0 ≤qB
j≤qA
jand payo s xja e linea o e e y asse jas discussed in A aujo,
Cha eauneu and Fa o (2018) and he appendix o Beissne and Riedel (2019). The iskless
asse is ic ionless wi h qB
1=qA
1=x1= 1. W i e θ+
j= max{θj,0}and θ−
j= max{−θj,0}
o he long and sho posi ions, espec i ely. The p ice o a po olio a ime 0 is gi en by
he sublinea unc ional
p(θ) =
J
X
j=1θ+
jqA
j−θ−
jqB
j
and he payo is gi en by he linea unc ional G(θ) = PJ
j=1 θjxj.△
5
Example 2.2 (Incomple e and Ambiguous Ma ke s).Ambiguous p icing ules ha e ecen ly
been s udied by Beissne and Riedel (2019) who de elop he gene al heo y o equilib ium
o such ma ke s and A aujo, Cha eauneu and Fa o (2012) discuss he s uc u e o ic ion-
less pa s o he ma ke . Bo h pape s immedia ely s a wi h he (sublinea ) supe hedging
unc ional ha we discuss below. △
Example 2.3 (En opy P icing).Conside he si ua ion when he e a e h ee s a es o he
na u e, Ω = {ω1, ω2, ω3}, h ee asse s (J= 3) aded on he ma ke s and he p ice o a
po olio a ime 0 is gi en by he unc ional
p(θ) = log1
3eθ1+θ2+θ3+1
3eθ1+2θ2+θ3+1
3eθ1+θ2+2θ3,
o all θ∈RJ, while he payo s o po olios a e linea and gi en by he mapping G(θ) =
(θ1+θ2+θ3, θ1+ 2θ2+θ3, θ1+θ2+ 2θ3). In his case, he ma ke s uc u e (p, G) desc ibes
ic ions only ela ed o p icing. No ading a ime 0 is no cos ly, i.e. p(0) = log(1) = 0.
Mo eo e , we ha e ha
p(θ+ke1) = log1
3e1+θ1+θ2+θ3+1
3e1+θ1+2θ2+θ3+1
3e1+θ1+θ2+2θ3
= log1
3eθ1+θ2+θ3+1
3eθ1+2θ2+θ3+1
3eθ1+θ2+2θ3ek
= log1
3eθ1+θ2+θ3+1
3eθ1+2θ2+θ3+1
3eθ1+θ2+2θ3+k
=p(θ) + k.
Finally, obse e ha pis he composi ion o he con ex unc ion X7→ logP3
i=1 1
3eX(ωi)
wi h he linea unc ion G(θ)=(θ1+θ2+θ3, θ1+ 2θ2+θ3, θ1+θ2+ 2θ3), so ha i is con ex
as well. Hence, we can conclude ha (p, G) is con ex and cash-addi i e. △
3 A bi age and he Fundamen al Theo em
In his sec ion, we gi e a de ini ion o a bi age o ou ma ke s uc u es and p o ide a
e sion o he undamen al heo em o asse p icing. We adop he ollowing s eng hening
s anda d no ion o absence o a bi age (e.g., see Ross (2005)).
De ini ion 3.1 (Robus no-a bi age L´ecuye and Ma ins-da Rocha (2021)).The con ex
ma ke s uc u e (p, G) sa is ies he obus no-a bi age p ope y i pand Ga e Ga eaux
6
di e en iable and he e exis s a po olio θ0∈RJsuch ha o any di ec ion η∈RJ, he
condi ions
G′θ0;η⩾0 and p′θ0;η⩽0,
imply
G′θ0;η= 0 and p′θ0;η= 0.
As discussed by L´ecuye and Ma ins-da Rocha (2021), his no ion o no-a bi age is a
s onge equi emen han he s anda d no-a bi age condi ion, bu hey coincide when pand
Ga e linea . Indeed, obse e ha when pand Ga e bo h linea , we ha e ha p′(θ0;η) = p(η)
and G′(θ0;η) = G(η), so ha he obus no-a bi age condi ion becomes equi alen o
[G(η)⩾0 and p(η)⩽0] =⇒[G(η) = 0 and p(η) = 0],
which is exac ly he s anda d no ion o he absence o a bi age.
We in oduce a new cha ac e iza ion o his obus no-a bi age speci ic o he cash-
in a ian case.
Theo em 3.1 (FTAP o con ex and cash addi i e ma ke s uc u es).Conside a con ex
ma ke s uc u e (p, G)such ha bo h pand Ga e Ga eaux di e en iable. Then (p, G)
sa is ies he obus no-a bi age p ope y i and only i he e exis s a measu e µ∈∆(Ω) wi h
ull suppo such ha o all po olios θ∈RJ
EµG(θ)≤p(θ).
No ice ha ou cha ac e iza ion is s onge han ha in L´ecuye and Ma ins-da Rocha
(2021) ob ained o gene al con ex ma ke . Indeed, he e µis a p obabili y measu e—wha
is commonly e e ed o as a ma ingale p obabili y measu e, while in hei case i is only
es ic ed o be a s ic ly posi i e ec o . Hence, con ex cash addi i e ma ke s uc u es
ha e a majp ad an age compa ed o a bi a y con ex ma ke s uc u e. We illus a e his
esul in he ollowing example.
Example 3.1 (Example 2.3 con inued).In his case we ha e ha he p ice unc ion is
p(θ) = log1
3eθ1+θ2+θ3+1
3eθ1+2θ2+θ3+1
3eθ1+θ2+2θ3,
7
5 Conclusion
This pape discusses hedging and he absence o a bi age in con ex cash-addi i e ma -
ke s. We p o ide a e sion o he undamen al heo em o asse p icing in his se ing. By
ocusing on he supe - eplica ion p ices, we in oduce he concep o a con idence unc ion
ha e lec s ma ke ic ions, such as ansac ion cos s and axes. As a majo special case,
we cha ac e ize he case o en opy p icing supe - eplica ion unc ionals. Hence, ou ind-
ings ex end adi ional asse p icing amewo ks o be e accommoda e eal-wo ld ma ke
impe ec ions.
6 Appendix
6.1 P oo s
P oo o Theo em 3.1.By Theo em 3.3 in L´ecuye and Ma ins-da Rocha (2021) i ol-
lows ha (p, G) sa is ies obus no-a bi age i and only i he e exis s a s ic ly posi i e
ec o µ∈RΩ
++ and a po olio θ⋆∈RJsuch ha
p(θ)−p(θ⋆)⩾µ·[G(θ)−G(θ⋆)],(3)
o all θ∈RJ.
Hence, i we assume ha (p, G) sa is ies he no-a bi age condi ion, hen by (3) i ollows
ha i we se θ=θ⋆−1, hen
1⩾X
ω∈Ω
µ(ω).
I we subs i u e in o (3) θ=θ⋆+ 1, hen by he same easoning we ob ain ha
1⩽X
ω∈Ω
µ(ω).
We can he e o e conclude ha µ∈∆(Ω).
Con e sely, i he e exis s a ma ingale measu e µwi h ull suppo such ha o all
po olios θ∈RJ
EµG(θ)≤p(θ),
hen (3) holds wi h θ⋆= 0, deli e ing he desi ed esul .
14
P oo o P oposi ion 3.1.Mono onici y o he supe - eplica ion p ice was showed in
L´ecuye and Ma ins-da Rocha (2021). Fi s , we a e going o show ha πpis cash-addi i e
when (p, G) is cash-addi i e. Le X∈RΩand k∈R. Then by de ini ion,
πp(X+k1Ω) = in {p(η) : G(η)⩾X+k1Ω}.
This is equi alen o
πp(X+k1Ω) = in {p(θ) : G(θ)−k1Ω⩾X}.
By cash-addi i i y o G, we ha e
πp(X+k1Ω) = in {p(θ) : G(θ−ke1)⩾X}.
We pose η1=θ1−kand ηj=θj o all j∈J {1}, hen
πp(X+k1Ω) = in {p(η+ke1) : G(η)⩾X}.
By cash-addi i i y o p, i is equi alen o
πp(X+k1Ω) = in {p(η) + k:G(η)⩾X}.
Hence he desi ed esul
πp(X+k1Ω) = πp(X) + k.
Now, we a e going o show ha he supe - eplica ion p ice only akes ini e alues. Le
k > 0, by cash addi i i y we ha e πp(k) = k. Assume he e exis s X∈RΩsuch ha
π(X) = −∞ and le X′= 2k−X. Since πdoes no ake he alue +∞, he e exis s α1∈R
such ha α > π(X′). We ha e π(X)<−αbu con exi y o πimplies ha
π(k)⩽1
2π(X′) + 1
2π(X),
I is equi alen o
k⩽1
2α−1
2α,
which con adic s he ac ha k > 0. Hence, πpdoes no ake he alue −∞ on RΩ.
P oo o Theo em 3.2.Le α:RΩ→R∪ {+∞} be he Legend e-Fenchel ans o m o
πp, ha is
α(P) := sup
X∈RΩ
(P·X−πp(X)).
15
Obse e ha α(P)∈[0,∞] o e e y P∈∆(Ω). To see his, obse e ha
α(P)⩾P·0−πp(0) = 0.
The supe - eplica ion p ice πis con ex, mo eo e , we showed ha i only akes ini e al-
ues and is con inuous. The Fenchel-Mo eau heo em s a es ha a unc ion is equal o i s
biconjuga e i i is p ope 2, lowe semi-con inuous and con ex. Hence, in pa icula , i is
equal o i s biconjuga e i i is con ex, con inuous and only akes ini e alues. Hence, he
supe - eplica ion p ice is equal o
πp(X) = sup
P∈RΩ
(P·X−α(P)).
We a e going o show ha α(P) = +∞i Pis no a p obabili y.
Assume i s ha P(ω)<0 o some ω∈Ω and choose an a bi a y n > 0. Le
Xn=−n1ωwhe e 1ωis he ec o o RΩwi h coo dina e ωequal o 1 and 0 o he wise.
L´ecuye and Ma ins-da Rocha (2021) show ha he supe - eplica ion p ice πpis mono one
hence since 0 ⩾Xnwe ha e 0 ⩾π(Xn). Mo eo e , by cons uc ion, P·Xn=−nP(ω)>0
hence we ha e
α(P)⩾P·Xn−π(Xn)⩾P·Xn,
ha is,
α(P)⩾−nP(ω)>0.
Since ncan be a bi a ily la ge, we ha e α(P)=+∞when P(ω)<0 o some ω∈Ω.
Now, assume ha P⩾0 and Pis no a p obabili y. Fi s assume ha Pω∈ΩP(ω)>1
and choose an a bi a y k > 0. By de ini ion, αsa is ies
α(P)⩾kX
ω∈Ω
P(ω)−π(k1Ω).
Cash-addi i i y implies
α(P)⩾k
X
ω∈Ω
P(ω)−1
.
Since kwas aken a bi a ily, i implies α(P)=+∞when P⩾0 and PωΩP(ω)>1.
Assume now ha Pω∈ΩP(ω)<1 and choose an a bi a y k > 0. Then, by de ini ion, α
sa is ies
α(P)⩾−kX
ω∈Ω
P(ω)−π(−k1Ω).
2We say ha a con ex unc ion :Rn→R∪ {−∞,+∞} is p ope when (X)<+∞ o a leas one
X∈Rnand (X)>−∞ o all X∈Rn.
16
The e o e, by cash-addi i i y,
α(P)⩾k
1−X
ω∈Ω
P(ω)
.
Since kwas aken a bi a ily, i implies α(P)=+∞when P⩾0 and PωΩP(ω)<1.
Hence, he supe - eplica ion p ice sa is ies
πp(X) = sup
P∈∆(Ω)
(EP(X)−α(P)),
whe e Pis a p obabili y. Since ∆(Ω) is a compac se and by he ex eme alue heo em, a
con inuous unc ion a ains i s sup emum on a compac se , we ob ain ha
πp(X) = max
P∈∆(Ω)(EP(X)−α(P)),
as desi ed.
P oo o P oposi ion 3.2.The p oo o his p oposi ion is a small a ia ion o he p oo
o F¨ollme and Schied (2016) Theo em 4.15. We w i e i o ac abili y. Le X∈RΩand
le X′=X−πp(X)1Ω, hen, we ha e
αmin(P)⩾EP(X′) = EP(X)−πp(X),
o all P∈∆(Ω). Hence,
πp(X)⩾sup
P∈∆(Ω)
(EP(X)−αmin(P)).
Fo all X∈RΩ, we wan o cons uc QX∈∆(Ω) such ha
πp(X)⩽EQX(X)−αmin(QX).(4)
as i would imply
πp(X) = max
P∈∆(Ω)(EP(X)−αmin(P)).
We is ix X∈RΩsuch ha πp(X) = 0. The payo Xis no con ained in he se
B={X∈RΩs. . πp(X)<0},
17
which is open. Hence we can sepa a e Xand B, ha is, he e exis s a non-ze o con inuous
linea unc ional ℓon RΩsuch ha
ℓ(X)⩾sup
Y∈B
ℓ(Y).
Le Y⩽0, by cash-addi i i y and mono onici y, −1Ω+λY ∈ B o all λ > 0. Hence,
ℓ(X)⩾ℓ(−1Ω) + λℓ(Y).
I implies ℓ(Y)⩽0. Since ℓis non-ze o, he e exis s Y∈RΩsuch ha ℓ(Y)<0, mo eo e
since i is nega i e we ha e ℓ(Y−)<0. Wi hou loss o gene ali y we can ake Ysuch ha
maxω∈Ω|Y(ω)|<1 and we ha e ℓ(−1Ω−Y−)⩽0. I implies ℓ(−1Ω) = ℓ(−1Ω−Y−) +
ℓ(Y−)<0. We le QXbe such ha
EQX(Y) = ℓ(Y)
ℓ(1Ω) o all Y∈RΩ.
We ha e
αmin (QX) = sup
Y∈Aπp
EQX(Y)⩾sup
Y∈B
EQX(Y) = 1
ℓ(1Ω)sup
Y∈B
ℓ(Y).
Mo eo e , o all Y∈ Aπp, we ha e Y−ϵ∈ B o all ϵ > 0. Hence Aπp−ϵ⊆ B o all ϵ > 0
which implies
αmin (QX) = 1
ℓ(1Ω)sup
Y∈B
ℓ(Y).
We ha e
EQX(X)−αmin (QX) = 1
ℓ(1Ω)(ℓ(X)−sup
Y∈B
ℓ(Y)) ⩾0 = πp(X).
By cash-addi i i y, o e e y Y∈RΩ, we can cons uc QYsa is ying Equa ion 4 ollowing
he same s eps by le ing X=Y−πp(Y)1Ω. I en ails
πp(X) = max
P∈∆(Ω)(EP(X)−αmin (P)) o all X∈RΩ.
Mo eo e , αmin is he minimal con idence unc ion sa is ying Equa ion 1. Indeed, o
e e y αsa is ying Equa ion 1, we ha e
α(P) = sup
X∈RΩ
EP(X)−πp(X)⩾sup
X∈Aπp
EP(X)−πp(X)⩾αmin (P),
o all P∈∆(Ω).
18
P oo o Theo em 3.3.We i s show ha i he ma ke s uc u e (p, G) does no admi
a bi age hen he e exis s a payo X0∈RΩsuch ha o all payo s X∈RΩ, we ha e
[X⩾0 and πp(X+X0)−πp(X0)⩽0] =⇒X= 0.
Fi s , we assume ha he e is no a bi age oppo uni y and we a e going o show ha he e
exis a payo X0∈RΩand a p obabili y µ∈∆(Ω) such ha
α(µ)=0, πp(X0) = Eµ(X0),
and
πp(X)⩾Eµ(X) o all X∈RΩ.
.
L´ecuye and Ma ins-da Rocha (2021) show ha a lowe -semicon inuous supe - eplica ion
p ice πpde ined on a closed se is consis en wi h he absence o a bi age i , and only i
he e exis a payo X0∈RΩand a s ic ly posi i e ec o µ∈RΩ
++ such ha
πp(X)−πp(X0)⩾µ·(X−X0),(5)
o all X∈RΩ. We can ew i e he condi ion as
πp(X+X0)−πp(X0)⩾µ·X.
o all X∈RΩ. Cash-addi i i y implies Pω∈Ωµ(ω) = 1. Hence, µis a p obabili y.
Taking X= 0 in Equa ion 5, i implies Eµ(X0)⩾πp(X0). Deno e PX0 he p obabili y
such ha
πp(X0) = EPX0(X0)−α(PX0).
We ha e
∇[Eµ(X0)−EPX0(X0) + α(PX0)] = µ−PX0⩾0.
I implies µ=PX0. The e o e α(µ) = 0 and πp(X0) = Eµ(X0). I en ails ha
πp(X)⩾µ·X(6)
o all X∈RΩ.
Con e sely, assume ha he e exis a payo X0∈RΩand a p obabili y µ∈∆(Ω) such
ha
α(µ)=0, πp(X0) = Eµ(X0),
19
and
πp(X)⩾Eµ(X) o all X∈RΩ.
We a e going o show ha he e is no a bi age oppo uni y. Le X∈RΩsuch ha
X⩾0 and πp(X+X0)−πp(X0)⩽0.
We ha e
0⩾πp(X+X0)−Eµ(X0)⩾Eµ(X+X0)−Eµ(X0) = Eµ(X).
Since µ > 0, we ha e 0 ⩾Eµ(X)⩾0. Hence X= 0.
6.2 P oo o Theo em 4.1
We in oduce i s p elimina y de ini ions.
De ini ion 6.1. A unc ion π:RΩ→Ris
1 con inuous i o e e y sequence (X)∞
n=0 in RΩ, limnXn=Ximplies limnπ(Xn) =
π(X);
2 disjoin exponen ial-addi i e i o e e y X, Y ∈RΩ
supp X∩supp Y=∅=⇒eπ(X+Y)=eπ(X)+eπ(Y)−1;
3 ansla ion in a ian i π(X+k) = π(X) + k;
4 no malized i π(k) = k o e e y k∈R;
5 s ic ly mono one i X⩾Y=⇒π(X)⩾π(Y) and π(X)> π(Y) i i u he holds
ha X(k)> Y (k) o some k.
The nex no el key lemma cha ac e izes unc ionals ha admi an en opic ep esen a-
ion.
Lemma 6.1. A unc ional π:RΩ→Rsa is ies (1)-(5) i and only i he e exis s Q∈∆(Ω)
wi h ull suppo such ha
π(X) = max
P∈∆(Ω)
EPX−R(P||Q) = log(EQ(eX)),
20
P oo o Theo em 4.1.
Su iciency. De ine πe:RΩ→Rby πe(X) = eπ(X)−1 o e e y X∈RΩ. Then condi ions
(1), (2), and (5) imply ha πesa is ies he condi ions o Theo em 2 in S anca (2020), so
ha he e exis s Q∈∆(Ω) and a con inuous and s ic ly inc easing unc ion ϕ:R→R
sa is ying ϕ(0) = 0 such ha
πe(X) = EQϕ(X),
o e e y X∈RΩ. Mo eo e , Qmus ha e ull suppo because o (5). To see his, obse e
ha Qsa is ies Q(A) = πe(1A) o e e y A⊆Ω.
Finally, obse e ha since πis no malized we ob ain πe(k) = ϕ(k) = ek−1 o e e y
k∈R. Hence, since ϕ−1(k) = log(k+ 1), we ob ain
π(X) = ϕ−1(πe(X)) = ϕ−1(EQϕ(X)) = logEQeX.
Hnece by P oposi ion 1.4.2 in Dupuis and Ellis (2011), we ob ain ha
π(X) = max
P∈∆(Ω)
EX−R(P||Q),
which concludes his pa o he p oo .
Necessi y. We only p o e disjoin addi i i y, as checking he necessi y o he o he condi ions
is ou ine. Conside X, Y ∈RΩwi h supp X∩supp Y=∅. We ha e
eπ(X+Y)=EQeX+Y
=EQeXeaY
=EQeX1supp X+EQeY1supp Y−1
=eπ(X)+eπ(Y)−1,
as desi ed.
Lemma 6.2. Assume ha Gis mono one, linea , G(RJ) = RΩ, and ha pis s ic ly
mono one and sa is ies segmen ed addi i i y. Then πsa is ies (1)-(6).
P oo . We show ha πsa is ies (2) as he checking ha he o he p ope ies a e sa is ied is
ou ine. Recall ha he supe - eplica ion p ice is gi en by
π(X) = in {p(θ) : θ∈RJand X⩽G(θ)}.
21
Take X, Y ∈RΩsuch ha supp Xsupp Y=∅. Since pis con inuous, s ic ly mono one and
Gis linea , mono one and sa is ies G(RJ) = RΩwe can ind po olios ˜
θand ˜
θ′such ha
supp ˜
θ∩supp ˜
θ′=∅,
G(˜
θ) = X,G(˜
θ′) = Y,p(˜
θ) = π(X) and p(˜
θ′) = π(Y). I ollows ha
π(X+Y) = in {p(θ) : θ∈RJ, X +Y⩽G(θ)}
= in {p(θ+θ′) : θ, θ′∈RJ,supp θ∩supp θ′=∅, X =G(θ), Y =G(θ′)}.
Hence, because x7→ exis a s ic ly inc easing and con inuous unc ion, and since supp G(θ)∩
supp G(θ′) = ∅ o any G(θ) = X,G(θ′) = Y, by segmen ed addi i i y we ob ain
eπ(X+Y)= expnin {p(θ+θ′) : θ, θ′∈RJ,supp θ∩supp θ′=∅, X =G(θ), Y =G(θ′)}o
= in {ep(θ+θ′):θ, θ′∈RJ,supp θ∩supp θ′=∅, X =G(θ), Y =G(θ′)}
= in {ep(θ)+ep(θ′)−1 : θ, θ′∈RJ,supp θ∩supp θ′=∅, X =G(θ), Y =G(θ′)}
= in {ep(θ)+ep(θ′):θ, θ′∈RJ,supp θ∩supp θ′=∅, X =G(θ), Y =G(θ′)} − 1
= in {ep(θ):θ∈RJ, X =G(θ)}+ in {ep(θ′):θ′∈RJ, Y =G(θ′)} − 1.
We can he e o e conclude ha
eπ(X+Y)=ep(˜
θ)+ep(˜
θ′)=eπ(X)+eπ(Y),
as desi ed.
We can now deli e he p oo o he esul .
P oo o Theo em 4.1. By Lemma 6.2, πsa is ies (1)-(5). The e o e he esul ollows by
applying Lemma 6.1.
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