No-Arbitrage Pricing for Dividend-Paying Securities in Discrete-Time Markets with Transaction Costs

Transcription

1 No-Arbirage Pricing for Dividend-Paying Secriies in Discree-Time Markes wih Transacion Coss Tomasz R. Bielecki Deparmen of Applied Mahemaics, Illinois Insie of Technology, Chicago, IL, USA Rodrigo Rodrigez Deparmen of Applied Mahemaics, Illinois Insie of Technology, Chicago, IL, USA Firs Circlaed: May 25, 2012 Igor Cialenco Deparmen of Applied Mahemaics, Illinois Insie of Technology, Chicago, IL, USA Absrac We prove a version of Firs Fndamenal Theorem of Asse Pricing nder ransacion coss for discree-ime markes wih dividend-paying secriies. Specifically, we show ha he no-arbirage condiion nder he efficien fricion assmpion is eqivalen o he exisence of a risk-neral measre. We derive dal represenaions for he sperhedging ask and sbhedging bid price processes of a derivaive conrac. Or resls are illsraed wih a vanilla credi defal swap conrac. Keywords: arbirage, fndamenal heorem of asse pricing, ransacion coss, consisen pricing sysem, liqidiy, dividends, credi defal swaps MSC2010: 91B25; 60G42; 46N10; 46A20 Tomasz R. Bielecki and Igor Cialenco acknowledge sppor from he NSF gran DMS

2 1 Inrodcion One of he cenral hemes in mahemaical finance is no-arbirage pricing and is applicaions. A he fondaion of no-arbirage pricing is he Firs Fndamenal Theorem of Asse Pricing FFTAP. We prove a version of he FFTAP nder ransacion coss 1 for discree-ime markes wih dividend-paying secriies. The FFTAP has been proved in varying levels of generaliy for fricionless markes. In a discree-ime seing for a finie sae space, he heorem was firs proved in Harrison and Pliska HP81]. Almos a decade laer, Dalang, Moron, and Willinger DMW90] proved he FFTAP for he more echnically challenging seing in which he sae space is general. Their approach reqires he se of advanced, measrable selecion argmens, which moivaed several ahors o provide alernaive proofs sing more accessible echniqes see Schachermayer Sch92], Kabanov and Kramkov KK94], Rogers Rog94], Jacod and Shiryaev JS98], and Kabanov and Sricker KS01b]. Using advanced conceps from fncional and sochasic analysis, he FFTAP was firs proved in a general coninos-ime se-p in he celebraed paper by Delbaen and Schachermayer DS94]. A comprehensive review of he lierare peraining o no-arbirage pricing heory in fricionless markes can be fond in Delbaen and Schachermayer DS06]. The firs rigoros sdy of he FFTAP for markes wih ransacion coss in a discreeime seing was carried o by Kabanov and Sricker KS01a]. Under he assmpion ha he sae space is finie, i was proved ha NA is eqivalen o he exisence of a consisen pricing sysem sing he erminology inrodced in Schachermayer Sch04]. However, heir resls did no exend o he case of a general sae space. As in he fricionless case, he ransiion from a finie sae space o a general sae space is nonrivial de o measre-heoreic and opological relaed difficlies. These difficlies were overcome in Kabanov, Rásonyi, and Sricker KRS02], where a version of he FFTAP was proven nder he efficien fricion assmpion EF. I was shown ha he sric no-arbirage condiion, a condiion which is sronger han NA, is eqivalen o he exisence of a sricly consisen pricing sysem. Therein, i was asked wheher EF can be discarded. Schachermayer Sch04] answered his qesion negaively by showing ha neiher NA nor he sric no-arbirage condiion alone is sfficienly srong o yield a version of he FFTAP. More imporanly, Schachermayer Sch04] proved a new version of he FFTAP ha does no reqire EF was proved. Specifically, he proved ha he robs no-arbirage condiion, which is sronger han he sric no-arbirage condiion, is eqivalen o he exisence of a sricly consisen pricing sysem. Sbseqen sdies ha rea he robs no-arbirage condiion are Bochard Bo06], Vallière, Kabanov, and Sricker DVK07], Jacka, Berkaoi, and Warren JBW08]. Recenly, Pennanen Pen11d, Pen11a, Pen11b, Pen11c] sdied no-arbirage pricing in a general conex in which markes can have consrains and ransacion coss may depend nonlinearly on raded amons. Therein, he problem of sperhedging a claims 1 In his sdy, a ransacion cos is defined as he cos incrred in rading in a marke in which secriies qoed prices have a bid-ask spread. We do no consider oher coss sch as broker s fees and axes. 2

3 process e.g. swaps is also invesigaed. An excellen srvey of he lierare peraining o no-arbirage pricing in markes wih ransacion coss can be fond in Kabanov and Safarian KS09]. Le s menion ha versions of he FFTAP for markes wih ransacion coss in a coninos-ime seing have also been sdied in he lierare. This lierare considers sronger condiions han NA see for insance Joini and Kallal JK95], Cherny Che07], Gasoni, Rásonyi, and Schachermayer GMS10], Denis, Gasoni, and Rásonyi DGR11], Denis and Kabanov DK12]. The fndamenal difference beween no-arbirage pricing heory for dividend-paying secriies and non-dividend paying secriies is ha ransacion coss de o rading in non-dividend paying secriies may no proporional o he nmber of nis of secriies prchased or sold. For insance, ransacion coss associaed wih ineres rae swap conracs and credi defal swap conracs accre over ime by merely holding he secriy for a non-dividend paying secriy ransacion coss are only charged whenever he secriy is bogh or sold. Or consideraion of ransacion coss on dividends disingishes his sdy. The conribion of his paper is smmarized as follows: We define and sdy he vale process and he self-financing condiion nder ransacion coss for discree-ime markes wih dividend-paying secriies Secion 2. We define and invesigae NA and EF in or conex Secion 3. We prove a key closedness propery of he se of vales ha can be sperhedged a zero cos Secion 3.1. Using classic separaion argmens, we prove a version of he FFTAP ha is relevan o or se-p. Specifically, we prove ha NA nder EF is saisfied if and only if here exiss a risk-neral measre Secion 3.2. We inrodce an appropriae noion of consisen pricing sysems in or se-p, and we sdy he relaionship beween hem and NA nder EF Secion 4. We demonsrae ha, if here are no ransacion coss on he dividends paid by secriies, NA nder EF is eqivalen o he exisence of a consisen pricing sysem Secion 4.1. We derive a dal represenaion for he sperhedging ask and sbhedging bid price processes for a derivaive conrac Secion 5. 2 The vale process and he self-financing condiion Le T be a fixed ime horizon, and le T := {0, 1,..., T }. Nex, le Ω, F T, F = F T, P be he nderlying filered probabiliy space. On his probabiliy space, we consider a marke consising of a savings accon B and of N raded secriies saisfying he following properies: 3

4 The savings accon can be prchased and sold according o he price process B := s=0 1 r s T, where r T =0 is a nonnegaive process specifying he risk-free =0 rae. The N secriies can be prchased according o he ex-dividend price process P ask := P ask,1,..., P ask,n T ; he associaed cmlaive dividend process is denoed by =0 A ask := A ask,1,..., A ask,n T =1. The N secriies can be sold according o he ex-dividend price process P bid := P bid,1,..., P bid,n T ; he associaed cmlaive dividend process is denoed by =0 A bid := A bid,1,..., A bid,n T =1. We assme ha he processes inrodced above adaped. In wha follows, we shall denoe by he backward difference operaor: X := X X 1, and we ake he convenion ha A ask 0 = A bid 0 = 0. I is easy o verify he following prodc rle for : X Y = X 1 Y Y X = X Y Y 1 X. Remark 2.1. For any = 1, 2,..., T and = 1, 2,..., N, he random variable A ask, is inerpreed as amon of dividend associaed wih holding a long posiion in secriy from ime 1 o ime, and he random variable A bid, is inerpreed as amon of dividend associaed wih holding a shor posiion in secriy from ime 1 o ime. We now illsrae he processes inrodced above in he conex of a vanilla Credi Defal Swap CDS conrac. Example 2.1. A CDS conrac is a conrac beween wo paries, a proecion byer and a proecion seller, in which he proecion byer pays periodic fees o he proecion seller in exchange for some paymen made by he proecion seller o he proecion byer if a pre-specified credi even of a reference eniy occrs. Le τ be he nonnegaive random variable specifying he ime of he credi even of he reference eniy. Sppose he CDS conrac admis he following specificaions: iniiaion dae = 0, expiraion dae = T, and nominal vale $1. For simpliciy, we assme ha he loss-given-defal is a nonnegaive scalar δ and is paid a defal. Typically, CDS conracs are raded on over-he-coner markes in which dealers qoe CDS spreads o invesors. Sppose ha he CDS spread qoed by he dealer o sell a CDS conrac wih above specificaions is κ bid o be received every ni of ime, and he CDS spread qoed by he dealer o by a CDS conrac wih above specificaions is κ ask o be paid every ni of ime. We remark ha he CDS spreads κ ask and κ bid are specified in he CDS conrac, so he CDS conrac o sell proecion is echnically a differen conrac han he CDS conrac o by proecion. The cmlaive dividend processes A ask and A bid associaed wih bying and selling he CDS wih specificaions above, respecively, are defined as A ask := 1 {τ } δ κ ask 1 {<τ}, A bid := 1 {τ } δ κ bid 1 {<τ} 4

5 for T. In his case, he ex-dividend ask and bid price processes P bid and P ask specify he mark-o-marke vales of he CDS for he proecion seller and proecion byer, respecively, from he perspecive of he proecion byer. From now on, we make he following sanding assmpion. Bid-Ask Assmpion: P ask P bid and A ask A bid. Noe ha if his assmpion is violaed, hen marke exhibis arbirage by simlaneosly bying and selling he corresponding secriy. For convenience, we define T := {1, 2,..., T }, J := {0, 1,..., N}, and J := {1, 2,..., N}. Unless saed oherwise, all ineqaliies and eqaliies beween processes and random variables are ndersood P-a.s. and coordinae-wise. 2.1 The vale process and self-financing condiion A rading sraegy is a predicable process φ := φ 0, φ 1,..., φ N T =1, where φ is inerpreed as he nmber of nis of secriy held from ime 1 o ime. Processes φ 1,..., φ N correspond o he holdings in he N secriies, and process φ 0 corresponds o he holdings in he savings accon B. We ake he convenion φ 0 = 0,..., 0. Definiion 2.1. The vale process V φ T =0 associaed wih a rading sraegy φ is defined as φ 0 1 N =1 φ 1 1 ask, bid, {φ 1 0}P0 1 {φ 1 <0}P0, if = 0, V φ = φ 0 B N =1 φ 1 bid, ask, {φ 0}P 1 {φ <0}P N =1 φ 1 {φ <0} Abid, 1 {φ 0} Abid,, if 1 T. For = 0, V 0 φ is inerpreed as he cos of he porfolio φ, and for φ i is inerpreed as he liqidaion vale of he porfolio before any ime ransacions, inclding any dividends acqired from ime 1 o ime. Remark 2.2. i I is imporan o noe he difference in he se of bid and ask prices, in he above definiion, beween he ime = 0 and he ime {1,..., T }. A ime = 0, V 0 φ is inerpreed as he cos of seing p he porfolio associaed wih φ. For = 1,..., T, he vale process V φ eqals he sm of he liqidaion vale of he porfolio associaed wih rading sraegy φ before any ime ransacions and he dividends associaed wih φ from ime 1 o. ii Also noe ha, de o he presence of ransacion coss, he vale process V may no be linear in is argmen, i.e. V φ V ψ V φ ψ, and V αφ αv φ for α R, and some rading sraegies φ, ψ, some ime T. This is he maor difference from he fricionless seing. 5

6 Nex, we inrodce he self-financing condiion, which is appropriae in he conex of his paper. Definiion 2.2. A rading sraegy φ is self-financing if B φ 0 1 for = 1, 2,..., T 1. = =1 =1 φ 1 1 ask, bid, { φ 1 0}P 1 { φ 1 <0}P φ 1 {φ 0} Aask, 1 {φ <0} Abid, 1 The self-financing condiion imposes he resricion ha no money can flow in or o of he porfolio. We noe ha if P := P ask = P bid and A := A ask = A bid, hen he self-financing condiion in he fricionless case is recovered. Remark 2.3. Noe ha he self-financing condiion no only akes ino accon ransacion coss de prchases and sales of secriies lef hand side of 1, b also ransacion coss accred hrogh he dividends righ hand side of 1. The nex resl gives a sefl characerizaion of he self-financing condiion in erms of he vale process. Proposiion 2.4. A rading sraegy φ is self-financing if and only if he vale process V φ saisfies V φ = V 0 φ φ 0 B =1 =1 φ φ =1 φ 1 { φ 0} P ask, 1 1 {φ 0} Aask, bid, 1 {φ 0}P ask, 1 {φ <0}P 1 { φ <0} P bid, 1 1 {φ <0} Abid, 2 for all T. Proof. By he definiion of V φ, and applying he prodc rle for he backwards difference 6

7 operaor, we obain V φ = V 0 φ =1 φ 1 =1 =1 =2 φ 0 B bid, 1 {φ 1 <0}P φ φ 1 B 1 φ 0 3 =2 0 1 {φ 1 {φ 0} Aask, ask, 1 0}P0 =1 φ 1 {φ <0} Abid, 1 {φ 1 <0} Abid, 1 1 {φ 1 0} Aask, 1. bid, 1 {φ 0}P ask, 1 {φ <0}P If φ is self-financing, hen we see ha 3 redces o 2. Conversely, assme ha he vale process saisfies 2. Sbracing 2 from 3 and applying he prodc rle for he backwards difference o boh sides yields ha φ is self-financing. The nex lemma exends he previos resl in erms of or nméraire B. For convenience, we le V φ := B 1 V φ for all rading sraegies φ. Proposiion 2.5. A rading sraegy φ is self-financing if and only if he disconed vale process V φ saisfies V φ = V 0 φ =1 =1 =1 φ B 1 bid, 1 {φ 0}P φ B { φ 0} P ask, 1 φ B 1 1 {φ 0} Aask, ask, 1 {φ <0}P 1 { φ <0} P bid, 1 1 {φ <0} Abid, 4 for all T. Proof. Sppose ha φ is self-financing. We may apply Proposiion 2.4 and he prodc rle for he backwards difference o see ha B 1 V φ = φ B 1 bid, ask, 1 {φ 0}P 1 {φ <0}P =1 =1 =1 φ B 1 1 ask, 1 { φ 0}P 1 bid, 1 { φ <0}P 1 φ B 1 1 {φ 0} Aask, 1 {φ <0} Abid, 7

8 for all T. Smming boh sides of he eqaion from = 1 o = shows ha necessiy holds. Conversely, if he vale process V φ saisfies 4, we may apply he prodc rle for he backwards difference o BB 1 V φ o dedce ha V φ = φ 0 B =1 =1 φ φ =1 φ ask, 1 { φ 0}P 1 bid, 1 {φ 0}P ask, 1 {φ <0}P bid, 1 { φ <0}P 1 1 {φ 0} Aask, 1 {φ <0} Abid,. Afer smming boh sides of he eqaion above from = 1 o = and applying Proposiion 2.4, we see ha φ is self-financing. Remark 2.6. If P = P ask = P bid and A = A ask = A bid, hen we recover he classic resl: a rading sraegy φ is self-financing if and only if he vale process saisfies for all T. V φ = V 0 φ =1 φ B 1 P w=1 Bw 1 A w For convenience, we define P ask, := B 1 P ask, P bid, := B 1 P bid, A ask, := B 1 A ask, and A bid, := B 1 A bid,. In fricionless markes, he se of all self-financing rading sraegies is a linear space becase secriies prices are no inflenced by he direcion of rading. This is no longer he case if he direcion of rading maers: he sraegy φ ψ may no be self-financing whenever φ and ψ are self-financing. Iniively his is re becase ransacion coss can be avoided whenever φ ψ < 0 by combining orders. However, he sraegy θ0, φ 1 ψ 1, φ 2 ψ 2,..., φ N ψ N can enoy he self-financing propery if he nis in he savings accon θ 0 are properly adsed. The nex lemma shows ha sch θ 0 exiss, is niqe, and saisfies φ 0 ψ 0 θ 0. Proposiion 2.7. Le ψ and φ be any wo self-financing rading sraegies wih V 0 ψ = V 0 φ = 0. Then here exiss a niqe predicable process θ 0 sch ha he rading sraegy θ defined as θ := θ 0, φ 1 ψ 1,..., φ N ψ N is self-financing wih V 0 θ = 0. Moreover, φ 0 ψ 0 θ 0. 8

9 Proof. The rading sraegies φ and ψ are self-financing, so by definiion we have ha and B 1 φ 0 B 1 ψ 0 =1 =1 = = φ =1 ψ =1 ask, 1 { φ 0}P 1 φ 1 1 {φ 1 0} Aask, 1 ask, 1 { ψ 0}P 1 ψ 1 1 {ψ 1 0} Aask, 1 bid, 1 { φ <0}P 1 1 {φ 1 <0} Abid, 1 bid, 1 { ψ <0}P 1 1 {ψ 1 <0} Abid, for = 2, 3,..., T. By adding eqaions 5 and 6, and rearranging erms we see ha ψ 0 φ 0 = ψ 1 0 φ 0 1 B 1 1 P ask, 1 φ ψ 7 =1 =1 =1 =1 φ 1 ψ 1 A ask, 1 P ask, 1 P bid, 1 1 { φ <0} φ 1 { ψ <0} ψ A 1 {φ 1 <0}φ 1 1 bid, {ψ 1 <0}ψ 1 1 for = 2, 3,..., T. Now recrsively define he process θ 0 as θ 0 1 : = =1 1 {φ 1 0}φ 1 1 {ψ 1 0}ψ 1 P ask, 0 =1 Aask, 1 1 {φ 1 <0}φ 1 1 {ψ 1 <0}ψ 1 P bid, 0, and θ 0 := θ 0 1 B 1 1 =1 =1 =1 φ ψ P ask, 1 1 { φ ψ<0} φ ψ P ask, 1 =1 1 {φ φ 1 ψ 1 <0} 1 ψ 1 A bid, 1 φ 1 ψ 1 A ask, 1 8 P bid, 1 Aask, 1 9

10 for = 2, 3,..., T. I follows ha θ 0 is niqe and is self-financing By definiion, he rading sraegy θ := θ 0, φ 1 ψ 1,..., φ N ψ N is self-financing. Sbracing 7 from 8 yields θ 0 φ 0 ψ 0 = θ 1 0 φ 0 1 ψ 1 0 =1 =1 =1 =1 1 { φ ψ<0} φ ψ P ask, 1 P bid, 1 P 1 { φ <0} φ 1 ask, { ψ <0} ψ 1 φ 1 1 ψ 1{φ A bid, 1 ψ 1 <0} 1 φ 1 1 {φ 1 <0} ψ 1 1 {ψ A bid, 1 <0} 1 for = 2, 3,..., T. I is sraighforward o verify ha he ineqaliy P bid, 1 Aask, 1 Aask, 1 1 {X<0} X 1 {Y <0} Y 1 {XY <0} X Y 10 holds for any random variables X and Y. Moreover, he ineqaliies P bid P ask and A ask A bid hold by assmpion. Hence, 9 redces o θ 0 φ 0 ψ 0 θ 0 1 φ 0 1 ψ 1 0 for = 2, 3,..., T. Since V 0 φ = V 0 ψ = 0, i follows ha θ 0 1 = φ0 1 ψ0 1 and V 0θ = V 0 φ V 0 ψ = 0. Afer recrsively solving 11, we conclde ha θ 0 φ 0 ψ 0 for all T. The nex resl is he naral exension of he previos proposiion o vale processes. I is iniively re since some ransacion coss may be avoided by combining orders. Theorem 2.8. Le φ and ψ be any wo self-financing rading sraegies sch ha V 0 φ = V 0 ψ = 0. There exiss a niqe predicable process θ 0 sch ha he rading sraegy defined as θ := θ 0, φ 1 ψ 1,..., φ N ψ N is self-financing wih V 0 θ = 0, and V T θ saisfies V T φ V T ψ V T θ. Proof. Le φ and ψ be self-financing rading sraegies. By applying Proposiion 2.4 and

11 rearranging erms, we may wrie V T φ V T ψ = φ 0 ψ B 0 =1 =1 φ T ψ bid, ask, T 1 {φ T ψ T 0}PT 1 {φ T ψ T <0}PT =1 φ ψ 1 { φ ψ 0} P ask, 1 φ ψ 1 {φ ψ 0} Aask, 1 { φ ψ<0} P bid, 1 1 {φ ψ<0} Abid, C 1 C 2, where C 1 is defined as C 1 := 1{φ T 0}φ T 1 {ψ T 0}ψ T P ask, T 1 {φ T <0}φ T 1 {ψ T <0}ψ T P bid, T =1 =1 =1 =1 and C 2 is defined as C 2 := 1 { φ <0} φ 1 { ψ <0} ψ P ask, 1 1 { φ 0} φ 1 { ψ 0} ψ P bid, 1 1{φ 0} φ 1 {ψ 0} ψ A bid, 1 {φ <0} φ 1 {ψ <0} ψ A ask,, =1 =1 =1 φ T ψ bid, ask, T 1 {φ T ψ T <0}PT 1 {φ T ψ T 0}PT φ ψ 1 { φ ψ 0} P bid, 1 1 { φ ψ<0} P ask, 1 φ ψ 1 {φ ψ 0} Abid, 1 {φ ψ<0} Aask,. By Proposiion 2.7, here exiss a niqe predicable process θ 0 sch ha he rading sraegy defined as θ := θ 0, φ 1 ψ 1,..., φ N ψ N is self-financing wih V 0 θ = 0 and 12 11

12 saisfies φ 0 ψ 0 θ 0. In view of Proposiion 2.5, since θ is self-financing, i follows ha V T θ = θ B 0 =1 =1 =1 φ T ψ bid, ask, T 1 {φ T ψ T 0}PT 1 {φ T ψ T <0}PT φ ψ 1 { φ ψ 0} P ask, 1 φ ψ 1 {φ ψ 0} Aask, Comparing eqaions 12 and 13 we see ha V T φ V T ψ = V T θ 1 { φ ψ<0} P bid, 1 1 {φ ψ<0} Abid,. 13 φ 0 ψ 0 θ B 0 C 1 C According o 10, he random variable C 1 C 2 is nonnegaive. Moreover, since φ 0 ψ 0 θ 0 and B 0, i follows ha T φ0 ψ 0 θ B 0 0 From 14, we conclde ha V T φ V T ψ V T θ. The following echnical lemma, which easy o verify, will be sed in he nex secion. Lemma 2.1. The following hold: Le Y a and Y b be any random variables, and sppose X m is a seqence of R-valed random variables converging a.s. o X. Then 1 {X m 0}X m Y b 1 {X m <0}X m Y a converges a.s. o 1 {X 0} XY b 1 {X<0} XY a. If a seqence of rading sraegies φ m converges a.s. o φ, hen V φ m converges a.s. o V φ. 2.2 The se of vales ha can be sperhedged a zero cos For all T, denoe by L 0 Ω, F, P ; R N1 he space of all P-eqivalence classes of R N1 -valed, F -measrable random variables. We eqip L 0 Ω, F, P ; R wih he opology of convergence in measre P. Also, le s define S := { φ : φ is self-financing }. For he sake of conciseness, we will refer o ses ha are closed wih respec o convergence in measre P simply as P-closed. We define he ses K := { V T φ : φ S, V 0 φ = 0 }, L 0 Ω, F T, P ; R := { X L 0 Ω, F T, P ; R : X 0 }, L 0 Ω, F T, P ; R := { X L 0 Ω, F T, P ; R : X 0 }, K L 0 Ω, F T, P ; R := { Y X : Y K and X L 0 Ω, F T, P ; R }. 12

13 The se K is he se of aainable vales a zero cos. On he oher hand, K L 0 Ω, F T, P ; R is he se of vales ha can be sperhedged a zero cos: for any X K L 0 Ω, F T, P ; R, here exiss an aainable vale a zero cos K K so ha X K. The following lemma assers ha he se of vales ha can be sperhedged a zero cos is a convex cone. Lemma 2.2. The se K L 0 Ω, F T, P ; R is a convex cone. Proof. Le Y 1, Y 2 K L 0 Ω, F T, P ; R. Then here exis K 1, K 2 K and Z 1, Z 2 L 0 Ω, F T, P ; R sch ha Y 1 = K 1 Z 1 and Y 2 = K 2 Z 2. By definiion of K, here exiss φ, ψ S wih V 0 φ = V 0 ψ = 0 sch ha K 1 = VT φ and K2 = VT ψ. We will prove ha for any posiive scalars α 1 and α 2 he following holds α 1 V T φ Z 1 α 2 V T ψ Z 2 K L 0 Ω, F T, P ; R, or, eqivalenly, ha here exiss K K sch ha α 1 V T φ α 2 V T ψ α 1 Z 1 α 2 Z 2 K. The vale process is posiive homogeneos, so α 1 VT φ α 2VT ψ = V T α 1φ VT α 2ψ. According o Theorem 2.8, here exiss a niqe predicable process θ 0 sch ha he rading sraegy defined as θ := θ 0, α 1 φ 1 α 2 ψ 1,..., α 1 φ N α 2 ψ N is self-financing wih V 0 θ = 0, and saisfies VT α 1φ VT α 2ψ VT θ. By definiion, we have V T θ K. Since α 1 V T φ α 2 V T ψ α 1 Z 1 α 2 Z 2 = V T α 1 φ V T α 2 ψ α 1 Z 1 α 2 Z 2 we conclde ha he claim holds. V T θ α 1 Z 1 α 2 Z 2 V T θ, Remark 2.9. The se K is no necessarily a convex cone. To see his, les sppose ha T = 1, J = {0, 1}, and r = 0. Consider he rading sraegies φ = {φ 0, 1} and ψ = {ψ 0, 1}, where φ 0 and ψ 0 are chosen so ha V 0 φ = V 0 ψ = 0. By definiion, V 1 φ, V 1 ψ K. However, is generally no in he se K. V 1 φ V 1 ψ = P bid 1 P ask 1 A ask 1 A bid 1 P bid 0 P ask 0, 3 The no-arbirage condiion We begin by inrodcing he definiion of he no-arbirage condiion. Definiion 3.1. The no-arbirage condiion NA is saisfied if for each φ S sch ha V 0 φ = 0 and V T φ 0, we have V T φ = 0. 13

14 In he presen conex, NA has he sal inerpreaion ha i is impossible o make somehing o of nohing. The nex lemma provides s eqivalen condiions o NA in erms of he se of aainable vales a zero cos, and also in erms of he se of vales ha can be sperhedged a zero cos. They are sraighforward o verify. Lemma 3.1. The following condiions are eqivalen: i NA is saisfied. ii K L 0 Ω, F T, P ; R L 0 Ω, F T, P ; R = {0}. iii K L 0 Ω, F T, P ; R = {0}. We proceed by defining The Efficien Fricion Assmpion in or conex cf. Kabanov e al. KRS02]. The Efficien Fricion Assmpion EF: { φ S : V0 φ = V T φ = 0 } = {0}. 15 Noe ha if 15 is saisfied, hen for each φ S, we have V 0 φ = V T φ = 0 if and only if φ = 0. Nex, we exemplify he imporance of EF in he conex of markes wih ransacion coss. Example 3.1. Le T = {0, 1} and J = {0, 1}, and assme ha P bid,1 < P ask,1, for = 0, 1, and A ask,1 1 < A bid,1 1. Sppose ha here exiss a rading sraegy φ ha is nonzero a.s. so ha V 0 φ = V 1 φ = 0. By he definiion of he vale process, φ {φ 1 1 0} φ1 1P ask,1 0 1 {φ 1 1 <0} φ 1 P bid,1 0 = 0, and V 1 φ = 1 {φ 1 1 0} φ1 1 1 {φ 1 1 <0} φ1 1 P bid,1 1 A ask,1 1 P ask,1 0 P ask,1 1 A bid,1 1 P bid,1 0 = 0. Sppose ha here exiss anoher marke B, P bid,1, P ask,1, Ãask,1, Ãbid,1 saisfying P bid,1 1 < P bid,1 ask,1 1 P 1 < P ask,1 1 a.s., and Ãask,1 1 = A ask,1 1, Ã bid,1 1 = A bid,1 bid,1 1, P 0 = P bid,1 0, and P 0 ask = P0 ask a.s. If we denoe by Ṽ φ he vale process corresponding o he marke B, P bid, P ask, Ãask, Ãbid, we hen have Ṽ0φ = V 0 φ = 0, Ṽ1φ 0, and PṼ1φ > 0 > 0, which violaes NA for he marke B, P bid, P ask, Ãask, Ãbid. We will denoe by NAEF he no-arbirage condiion nder he efficien fricion assmpion. 14

15 In wha follows, we denoe by P he se of all R N -valed, F-predicable processes. Also, we define he mapping F φ := =1 =1 φ bid,, T 1{φ T 0}PT 1 {φ T =1 for all R N -valed sochasic processes and le K := {F φ : φ P}. Remark 3.1. φ 1{ φ 0} P ask,, 1 φ 1{φ 0} Aask,, ask,, <0}PT 1 { φ <0} P bid,, 1 1 {φ <0} Abid,, φ s T s=1 L 0 Ω, F T, P; R N L 0 Ω, F T, P; R N, i Noe ha F is defined on he se of all R N -valed sochasic processes. On he conrary, he vale process is defined on he se of rading sraegies, which are R N1 - valed predicable processes. ii The se K has he same financial inerpreaion as he se K. We inrodce he se K becase i is more convenien o work wih from he mahemaical poin of view. iii F αφ = αf φ for any nonnegaive random variable α and R N -valed sochasic processes φ. The nex resls provides an eqivalen condiion for EF o hold. Lemma 3.2. The efficien fricion assmpion EF is saisfied if and only if {ψ P : F ψ = 0} = {0}. Proof. Le ψ P be sch ha F ψ = 0. We define he rading sraegy φ := φ 0, ψ 1,..., ψ N, where φ 0 is chosen so ha ψ S and V 0 φ = 0. we see ha VT φ = F ψ, which gives s V T φ = 0. EF is saisfied, so φ = 0 for = 0,..., N, which in pariclar implies ha ψ = 0 for = 1,..., N. Conversely, sppose EF holds, and fix φ S so ha V 0 φ = VT φ = 0. Define he predicable process ψ := φ for = 1,..., N. By Proposiion 2.5 and he definiion of F, i is re ha F ψ = VT φ. Ths, F ψ = 0. By assmpion, we have ha ψ = 0 for = 1,..., N, which implies φ = 0 for = 1,..., N. From he definiion of V 0 φ and becase V 0 φ = 0, i follows ha φ 0 1 = 0. Since φ S, we may recrsively solve for φ 0 2,..., φ0 T o dedce ha φ0 = 0 for = 2,..., T. Hence, φ = 0 for = 0, 1,..., N. Lemma 3.3. We have ha K = K. Proof. The claim follows from Proposiion

16 3.1 Closedness propery of he se of vales ha can be sperhedged a zero cos In his secion, we prove ha he se of vales ha can be sperhedged a zero cos, K L 0 Ω, F T, P ; R, is P-closed whenever NAEF is saisfied. This propery plays a cenral role in he proof of he Firs Fndamenal Theorem of Asse Pricing Theorem 3.4. We will denoe by he Eclidean norm on R N. Le s firs recall he following lemma from Schachermayer Sch04], which is closely relaed o Lemma 2 in Kabanov and Sricker KS01b]. Lemma 3.4. For a seqence of random variables X m L 0 Ω, F, P; R N here is a sricly increasing seqence of posiive, ineger-valed, F-measrable random variables τ m sch ha X τ m converges a.s. in he one-poin-compacificaion R N { } o some random variable X L 0 Ω, F, P; R N { }. Moreover, we may find he sbseqence sch ha X = lim sp m X m, where =. The nex resl exends he previos lemma o processes. Lemma 3.5. Le F i be a σ-algebra, and Yi m L 0 Ω, F i, P; R N for i = 1,..., M. Sppose ha F i F for all i, and ha Yi m saisfies lim sp m Yi m < for i = 1,..., M. Then here is a sricly increasing seqence of posiive, ineger-valed, F M -measrable random variables τ m sch ha, for i = 1,..., M, he seqence Y τ m i converges a.s. o some Y i L 0 Ω, F i, P; R N. Proof. We firs apply Lemma 3.4 o he random variable Y1 m : here exiss a sricly increasing seqence of posiive, ineger-valed, F 1 -measrable random variables τ1 m sch ha {τ1 1ω, τ 1 2ω,..., } N for ω Ω, and Y τ 1 m 1 converges a.s. o some Y 1 L 0 Ω, F 1, P; R N. Since lim sp m Y2 m <, we also have ha lim sp m Y τ 1 m 2 <. Moreover, Y τ 1 m 2 L 0 Ω, F 2, P; R N since F 1 F 2. Therefore, we may apply Lemma 3.4 o he seqence Y τ m 1 2 o find a sricly increasing seqence of posiive, ineger-valed, F 2 -measrable random variables τ2 m sch ha {τ 1 2 ω, τ 2 2 ω,... } {τ 1 1 ω, τ 2 1 ω,... } N, a.e. ω Ω, 17 and Y τ 2 m 2 converges a.s. o some Y 2 L 0 Ω, F 2, P; R N. From 17, he seqence Y τ 2 m 1 converges a.s. o Y 1. We may conine by recrsively repeaing he argmen above o he seqences Yi m, for i = 3,..., M, o find sricly increasing seqences of posiive, ineger-valed, F i -measrable random variables τi m sch ha {τ 1 i ω, τ 2 i ω,... } {τ 1 1 ω, τ 2 1 ω,... } N, a.e. ω Ω, 18 and Y τ i m i converges a.s. o some Y i L 0 Ω, F i, P; R N. Becase of 18, we see ha Y τ M m i converges a.s. o Y i for i = 1,..., M. Therefore, τ m := τm m defines he desired seqence. 16

17 We proceed by proving a echnical lemma. Lemma 3.6. Le F i be a σ-algebra, and Yi m L 0 Ω, F i, P; R N for i = 1,..., M. Sppose ha F i F for all i, and ha here exiss k {1,..., M} and Ω Ω wih PΩ > 0 sch ha lim sp m Yk mω = for a.e. ω Ω, and lim sp m Yi m ω < for i = 1,..., k 1 and for a.e. ω Ω. Then here exiss a sricly increasing seqence of posiive, ineger-valed, F k -measrable random variables τ m sch ha lim m Y τ m k ω =, for a.e. ω Ω, and 2 Xi m ω := 1 Ω ω Y τ m ω i ω Y τ, ω Ω, i = 1,..., M, m ω k ω saisfies lim m Xi m ω = 0, for i = 1,..., k 1 and for a.e. ω Ω Proof. Since lim sp m Yk mω = for a.e. ω Ω, we may apply Lemma 3.4 o he seqence Yk m o find a sricly increasing seqence of posiive, ineger-valed, F k -measrable random variables τ m so ha Y τ m ω k ω diverges for a.e. ω Ω. Becase lim sp m Yi m < for i = 1,..., k 1, we have lim sp m Y τ m i < for i = 1,..., k 1. Now since Y τ m ω k ω diverges for a.e. ω Ω, lim m Xm i ω = 1 Ω ω τ mω Yi ω Y τ m ω k ω = 0, a.e. ω Ω, i = 1,..., k 1. Ths, Xi m converges a.s. o 0 for i = 1,..., k 1, which implies ha Xi m o 0 for i = 1,..., k 1. Hence, he claim holds. We are now ready o prove he crcial resl in his paper. converges a.s. Theorem 3.2. If he no-arbirage condiion nder he efficien fricion assmpion NAEF is saisfied, hen he se K L 0 Ω, F T, P; R is P-closed. Proof. According o Lemma 3.3, we may eqivalenly prove ha K L 0 Ω, F T, P; R is P-closed. Sppose ha X m K L 0 Ω, F T, P; R converges in probabiliy o X. Then here exiss a sbseqence X km of X k so ha X km converges a.s. o X. Wih a sligh abse of noaion, we will denoe by X m he seqence X km in wha follows. By he definiion of K L 0 Ω, F T, P; R, here exiss Z m L 0 Ω, F T, P; R and φ m P so ha X m = F φ m Z m. 19 We proceed he proof in wo seps. In he firs sep, we show by conradicion ha lim sp m φ m s < for all s T. Sep 1a: Le s assme ha lim sp m φ m s < for all s T does no hold. Then { I 0 := s T : Ω Ω sch ha PΩ > 0, lim sp m φ m s ω = for a.e. ω Ω } 2 We ake Xi m ω = 0 whenever Y τ m ω k ω = 0. We will ake he convenion x/0 = 0 hrogho his secion. 17

18 is nonempy. Le 0 := min I 0, and define he F 0 1-measrable se E 0 := { ω Ω : lim sp φ m 0 ω = }. m Noe ha PE 0 > 0 by assmpion. We now apply Lemma 3.6 o φ m : here exiss a sricly increasing seqence of posiive, ineger-valed, F 0 1-measrable random variables τ0 m sch ha lim m m φτ 0 ω 0 ω =, a.e. ω E 0, 20 and ψ m,0 s := 1 E 0 saisfies lim m ψ m,0 s ω = 0, for s = 1,..., 0 1, for a.e. ω Ω. We proceed as follows. φ τ m 0 s φ τ m 0 0, s T, 21 Recrsively for i = 1,..., T If lim sp m ψs m,i 1 < for all s { i 1 1,..., T }, hen define k := i and ϕ m := ψ m,k 1, and proceed o Sep 1b. Else, define { i := min s { i 1 1,..., T } : Ω E i 1 s.. PΩ > 0, and lim sp ψs m,i 1 ω = for a.e. ω Ω }, m E i := { ω E i 1 : lim sp m ψ m,i 1 i ω = }. Nex, apply Lemma 3.6 o ψ m,i : here exiss a sricly increasing seqence of posiive, ineger-valed, F i 1-measrable random variables τi m sch ha {τ 1 i ω, τ 2 i ω,... } {τ 1 0 ω, τ 2 0 ω,... }, a.e. ω Ω, 22 he seqence ψ τ m i i,i 1 saisfies lim m ψτ m i ω,i 1 i ω =, a.e. ω E i, 23 and he seqence ψ m,i defined as ψ m,i s ψ τ i m s := 1 E i ψ τ i m i,i 1,i 1, s T, 24 saisfies lim m ψ m,i s ω = 0 for s = 1,..., i 1, for a.e. ω Ω. Repea: i i 1. 18

19 Given his consrcion, we define βi m ω := τ i τ i1 τk m ω, i {0,..., k}, ω Ω, k U m ω : = φ βm 0 ω 0 ω ψ βm i ω,i 1 i ω, ω Ω. i=1 We make he following observaions on his consrcion: i The consrcion always prodces a seqence ϕ m sch ha lim sp m ϕ m s < for all s T. Indeed, if i = T for some i = 1,..., T, hen lim m ψs m,i ω = 0 for s = 1,..., T 1, for a.e. ω Ω, and lim m ψ m,i T ω = 1 E iω, for a.e. ω Ω. The seqence ψ m,i clearly saisfies lim sp m ψs m,i < for all s T. ii We have ha ϕ m s L 0 Ω, F k 1, P, R N for s = 1,..., k 1, and ϕ m s L 0 Ω, F s 1, P, R N for s = k,..., T. Hence, he seqence ϕ m is no a seqence of predicable processes. However, he limi of any a.s. convergen sbseqence of ϕ m is predicable becase ϕ m s converges a.s. o 0 for s = 1,..., k 1. iii E k E 0, and PE k > 0. iv Any a.s. convergen sbseqence of ϕ m converges a.s. o a nonzero process since ϕ m k converges a.s. o 1 E k, which is nonzero a.s. since PE k > 0. v From 21 and 24, we have ϕ m s = 1 E φ βm 0 s Becase E k E 0, /U m for all s T, where E := k i=1 Ei. ϕ m s = 1 E k φ βm 0 s U m, s T. 25 vi U m ω diverges for a.e. ω E k since 20, 22, and 23 hold. Sep 1b: By he previos sep, lim sp m ϕ m s < for all s T. We apply Lemma 3.5 o ϕ m o find a sricly increasing seqence of posiive, ineger-valed, F T 1 -measrable random variables ρ m so ha ϕ ρm converges a.s. o some process ϕ sch ha 3 ϕ s L 0 Ω, F k 1, P; R N for s = 1,..., k 1, and ϕ s L 0 Ω, F s 1, P; R N for s = k,..., T. By observaion ii in Sep 1a, we have ha ϕ is predicable. Sep 1c: We proceed by showing ha NAEF implies PE 0 = 0. Towards his, we firs show ha he process ϕ consrced in Sep 1b saisfies F ϕ K. For he sake of noaion, we define η m := β ρm 0. From 25, we have ϕρm = 1 E kφ ηm /U ρm. Since 1 E k and U ρm are nonnegaive, R-valed random variables, 3 See observaion ii in Sep 1a. F φ ηm 1 E k = F U ρm φ ηm 1 E k = F ϕ ρm. 26 U ρm 19

20 Becase ϕ ρm converges a.s. o ϕ, we may apply Lemma 2.1 o see ha F ϕ ρm converges a.s. o F ϕ. Since ϕ is predicable, we have from he definiion of K ha F ϕ K. We proceed by showing ha F ϕ L 0 Ω, F T, P; R. Les begin by defining X m := X ηm /U ρm and Z m := Z ηm /U ρm. From 19, F φ ηm = X ηm Z ηm. 27 By mliplying boh sides of 27 by 1 E k/u ρm, we see from 26 ha F ϕ ρm = 1 E k X m Z m. 28 The seqence X m converges a.s. by assmpion, so he seqence X ηm also converges a.s. Recall ha he seqence U m ω diverges 4 for a.e. ω E k, so U ρm ω diverges for a.e. ω E k since {ρ 1 ω, ρ 2 ω,..., } N for a.e. ω Ω. Hence, 1 X m E k converges a.s. o 0. Since F ϕ ρm and 1 X m E k converge a.s., he seqence 1 Z m E k also converges a.s. o some Z L 0 Ω, F T, P; R. Ths, F ϕ ρm converges a.s. o Z, which implies F ϕ L 0 Ω, F T, P; R. Since F ϕ K, we immediaely see ha F ϕ K L 0 Ω, F T, P; R. I is assmed ha NA is saisfied, so by Lemmas 3.1 and 3.3 we dedce ha F ϕ = 0. We are spposing ha EF holds, so according o Lemma 3.2 we have ϕ = 0. This canno happen given or assmpion ha PE k > 0 becase 5 ϕ k = 1 E k. Therefore, we ms have ha PE k = 0. This conradics he consrcion in Sep 1a, so PE 0 = 0. Sep 2: By he conclsion in Sep 1, we obain ha lim sp m φ m s < for s T. By applying Lemma 3.5 o φ m, we may find a sricly increasing seqence of posiive, ineger-valed, F T 1 -measrable random variables σ m sch ha φ σm converges a.s. o some predicable process φ. By Lemma 2.1, he seqence F φ σm converges a.s. o F φ. Since φ P, we have F φ K. Becase X m converges a.s. o X, he seqence X σm also converges a.s. o X. From 19, i is re ha X σm = F φ σm Z σm. Since X σm and F φ σm converges a.s., he seqence Z σm also converges a.s. Ths, F φ σm X σm converges a.s. o some nonnegaive random variable Z := F φ X, which gives s ha X = F φ Z. We conclde ha X K L 0 Ω, F T, P; R. 3.2 The Firs Fndamenal Theorem of Asse Pricing In his secion, we formlae and prove a version of he Firs Fndamenal Theorem of Asse Pricing FFTAP. We define he following se for convenience: Z := {Q : Q P, P ask,, P bid,, A ask,, A bid, are Q-inegrable}. We now define a risk-neral measre in or conex. 4 See observaions vi in Sep 1a. 5 See observaions iv in Sep 1a. 20

21 Definiion 3.2. A probabiliy measre Q is a risk-neral measre if Q Z, and if E Q V T φ] 0 for all φ S sch ha φ is bonded a.s., for J, and V 0 φ = 0. A naral qesion o ask is wheher he expecaion appearing in he definiion above exiss. The following lemma shows ha, indeed, i does. Lemma 3.7. Sppose ha Q Z, and le φ S be sch ha φ is bonded a.s., for J, and V 0 φ = 0. Then VT φ is Q-inegrable. Proof. From he definiion of Z, he processes P ask,, P bid,, A ask,, A bid, are Q-inegrable. Becase Q is eqivalen o P, and since φ is bonded P-a.s. for J, we have ha φ is bonded Q-a.s. for J. Therefore, we see from Proposiion 2.5 ha E Q V T φ ] < holds. The nex lemma provides a mahemaically convenien condiion ha is eqivalen o NA. Lemma 3.8. The no-arbirage condiion NA is saisfied if and only if for each φ S sch ha φ is bonded a.s. for J, V 0 φ = 0, and V T φ 0, we have V T φ = 0. Proof. Necessiy holds immediaely, so we only show sfficiency. Le φ S be a rading sraegy so ha V 0 φ = 0 and V T φ 0. We will show ha V T φ = 0. Firs, define he F 1 measrable se Ω m, := {ω Ω : φ ω m} for m N, T, and J, and define he seqence of rading sraegies ψ m as ψ m, := 1 Ω m, φ for T and J, where ψ m,0 is chosen so ha ψ m is self-financing and V 0 ψ m = 0. Since 1 Ω m, converges a.s. o 1 for all T and J, we have ha ψ m, converges a.s. o φ for all T and J. Now we prove ha V 0 ψ m converges a.s. o V 0 φ. Towards his, we firs show ha ψ m, 1 converges a.s. o φ 1 for all J. By he definiion of V 0ψ m, V 0 ψ m = ψ m,0 1 =1 Since ψ m,0 is chosen so ha V 0 ψ m = 0, we have ψ m,0 1 = =1 ψ m, 1 1{ψ m, 1 0} P ask, 0 1 {ψ m, 1 <0} P bid, 0. ψ m, 1 1{ψ m, 1 0} P ask, 0 1 {ψ m, 1 <0} P bid, 0. The seqence ψ m, 1 converges a.s. o φ 1 for all J, so by Lemma 2.1 he seqence ψ m,0 1 converges a.s. o =1 φ ask, 1 1{φ 1 0}P0 1 {φ 1 bid, <0}P0 However, V 0 φ = 0, so ψ m,0 1 converges a.s. o φ 0 1 Ths, ψm, 1 converges a.s. o φ 1 J. By Lemma 2.1, we have ha V 0 ψ m converges o V 0 φ. 21. for all

22 According o Lemma 2.1, VT ψm converges a.s. o VT φ since ψm, converges a.s. o φ for all T and J. Nex, since ψ m is self-financing and ψ m, is bonded a.s. for all J and all m N, we obain V 0 ψ m = 0, VT ψ m 0 = VT ψ m = 0, m N. Since V 0 ψ m converges a.s. o V 0 φ, and VT ψm converges a.s. o VT φ we have V 0 φ = 0, V T φ 0 = V T φ = 0. Since V 0 φ = 0 and V T φ 0, we conclde ha V T φ = 0, so NA holds. Nex, we recall he well-known Kreps-Yan Theorem. I was firs proved by Yan Yan80], and hen obained independenly by Kreps Kre81] in he conex of financial mahemaics. For a proof of he version presened in his paper, see Schachermayer Sch92]. Theorem 3.2 and he Kreps-Yan Theorem will essenially imply he FFTAP Theorem 3.4. Theorem 3.3 Kreps-Yan. Le C be a closed convex cone in L 1 Ω, F, P; R conaining L 1 Ω, F, P; R sch ha C L 1 Ω, F, P; R = {0}. Then here exiss a fncional f L Ω, F, P; R sch ha, for each h L 1 Ω, F, P; R wih h 0, we have ha E P fh] > 0 and E P fg] 0 for any g C. We are now ready prove he following version of he FFTAP. Theorem 3.4 Firs Fndamenal Theorem of Asse Pricing. The following condiions are eqivalen: i The no-arbirage condiion nder he efficien fricion assmpion NAEF is saisfied. ii There exiss a risk-neral measre. iii There exiss a risk-neral measre Q so ha dq/dp L Ω, F T, P; R. Proof. In order o prove hese eqivalences, we show ha ii i, i iii, and iii ii. The implicaion iii ii is immediae, so we only show he remaining wo. ii i: We prove by conradicion. Assme here exiss a risk-neral measre Q, and ha NA does no hold. By Lemma 3.8, here exiss φ S so ha φ is bonded a.s., for J, V 0 φ = 0, VT φ 0, and PV T φω > 0 > 0. Since Q is eqivalen o P, we have V 0 φ = 0, VT φ 0 Q-a.s., and QV T φω > 0 > 0. So E QVT φ] > 0, which conradics ha Q is risk-neral. Hence, NA holds. i iii: We firs consrc a probabiliy measre P saisfying P Z and d P/dP L Ω, F T, P; R. Towards his, le s define he F T -measrable weigh fncion w := 1 =0 P ask, =0 P bid, 22 A ask, A bid,, 29

23 and le P be he measre on F T wih Radon-Nikodým derivaive d P/dP = c/w, where c is an appropriae normalizing consan. We see ha P is eqivalen o P, and d P/dP L Ω, F T, P; R. By he choice of he weigh fncion w, he processes P ask,, P bid,, A ask,, A bid, are P-inegrable. Ths, P Z. Nex, since P is eqivalen o P, i follows ha K L 0 Ω, F T, P ; R L 0 Ω, F T, P ; R = {0} by Lemma 3.1, he se K L 0 Ω, F T, P ; R is P-closed according o Theorem 3.2, and K L 0 Ω, F T, P ; R is a convex cone by Lemma 2.2. Le s now consider he se C := K L 0 Ω, F T, P ; R L 1 Ω, F T, P ; R. We observe he following ha C L 1 Ω, F T, P ; R = {0}, C is a convex cone, and C L 1 Ω, F T, P ; R. Moreover, since convergence in L 1 Ω, F T, P ; R implies convergence in measre P, we have ha C is closed in L 1 Ω, F T, P ; R. Ths, according o Theorem 3.3, here exiss a sricly posiive fncional 6 f L Ω, F T, P ; R sch ha E PKf] 0 for all K C. Becase 0 L 0 Ω, F T, P ; R, i follows from he definiion of C ha E PKf] 0, K K L 1 Ω, F T, P ; R. By he definiion of K, his implies ha E PV T φf] 0 for all φ S sch ha V 0φ = 0 and VT φ L1 Ω, F T, P ; R. In pariclar, E PV T φf] 0 for all φ S sch ha φ is bonded a.s., for J, V 0 φ = 0, and VT φ L1 Ω, F T, P ; R. Since P Z, we obain from Lemma 3.7 ha VT φ is P-inegrable. Ths, E PV T φf] 0 for all φ S sch ha φ is bonded a.s., for J, and V 0 φ = 0. We proceed by consrcing a risk-neral measre. Le Q be he measre on F T wih Radon-Nikodým derivaive dq/d P := cf, where c is an appropriae normalizing consan. Becase f is a sricly posiive fncional in L Ω, F T, P ; R, we have ha Q is eqivalen o P. Since P is eqivalen o P, i follows ha Q is eqivalen o P. Also, dq dp = dq d P d P dp = c c f w. 30 Ths, since w 1 and f L Ω, F T, P ; R, we have dq/dp L Ω, F T, P; R. This gives s dq/dp L Ω, F T, P; R since P is eqivalen o P. Moreover, we noe ha he processes P ask,, P bid,, A ask,, A bid, are Q-inegrable. Hence, Q Z. We conclde ha Q is a risk-neral measre since E Q VT φ] = c E PV T φf] 0 for all φ S sch ha φ is bonded a.s., for J, and V 0 φ = 0. Remark 3.5. i Noe ha EF is no needed o prove he implicaion ii i. ii In pracice, i is ypically reqired for a marke model o saisfy NA. According o Theorem 3.4, i is enogh o check ha here exiss a risk-neral measre. However, 6 For each h L 1 Ω, F T, P ; R wih h 0, we have E Pfh] > 0. 23

24 his is no sraighforward becase i has o be verified wheher here exiss a probabiliy measre Q Z so ha E Q V T φ] 0 for all φ S so ha φ is bonded a.s., for J, and V 0 φ = 0. We will show in he following secion ha consisen pricing sysems help solve his isse see Proposiion 4.2 and Theorem Consisen pricing sysems Consisen pricing sysems CPSs are insrmenal in he heory of arbirage in markes wih ransacion coss hey provide a bridge beween maringale heory in he heory of arbirage in fricionless markes and more general conceps in heory of arbirage in markes wih ransacion coss. Essenially, CPSs are inerpreed as corresponding axiliary fricionless markes. They are very sefl from he pracical poin of view becase hey provide a sraighforward way o verify wheher a financial marke model saisfies NA. In his secion, we explore he relaionship beween CPSs and NA. We begin by defining a CPS in or conex. Definiion 4.1. A consisen pricing sysem CPS corresponding o he marke B, P ask, P bid, A ask, A bid is a qadrple {Q, P, A, M} consising of i a probabiliy measre Q Z; ii an adaped process P saisfying P bid, P P ask, ; iii an adaped process A saisfying A ask, A A bid, ; iv a maringale M nder Q saisfying M = P A for all T. Remark 4.1. Since or marke is fixed hrogho he paper, we shall simply refer o {Q, P, A, M} as a CPS, raher han a CPS corresponding o he marke B, P ask, P bid, A ask, A bid. For a CPS {Q, P, A, M}, he process P is inerpreed as he corresponding axiliary fricionless ex-dividend price process, and he process A has he inerpreaion of he corresponding axiliary fricionless cmlaive dividend process, whereas M is viewed as he corresponding axiliary fricionless cmlaive price process. The nex resl esablishes a relaionship beween NA and CPSs in or conex. Proposiion 4.2. If here exiss a consisen pricing sysem CPS, hen he no-arbirage condiion NA is saisfied. Proof. Sppose here exiss a CPS, call i {Q, P, A, M}, and sppose φ S is a rading sraegy sch ha φ is bonded a.s., for J, and V 0 φ = 0. In view of Proposiion 2.5, and becase P bid P P ask and A ask A A bid, we dedce ha V T φ φ T T P T φ P 1 φ A. =1 24

25 Since M = P A is a maringale nder Q, and becase φ is bonded a.s., for J, we have E Q V T φ] = = T E Q φ T P T ] φ P 1 φ A =1 = 0. =1 =1 E Q φ E Q P T T w=1 1 A w P 1 ] E Q φ E Q M ] T M 1F 1 w=1 A w ] ] F 1 Therefore Q is a risk-neral measre. According o Theorem 3.4, NA holds. A his poin, a naral qesion o ask is wheher here exiss a CPS whenever NA is saisfied. In general, his is sill an open qesion. However, for he special case in which here are no ransacion coss in he dividends paid by he secriies, A ask = A bid, will show in Theorem 4.3 ha here exiss a CPS if and only if NAEF is saisfied. Proposiion 4.2 is imporan from he modeling poin of view becase i provides a sfficien condiion for a model o saisfy NA. As he nex example below illsraes, i is sally sraighforward o check wheher here exiss a CPS. Example 4.1. Les consider he CDS specified in Example 2.1. Recall ha he cmlaive dividend processes A ask and A bid corresponding o he CDS are defined as A ask := 1 {τ } δ κ ask 1 {<τ}, A bid := 1 {τ } δ κ bid 1 {<τ} for all T. Le s fix any probabiliy measre Q eqivalen o P. We poslae ha he ex-dividend prices P ask and P bid saisfy P ask, = E Q T A bid, =1 ] T F, P bid, = E Q A ask, =1 F ], for all T. By sbsiing A ask, and A bid, ino he eqaions for P ask, and P bid, above, we see ha P ask, = E Q 1 {<τ T } Bτ 1 δ κ bid P bid, = E Q 1 {<τ T } Bτ 1 δ κ ask =1 =1 B 1 1 {<τ} F ], B 1 1 {<τ} F ]. 25

26 For a fixed κ κ bid, κ ask ], we define A := B 1 1{τ=} δ κ1 {<τ}, T, T ] F P := E Q A = E Q 1 {<τ T } Bτ 1 δ κ B 1 F 1 {<τ} ], T, M := P =1 =1 A, T. The qadrple {Q, P, A, M} is a CPS. To see his, firs observe ha A and P are Q- inegrable since A is bonded Q-a.s. Ths, Q Z. Nex, M saisfies M = E Q 1 {τ T } B 1 δ κ τ B 1 1 {<τ} F ], T, so M is a Doob maringale nder Q. Also, since κ κ bid, κ ask ], we have A ask, A A bid, and P bid, P P ask,. Ths, {Q, P, A, M} is a CPS. According o Proposiion 4.2, we may addiionally conclde ha he financial marke model {B, P ask, P bid, A ask, A bid } saisfies NA. 4.1 Consisen pricing sysems nder he assmpion A ask = A bid In his secion we invesigae he relaionship beween risk-neral measres and CPSs nder he assmpion A ask = A bid. Le s denoe by A he process A ask. We begin by proving wo preliminary lemmas ha hold in general wiho he assmpion A ask = A bid. Lemma 4.1. If Q is a risk-neral measre, hen P bid,, σ 1 E Q P ask,, σ 2 σ 2 A bid,, =σ 1 1 ] F σ1, Pσ ask,, 1 E Q for all J and sopping imes 0 σ 1 < σ 2 T. Pσ bid,, 2 σ 2 A ask,, =σ 1 1 F σ1 ], Proof. Sppose Q is a risk-neral measre. For sopping imes 0 σ 1 < σ 2 T and random variables ξ σ1 L Ω, F σ1, P; R N, we define he rading sraegy θσ 1, σ 2, ξ σ1 := θ 0 σ 1, σ 2, ξ σ1, 1 {σ1 1 σ 2 }ξ 1 σ 1,..., 1 {σ1 1 σ 2 }ξ N σ 1 T =1, where θ 0 σ 1, σ 2, ξ σ1 is chosen sch ha θσ 1, σ 2, ξ σ1 is self-financing and V 0 θσ 1, σ 2, ξ σ1 = 0. De o Proposiion 2.5, he vale process associaed wih θ saisfies V T θσ 1, σ 2, ξ σ1 = =1 1 {ξ σ1 0} ξ σ 1 P bid,, σ 2 =1 σ 2 =σ {ξ σ1 <0} ξ σ 1 P ask,, σ 2 A ask,, σ 2 =σ 1 1 Pσ ask,, 1 A bid,, P bid,, σ 1. 26

27 Since Q is a risk-neral measre, we have E Q V T θσ 1, σ 2, ξ σ1 ] 0 for all sopping imes 0 σ 1 < σ 2 T and ξ σ1 L Ω, F σ1, P; R N. Hence, we are able o obain E Q N =1 1 {ξ σ1 0} ξ σ 1 P bid,, σ 2 =1 σ 2 =σ {ξ σ1 <0} ξ σ 1 P ask,, σ 2 σ 2 =σ 1 1 A ask,, A bid,, Pσ ask,, 1 P bid,, σ 1 ] 0, for all sopping imes 0 σ 1 < σ 2 T and random variables ξ σ1 L Ω, F σ1, P; R N. By he ower propery of condiional expecaions, we ge ha E Q N =1 1 {ξ σ1 0} ξ σ 1 E Q P bid,, σ 2 =1 1 {ξ σ1 <0} ξ σ 1 E Q P ask,, σ 2 σ 2 =σ 1 1 σ 2 =σ 1 1 A ask,, A bid,, ] Pσ ask,, Fσ1 1 P bid,, σ 1 Fσ1 ] ] 0. for all sopping imes 0 σ 1 < σ 2 T and random variables ξ σ1 L Ω, F σ1, P; R N. This implies ha he claim is saisfied. The nex resl is moivaed by Theorem 4.5 in Cherny Che07]. We will denoe by T he se of sopping imes in {, 1,..., T }, for all T. Lemma 4.2. Sppose Q is a risk-neral measre, and le X b, s X a, s := ess sp E Q Pσ bid,, σ T s := ess inf E Q Pσ ask,, σ T s σ A ask,, σ A bid,, F s ], F s ], for all J and s T. Then X b is a spermaringale and X a is a sbmaringale, boh nder Q, and saisfy X b X a. Proof. Le s fix J. The processes X b, and X a, are Snell envelopes, so X a, is a spermaringale and X b. is a sbmaringale, boh nder Q see for insance, Föllmer and Schied FS04]. We now show ha X b, X a,. Le s define he process X := E Q τ 1 Pτ bid,, 1 A ask,, ] F E Q P ask,, τ 2 τ 2 A bid,, F ], T. 27

28 For any sopping imes τ 1, τ 2 T, we see ha X = E Q E Q = E Q 1 {τ1 τ 2 } E Q 1 {τ1 >τ 2 } τ 1 Pτ bid,, 1 A ask,, E Q P ask,, τ 2 τ 1 Pτ bid,, 1 τ 1 Pτ bid,, 1 A ask,, Afer rearranging erms, we dedce ha X = E Q 1 {τ1 τ 2 }P bid,, 1 {τ1 τ 2 }E Q P ask,, τ 2 τ 2 A ask,, E Q P ask,, τ 2 ] F τ2 A bid,, ] ] F F τ1 τ 2 P ask,, τ 2 τ 1 τ 1 A ask,, τ 2 A bid,, =τ 1 1 τ 1 E Q 1 {τ1 >τ 2 }E Q P bid,, τ 1 1 {τ1 >τ 2 }P ask,, τ 2 Becase A ask, A bid,, we are able o obain X E Q 1 {τ1 τ 2 } E Q 1 {τ1 >τ 2 } τ 1 E Q =τ 2 1 τ 2 A bid,, τ 2 A bid,, τ 2 A bid,, ] ] F F τ1 A ask,, ] F τ2 A ask,, F ]. τ 2 P bid,, Pτ ask,, 2 A bid,, =τ 1 1 τ 1 ] E Q Pτ bid,, 1 A ask,, F τ2 =τ 2 1 ] ] F F τ1 ] A bid,, F. ] ] F F τ1 P ask,, τ 2 F ]. 31 Since Q is a risk-neral measre, we see from Lemma 4.1 and 31 ha X 0. The sopping imes τ 1 and τ 2 are arbirary in he definiion of X, so we conclde ha X b, X a,. The nex heorem gives sfficien and necessary condiions for here o exis a CPS cf. Cherny Che07]; Denis, Gasoni, and Rásonyi DGR11]. Theorem 4.3. Under he assmpion ha A ask = A bid, here exiss a consisen pricing sysem CPS if and only if he no-arbirage condiion nder he efficien condiion NAEF is saisfied. Proof. Necessiy is shown in Proposiion 4.2, so we only prove sfficiency. Sppose ha NAEF is saisfied. According o Theorem 3.4, here exiss a risk-neral measre Q. By 28

29 Lemma 4.1, P bid,, σ 1 E Q P ask,, σ 2 σ 2 A, =σ 1 1 ] F σ1, Pσ ask,, 1 E Q Pσ bid,, 2 σ 2 A, =σ 1 1 for all J and sopping imes 0 σ 1 < σ 2 T. Now, le s define he processes Y b, Y a, X b, := ess sp E Q Pσ bid,, σ T := ess inf E Q Pσ ask,, σ T := Y b, σ A, =1 σ A, =1 A,, X a, := Y a, F ], F ], F σ1 ], A,, 32 for all T and J. From Lemma 4.2, we know ha nder Q he process X a is a sbmaringale and he process X b is a spermaringale, and ha hey saisfy X b X a. For = 0, 1,..., T 1 and J, recrsively define M 0 := Y a, 0, P 0 1 M 1 := P 1 A, a, := Y0, P 1 := λ Y a, 1 1 λ b, Y1, 33 where λ saisfies M E QX b, 1 F ] λ = E Q X a, 1 Xb, 1 F ], if E QX a, 1 F ] E Q X b, 1 F ], 1 2, oherwise. Les fix J for he res of he proof. Sep 1: In his sep, we show ha he processes above are well defined and adaped. Firs, noe ha P 0 and M 0 are well defined, and ha, by 34, λ 0 = M 0 E QX b, 1 F 0] E Q X a, 1 X b, 1 F 0], or λ 0 = 1 2. Ths, λ 0 is well defined and F 0-measrable. Nex, we compe P 1 and M 1, and conseqenly we compe λ 1 ; all of hem being F 1-measrable. Indcively, we see ha P, M, and λ, for = 2,..., T are well defined and F -measrable. Sep 2: We indcively show ha λ 0, 1] for = 0, 1,..., T 1. We firs show ha λ 0 0, 1]. If E QX a, 1 X b, 1 F 0] = 0, hen λ 0 0, 1] aomaically, so sppose ha 34 29

30 E Q X a, 1 X b, 1 F 0] > 0. Now, by he definiion of M, we have ha M 0 = Xa, 0 gives ha λ 0 = Xa, 0 E Q X b, 1 F 0] E Q X a, 1 X b,, so 34 1 F 0]. 35 The process X a, is a sbmaringale nder Q, so i immediaely follows ha λ 0 1. On he oher hand, since X b, is a spermaringale nder Q, λ 0 X a, 0 X b, 0 E Q X a, 1 X b, 1 F 0]. Becase X a, 0 X b, 0, we dedce ha λ 0 0. Sppose ha λ 0, 1] for = 0, 1,..., T 2. We now prove ha λ T 1 E Q X a, T X b, T F T 1] = 0, hen λ T 1 = 1/2, so assme ha E QX a, T According o 34 and he definiion of M, we have ha λ T 1 = λ T 2 Xa, T 1 1 λ T 2 Xb, T 1 E QX b, T F T 1] E Q X a, T Xb, T F T 1] Since λ T 2 1, and becase Xb, is a spermaringale nder Q, we have ha Becase X a, X b,, we arrive a λ T 1 see from 36 ha λ T 1 λ T 2 Xa, T 1 Xb, T 1 E Q X a, T Xb, T F T 1]. 0, 1]. If X b, T F T 1] > Now, since Xa, T 1 Xb, T 1 and λ T 2 1, we λ T 1 Xa, T 1 E QX b, T F T 1] E Q X a, T Xb, T F T 1] The process X a, is a sbmaringale nder Q, so i follows ha λ T 1 1. We conclde ha λ 0, 1] for = 0, 1,..., T 1. Sep 3: Nex, we show ha M is a maringale nder Q. Firs we noe ha by 32 and 33 we have M 1 = λ Xa, 1 1 λ Xb, From here, he Q-inegrabiliy of M follows from Q-inegrabiliy of X a,, X b, and bondedness of λ. From 34 and 37, we ge ha E Q M 1 F ] = M, for = 0, 1,..., T 1. Hence, M is a maringale nder Q. Sep 4: We conine by showing ha P saisfies P bid,, P P ask,,. Le s firs show ha P bid, 0 P 0 P ask, 0. By definiion of P 0, we have ha P 0 = Y a, 0, and by 32 we see ha Y a, 0 = X a, bid, 0. Therefore, he claim holds since P0 X a, 0 P ask, 0. We proceed by proving ha P bid, P P ask, le {1,..., T }. By he definiion of P, we have P 32, i is re ha X a, X b, if and only if Y a, 30. for all {1,..., T }. Towards his, = λ 1 Y a, 1 λ b, 1 Y. From. Also, since T, we see from Y b,