1 Introducton he Fundamental heorem of Asset Prcng, whch orgnates n the Arrow- Debreu model (Debreu [1959]) and s further formalzed n (among others) C

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1 Arbtrage and vablty n securtes markets wth xed tradng costs Elyes Joun Hed Kallal y Clotlde Napp z July, 1999 Abstract hs paper studes foundatonal ssues n securtes markets models wth xed costs of tradng,.e. transactons costs that are bounded regardless of the transacton sze, such as: xed brokerage fees, nvestment taxes, operatonal and processng costs, or opportunty costs. We show that the absence of free lunches n such models s equvalent to the exstence of a famly of absolutely contnuous probablty measures for whch the normalzed securtes prce processes are martngales, condtonal on any possble future event. hs s a weaker condton than the absence of free lunches n frctonless models, whch s equvalent to the exstence of an equvalent martngale measure. We also show that the only arbtrage free prcng rules on the set of attanable contngent clams are those that are equal to the sum of an expected value wth respect to any absolutely contnuous martngale measure and of a bounded xed cost functonal. Moreover, these prcng rules are the only ones to be vable as models of economc equlbrum. Keywords :arbtrage - xed costs - absolutely contnuous martngale measure - contngent clams prcng - vablty CRES-ENSAE, 15, Bd Gabrel Per, Malako Cedex, France and CERMSEM- Unverste de Pars I. Joun s currently vstng the Stern School of Busness at NYU, 44 W 4th St, NY, y Salomon Brothers, 111 Buckngham Palace Road, London, SW1W OSB, U.K. Salomon s not responsble for any statement or conclusons heren, and no opnons, theores, or technques presented heren n any way represent the poston of Salomon Brothers Inc. z CRES, 15, Bd Gabrel Per, Malako Cedex, France. 1

2 1 Introducton he Fundamental heorem of Asset Prcng, whch orgnates n the Arrow- Debreu model (Debreu [1959]) and s further formalzed n (among others) Cox and Ross (1976), Harrson and Kreps (1979), Harrson and Plska (1981), Due and Huang (1986), Dybvg and Ross (1987), Dalang, Morton and Wllnger (1989), Back and Plska (199), and Delbaen and Schachermayer (1994), asserts that the absence of free lunch n a frctonless securtes market model s equvalent to the exstence of an equvalent martngale measure for the normalzed securtes prce processes. he only arbtrage free and vable prcng rule on the set of attanable contngent clams, whch s a lnear space, s then equal to the expected value wth respect to any equvalent martngale measure. In ths paper, we study some foundatonal ssues n the theory of asset prcng n securtes markets models wth xed tradng costs. ransacton costs are sad to be xed n the sense that they are bounded regardless of the transacton sze. Such xed costs nclude for example xed brokerage fees, brokerage arrangements where margnal fees go to zero beyond a gven volume that s reset perodcally (such arrangements are common n the ndustry), xed nvestment taxes to gan access to a market (such as a foregn market), operatonal and processng costs that typcally exhbt strong economes of scale (e.g. through automaton), xed costs nvolved n settng up an oce and obtanng access to nformaton, and the opportunty cost of lookng at a market or of dong a specc trade. We nd that the absence of free lunches n models wth xed tradng costs s equvalent to the exstence of a famly of \absolutely contnuous" probablty measures 1 for whch the normalzed (by a numerare) securtes prce processes are martngales, condtonal on any possble future event. Note that ths s a weaker condton than the exstence of an equvalent martngale measure (as n frctonless markets) because n ths case the martngale measures are only requred to be absolutely contnuous. As n the Fundamental heorem of Asset Prcng, we nd that the absence of free lunches s also equvalent to the exstence of a famly of nonnegatve state prce denstes and to the exstence of a famly 1 Let (;F;P)beagven probablty space. We say that another probablty measure Q dened on the same probablty space ^e (;F;P) s absolutely contnuous wth respect to P, and we shall wrte Q<<P;f for all event A n F satsfyng P (A) =we have Q (A) =: 2

3 of contnuous weakly postve lnear operators. We dene admssble prcng rules on the set of attanable contngent clams as the prce functonals that are arbtrage free and are lower than or equal to the surreplcaton cost (.e. the lowest cost of domnatng a gven payo). Indeed, no ratonal agent would pay more than ts surreplcaton cost for a contngent clam snce there sacheaper way toacheve at least the same payo usng a tradng strategy. We then show that the only admssble prcng rules on the set of attanable contngent clams are those that are equal to the sum of an expected value wth respect to any absolutely contnuous martngale measure and of a bounded xed cost functonal. Moreover, these prcng rules are the only ones to be vable as models of economc equlbrum,.e. such that there exst some prce-takng maxmzng agents who are happy wth ther ntal endowment, and hence for whom supply s equal to demand. A smple example can llustrate our man result. Consder a model where two securtes, denoted by A and B; can be traded at two dates and 1 and n two possble states of the world s 1 and s 2 at date 1: Securty A; the numerare, s normalzed to be always worth one unt of account and securty B has avalue of 1 at date and avalue of 1 or 2 at date 1 n state s 1 or s 2 respectvely (all n numerare unts). In the perfect market case, ths model yelds an arbtrage opportunty whch conssts n buyng one unt of B and sellng one unt of A at date at a zero nvestment cost, and closng the poston at date 1 at a prot n state s 1 and at no loss n state s 2 : If we ntroduce xed tradng costs, ths arbtrage opportunty dsappears snce the nvestment requred at date by the strategy s not zero anymore but s equal to the xed cost. Accordng to the Fundamental heorem of Asset Prcng, there cannot exst an equvalent martngale measure. Nevertheless, the probablty Q dened on the set S = fs 1 ;s 2 g of the possble states of the world at date 1 by Q (s 1 )=1andQ(s 2 ) = s an absolutely contnuous martngale measure for securtes A and B: here s an exstng body of lterature that studes transacton costs and other market frctons. For nstance, Joun and Kallal (1995a) studes proportonal transacton costs and nds that a bd-ask prce process s arbtrage free f and only f there exsts an equvalent probablty measure that transforms some process between the normalzed bd and ask prce processes nto a martngale. Joun and Kallal (1995b) studes the case of short sales constrants and shortsellng costs (as well as derent borrowng and lendng rates) and nds that the absence of arbtrage s equvalent to the exstence 3

4 of an equvalent supermartngale measure. he set of expected values of the payo of a contngent clam wth respect to all the martngale (resp. supermartngale) measures s an nterval and concdes wth the set of ts possble prces compatble wth arbtrage and economc equlbrum. he characterstc of ths class of frctons s that they lead to a prcng rule that s sublnear,.e. postvely homogeneous and subaddtve, and snce ths s not the case for xed transacton costs they requre a specc analyss. Also, Cvtanc and Karatzas (1993 and 1996) study the optmal hedgng problem n a dffuson model wth portfolos constraned to belong to a gven convex set and proportonal transacton costs respectvely. Pham and ouz (1996) studes the case of constrants that take the form of closed convex cones n nte dscrete tme. 2 As far as xed transacton costs are concerned Due and Sun (199), Grossman and Laroque (199) and Morton and Plska (1995), among others, have studed the optmal portfolo problem wth transacton fees that are proportonal to the sze of the overall portfolo (as opposed to the sze of the specc transacton). he remander of the paper s organzed as follows. Secton 2 descrbes our securtes markets model wth xed tradng costs. Secton 3 characterzes the absence of free lunches n such a model. Secton 4 characterzes the arbtrage free and vable prcng rules. Secton 5 concludes. 2 he model wth xed costs he securtes market model conssts of a set = [;] of tradng dates, where denotes the termnal date for all economc actvty; a complete probablty space (;F;P); where the set represents all possble states of the world; an nformaton structure whch descrbes how nformaton s revealed to agents, gven by a ltraton F = ff t g t2 wth F = f;; g and F = F ; n +1traded securtes ; :::; n and a (n + 1)-dmensonal, F - adapted process Z = fz t ; t 2 g wth component processes Z ; :::; Z n where Zt k represents the prce of securty k at tme t: We assume that for all t; Z t = 1; whch means that the rskless rate s equal to zero. Note that ths assumpton amounts to a normalzaton of all securtes prces by anumerare, 2 Other papers on market frctons nclude Magll and Constantndes (1976), Constantndes (1986), Dybvg and Ross (1986), Prsman (1986), Ross (1987), aksar et al. (1988), He and Pearson (1991), Bensad et al. (1992), Hndy (1995) and Joun and Kallal (1999). 4

5 and can be made wthout any loss of generalty as long as at least one of the securtes has a postve prce at any tme. In the remander of the paper we shall refer to the th securty as the rskless asset. We also make the techncal assumpton 3 that for any tradng date t n ; Z t s n L 1 (;F t ;P): A tradng strategy s a (n + 1)-dmensonal F -adapted process = f t ; t 2 g wth component processes t ; :::; n t where k t represents the quantty of securty k held at tme t. he vector t represents the agent's portfolo at tme t and ts components may take negatve as well as postve values. Hence, 4 V t = t Z t s the market value of the portfolo t at date t and we call the process V = n V t ; t 2 o the value process for the strategy : Let t denote for each date t the vector ( 1 t ; :::; n t ) of quanttes of rsky securtes held at tme t. As n Harrson and Kreps (1979), we only consder smple strateges,.e., strateges such that: for all t; t Z t s nl 1 (;F t ;P); agents may trade only at a nte number of dates (although that number can be arbtrarly large) that must be speced n advance. 5 Note that smple strateges are natural n our context because we shall assume that agents ncur a xed transacton cost each tme they trade. We denote by c t the postve xed transacton cost pad at date t f tradng has occurred n any of the rsky securtes and c = fc t ; t 2 g : If agents do not trade n any of the rsky securtes at tme t; then we assume that they do not ncur any transacton cost. he transacton cost s xed n the sense that t s bounded regardless of the amount of securtes traded,.e. for all t there exsts some real number C t such that <c t < C t P a.s.. We assume that the process c s F -adapted, whch means that agents only know at tme t the past and current values of the xed tradng cost but nothng more. We also allow the xed transacton costs to depend upon the tradng strategy (and not to be necessarly strctly postve ateach tradng date),.e., to each smple strategy wth tradng dates t ; :::; t N = s assocated a nonnegatve transacton cost process c = c t t2ft ;:::;t N g wth c t = C (t; ( t ) t t) such that: for any smple tradng strateges and, such that =, we have c = c and agents do not pay any xed transacton cost f they do not 3 We recall that L 1 (;F;P) denotes the set of measurable random varables wth nte expected value wth respect to P: 4 P For all (x; y) nrd R d d for some postve real number d, we let x y = =1 x y. 5 he extenson to tradng dates that are stoppng tmes (nstead of beng speced n advance) s straghtforward. 5

6 trade the rsky securtes,.e. for any smple strategy wth tradng dates t ; :::t N ; c t 1 ( t = t,1 ) = for all wth 1 N c t 1 ( t =) = ; c 1 ( =) = c t = for all t =2 ft ; :::; t N g. for any date t, there exsts a postve random varable c t such that for any smple strategy wth tradng dates t ; :::t N ; c t 1 ( t6=; t = for all t <t) c t1 ( t6=; t = for all t <t) for all wth 1 N;.e., the rst tme real tradng occurs, the xed cost must be postve. Or, there exsts a postve real number " such that for any smple strategy wth tradng dates t ; :::t N ; X ft g c t ".e., the cumulatve transacton cost from the rst to the last tradng date must be greater than some postve constant. for all t, there exsts a postve real number C t such that for any smple strategy c t C t.e., the transacton cost s bounded at each date. hs mples that for any smple strategy wth tradng dates t ; :::t N, the cumulatve transacton cost P ft g c t s smaller than or equal to some constant (that depends on the strategy only, and not on the state of the world). We could nderently assume that for any strategy and any tradng date t; the transacton cost at tme t s such that c t!!1, whch means that the transacton cost per unt of securty traded goes to zero as the amount traded becomes arbtrarly large. 6

7 Note that these condtons are consstent wth a large class of transacton costs that can be dented n nancal markets. hey nclude xed brokerage fees or brokerage arrangements where margnal fees go to zero beyond a gven volume that s reset perodcally (such arrangements are common n the ndustry), and xed nvestment taxes to gan access to a market such asa foregn market. hey also nclude operatonal and trade processng costs that typcally exhbt strong economes of scale (especally f these tasks have been automated), and xed costs ncurred n settng up an oce and obtanng access to prce or other relevant nformaton. Also, the opportunty cost of focusng on a market or of dong a specc trade can be vewed as a xed cost. In order to get some of our results, we shall need the followng addtonal assumpton 6 (that we shall menton each tme t s needed): Assumpton A : here exsts a real number C such that for every strategy ; P t2 c t <C. hs means that, under Assumpton A, the cumulatve transacton costs of any tradng strategy are assumed to be bounded by a constant. Note that ths condton s automatcally satsed n a dscrete tme model wth a nte or nnte number of states of the world (as long as transacton costs are bounded at each tme), but a nte number of possble tradng dates. It s also automatcally satsed n a model where there s a xed cost to access a market such as a xed nvestment tax, a xed cost for settng up nformaton technology or a trade processng department, or a xed opportunty cost of lookng at a market. It s also consstent wth a stuaton where the xed transacton costs consst n brokerage fees wth a brokerage arrangement where transactons go free beyond a certan volume whch s reset on a perodcal bass (ths type of arrangement s common n the ndustry). Agents transfer wealth from all dates and events (for contngent wealth) to the termnal date usng the traded securtes, subject to payng the xed transacton costs. In dong so they use self-nancng strateges dened as follows. Let be a date n and let B be an event n F (n the remander of the paper we shall always suppose that P (B) 6= ): We then have: 6 For nstance, we shall need Assumpton A when we use the same denton of free lunch as n Kreps (1981). However, we shall also ntroduce an alternatve denton of free lunch for whch Assumpton A s not requred for any of our results. 7

8 Denton 2.1 A self nancng smple strategy from the date and the event B s a strategy that s null before the date and outsde the event B = n 6= o ; and such that there exst tradng dates t ; :::; t N ; wth = t ::: t N = ; for whch (t;!) s a:s: constant over each nterval [t k,1 ; t k [ and satses tk Z tk + c t k tk,1 Z tk for k =1; :::; N, 1 and Z + c = tn,1 Z : hs means that a self-nancng smple strategy does not requre any addtonal nvestment beyond what s requred at the ntal date: purchases of securtes as well as transacton costs after the ntal date are nanced by the sale of other securtes. Let S ;B denote the set of such strateges. We also have: Denton 2.2 A frctonless self nancng smple strategy from the date and the event B s a strategy that s null before the date and outsde the event B and such that there exst tradng dates t ; :::; t N wth = t ::: t N = for whch (t;!) s a:s: constant over each nterval [t k,1 ; t k [ and satses tk Z tk = tk,1 Z tk a:s: P for k =1; :::; N. hs means that a frctonless self nancng smple strategy s a self - nancng smple strategy n an otherwse dentcal economy where there are no transacton costs. Let W ;B denote the set of such strateges. 3 Arbtrage opportuntes and free lunches 3.1 Arbtrage opportuntes An arbtrage opportunty s a tradng strategy that yelds a postve gan n some crcumstances wthout a countervalng threat of loss n any other crcumstances. A free lunch s the possblty of gettng arbtrarly close to an arbtrage opportunty. We shall dene two concepts of arbtrage opportuntes as follows : Denton An arbtrage opportunty wth xed costs (AO 1 ) s a strategy such that there exst (; j) n, j, an event 8

9 B n F, for whch s null after date j, belongs to S ;B, V + c on B, V j and ether V + c or V j s derent from. 2. A frctonless strong arbtrage opportunty (AO 2 ) s a strategy such that there exst (; j) n, j, an event B n F, for whch s null after date j; belongs to W ;B, V < on B and V j : hs means that an AO 1 s a tradng strategy that yelds, n our model wth xed transacton costs, a postve gan n some crcumstances wthout a threat of loss n other crcumstances. An AO 2 s a tradng strategy that yelds a postve gan at the startng date and event of the tradng strategy wthout a countervalng threat of loss n other crcumstances. We then have: Proposton here exsts an AO 1 f and only f there exsts a net gan arbtrage opportunty wth xed tradng costs,.e. a strategy such that there exst a date n and an event B n F for whch belongs to S ;B, and h V, V, c, 6= on B. 2. here exsts an AO 2 f and only f there exsts a frctonless "-net gan arbtrage opportunty,.e. a strategy such that there exst a date n ; an event B n F and a postve real number " for whch belongs to W ;B and V, V " on B. 3. here exsts an AO 1 f and only f there exsts an AO 2. hs means that the two notons of arbtrage opportuntes that we have ntroduced are equvalent. Also, an arbtrage opportunty n our model wth xed transacton costs corresponds to the possblty ofachevng a postve net gan. An arbtrage opportunty n the otherwse dentcal frctonless model corresponds to a net gan that s greater than some postve constantn all states of the world. It s hence clear that the set of arbtrage opportuntes n our model wth xed transacton costs s strctly smaller than the set of arbtrage opportuntes n the frctonless model, or equvalently that the assumpton of no arbtrage n our model wth xed transacton costs s less strngent than n the frctonless model. 9

10 3.2 Free lunches As n Kreps (1981), we dene a free lunch as the possblty ofgettng arbtrarly close to an arbtrage opportunty. More precsely, we have Denton Afree lunch wth xed costs (FL 1 ) sasequence ( n ) n2n of tradng strateges such that there exst n ; B n F ; sequences (x n ) n2n and (" n ) n2n of random varables belongng respectvely to L 1 (;F;P) and L 1 (;F ;P) and convergng n L 1 (;F;P) respectvely to x and " on B wth x + " 6=for whch for all n; n s n S ;B, V n + c n," n on B and V n x n : 2. A frctonless strong free lunch (FL 2 ) s a sequence ( n ) n2n of tradng strateges such that there exst n ; B n F and sequences (x n ) n2n and (r n ) n2n of random varables belongng respectvely to L 1 (;F;P) and L 1 (;F ;P) and convergng n L 1 (;F;P) respectvely to x and r> on B for whch for all n; n s n W ;B and satses V n,r n and V n x n : 3. An \asymptotc free lunch" (AsF L) sasequence ( n ) n2n of strateges of postve such that there exst n ; B n F, a sequence ( n ) n real numbers and sequences (x n ) n2n and (" n ) n2n of random varables belongng respectvely to L 1 (;F;P) and L 1 (;F ;P) and convergng n L 1 (;F;P) respectvely to x and " > on B for whch for all n; n s n S V n ;B + c n,," n on B V n and x n : n n hs means that a free lunch s a sequence of strateges wth a payo that converges to an arbtrage opportunty. A frctonless strong free lunch s a sequence of strateges wth a payo that converges to a frctonless strong arbtrage opportunty. An \asymptotc free lunch" s a sequence of strateges that are strong free lunches when renormalzed by a sequence of scalng factors. We ntroduce ths noton n order to avod usng Assumpton A n our characterzaton heorems n the next secton. Note that, as n the denton of arbtrage opportuntes, we could replace the date wth any date j, satfyng j for whch n s null after the date j. We then have 1

11 Proposton here exsts a FL 1 f and only f there exsts a net gan free lunch wth xed costs,.e. a sequence ( n ) n2n of strateges such that there exst n ; Bn F and a sequence (x n ) n2n of random varables belongng to L 1 (;F;P) and convergng n L 1 (;F;P) to some x ; 6= on B for whch for all n; n s n S ;B and V n, V n + c n x n. 2. here exsts a FL 2 f and only f there exsts a frctonless "-net gan free lunch,.e., a sequence ( n ) n2n of strateges such that there exst n ; Bn F ; apostve real number " and a sequence (x n ) n2n of random varables belongng to L 1 (;F;P) and convergng to some x " on B for whch for all n, n s n W ;B and satses V n, V n x n. hs means that a free lunch corresponds to a sequence of tradng strateges wth a payo that converges to a postve net gan. Smlarly a frctonless strong free lunch corresponds to a sequence of tradng strateges wth a payo that converges to a net gan that s strctly postve n all states of the world. We then have the followng characterzaton of the absence of frctonless strong free lunches: Corollary 3.1 Let K ;B = n V, V ; 2 W ;B o L1 (;F;P) ; the set of possble gans from date and event B n the frctonless model, and C ;B = K ;B, L 1 +; where the closure s taken n L 1 : Let A B = n f 2 L 1 +; 9" > such that f " on B o : he assumpton of no frctonless strong free lunch (NFL 2 ) s equvalent to the condton that for all n and B n F ; the two convex sets C ;B and A B have an empty ntersecton. We also have Lemma he absence of frctonless strong free lunch (NFL 2 ) mples the absence of free lunch n our model wth xed tradng costs (NFL 1 ) : 2. Under Assumpton A, the absence of free lunch n our model wth xed costs (NFL 1 ) and the absence of frctonless strong free lunch (NFL 2 ) are equvalent. 11

12 3. he absence of \asymptotc free lunch" (NAsF L) n our model wth xed tradng costs and the absence of frctonless strong free lunch (NFL 2 ) are equvalent. It s easy to see that the absence of frctonless strong free lunch mples the absence of free lunch wth xed tradng costs. But, unlke for arbtrage opportuntes, the converse s not necessarly true. Indeed, although the number of tradng dates for each tradng strategy n s nte, t can be arbtrarly large, and therefore so can the cumulatve tradng costs. Hence the need to bound the total tradng costs of any smple strategy as n Assumpton A or to consder the noton of \asymptotc free lunch". In both cases we obtan the equvalence between the absence of strong frctonless free lunches and the absence of free lunches n our model wth xed tradng costs. 3.3 Absolutely contnuous martngale measures Wth the notatons of corollary??, t s easy to see, usng the denton of the set of self nancng smple tradng strateges n n the frctonlesso model W ;B and the fact that Z t =1; that K ;B = V, V ; 2 W ;B ; the set of possble gans from date and event B n the frctonless model, s a vector space and that t s equal to K ;B = Ln n s Z t, Z s ; s 2 P s;b ; s t o ; where for all s ; Z s = (Z 1 s ; :::; Z n s ) and where P s;b denotes the set of n -dmensonal random varables s = ( 1 s; :::; n s ) that are F s -measurable, null outsde B and before and such that s Z s s n L 1 (;F s ;P) : he use of corollary?? and of a separaton theorem wll now enable us to obtan our man result: the characterzaton of the absence of frctonless strong free lunches n terms of absolutely contnuous martngale mesures. heorem 3.1 here exsts no frctonless strong free lunch f and only f for all n and all B n F ; there exsts an absolutely contnuous probablty measure P ;B dened on(;f), wth bounded densty, such that P ;B (B) =1 and E P ;B [Z t j F s ]=Z s for all (s; t) such that s t: We then obtan the Fundamental heorem of Asset Prcng for securtes markets models wth xed tradng costs. heorem 3.2 he followng are equvalent: 12

13 1. here exsts no \asymptotc free lunch" n our model wth xed tradng costs. 2. here exsts a famly of absolutely contnuous martngale measures: for all n and for all B n F ; there exsts an absolutely contnuous probablty measure wth bounded densty P ;B dened on (;F) such that P ;B (B) =1and satsfyng E P ;B [Z t j F s ]=Z s for all (s; t) wth s t: 3. here exsts a famly of nonnegatve state prce denstes: for all n and for all B n F, there exsts a bounded random varable g ;B L 1 (;F;P) wth g ;B ; 6= on B and such that for all (s; t) wth s t h h E g ;B Z t 1 A\B = E g ;B Z s 1 A\B for all A n F s : n 4. here exsts a famly of weakly postve 7 contnuous lnear operators: let R ;B denote the set of random varables null outsde B and belongng to L 1 (;F ;P) : For all n ; for all B n F ; there exsts a weakly postve contnuous lnear operator ;B dened on R ;B and takng values n R ;B, such that there exsts an event A n F wth A B and P (A) 6= for whch ;B V = V on A, for all n W ;B : Under Assumpton A, these statements are all equvalent to: 5. here exsts no free lunch n our model wth xed tradng costs. hs means that the absence of free lunches n our model wth xed tradng costs (or equvalently the absence of free lunches n the otherwse dentcal frctonless model) s equvalent to the exstence of a famly of absolutely contnuous martngale probablty measures: absolutely contnuous martngale measures condtonal on any possble future event. Note the derence wth the frctonless case where the absence of free lunches (a weaker condton than the absence of free lunches n the model wth xed tradng costs) s 7 Let X denote the set of random varables on (;F;P) : A functonal p dened on X s sad to be weakly postve f for all x n X such that P (x )=1; we have p (x) : 13

14 equvalent to the exstence of an equvalent martngale probablty measures (a stronger condton snce a famly of absolutely contnuous martngale measures can be derved from any equvalent martngale measure) as shown n Harrson and Kreps (1979). We can also obtan the slghtly more general results n the sprt of Yan's (198) theorems ( also see Strcker (199), among others, for an applcaton of Yan's theorem). heorem 3.3 Let K be a convex set n L 1 (;F;P) contanng : he followng condtons are equvalent : 1. For all n L 1 such that > ; there exsts a postve real number c for whch c s not n K, L 1 +: 2. here exsts a postve real number c such that c1 s not n K, L 1 +: 3. here exsts a random varable Z n L 1 (;F;P) satsfyng Z ; 6= and sup 2K E [Z] < 1 We also have Corollary 3.2 Let K denote K ; wth K ;B = n V, V ; 2 W ;Bo = Ln n s Z t, Z s ; s s n P s;b ;t s o where for all s ; Z s = (Z 1 s ; :::; Z n s ) and P s;b denotes the set of n - dmensonal random varables s = (s; 1 :::; n s ) that are F s -measurable, null outsde B and n before ; and such that s Z s s n L 1 (;F s ;P): Also, let A B denote f 2 L 1 +; 9" > such that f " on Bg : he followng condtons are equvalent : 1. he ntersecton A \ K, L 1 + s empty. 2. he random varable 1 does not belong to K, L 1 +: 3. here exsts an absolutely contnuous martngale measure for Z: 14

15 hs concludes our characterzaton of processes that admt an absolutely contnuous martngale measure - whch relates to the heorem of Asset Prcng n securtes markets models wth xed tradng costs (note that the mplcatons 2) ) 1) n heorems?? and Corollary?? are qute general and can be useful n other contexts as well). he characterzaton of processes that admt an equvalent martngale measure (or the Fundamental heorem of Asset Prcng n frctonless securtes markets models) can be found n Harrson and Kreps (1979), Yan (198), Kreps (1981), Due and Huang (1986), Strcker (199) or Delbaen and Schachermayer (1994 and 1998), as well as Back and Plska (1988) and Dalang et al. (198) for the dscrete tme case. We shall now exhbt an example of a process that admts a famly of absolutely contnuous martngale probablty measures but does not admt any equvalent martngale probablty measure. Example (Delbaen and Schachermayer (1994)): Let W be a standard Wener process, wth ts natural ltraton (G t ) t1 : We dene a local martngale of exponental type by: L t = exp(,(f W ) t, 1(R t f 2 2 (u)du)) f t<1; and L 1 =; where f(t) = p 1 1,t : We dene the stoppng tme by = nfft; L t 2g: We then dene the prce process S t by: ds t = dw t + p 1 1,t dt f t ; and ds t =ft ; and the ltraton (F t ) t1 =(G mn(t; ) ) t1 : Accordng to Delbaen and Schachermayer (1994), there exsts a unque probablty measure Q that s absolutely contnuous wth respect to P and makes the process S a martngale. It s gven by dq = L dp: Snce P [L = ] > ; Q s not equvalent to P: Moreover, for all t < 1; the measures Q and P are equvalent on F t snce the densty L s postve. It s now easy to see that for any date t and for all event B at that date, there exsts a probablty measure Q t;b gven by dq t;b = L 1 B dp such that E[L 1 B ] Qt;B (B) =1 and E Qt;B [S v jf u ]=S u for all (u; v) wth t u v: 15

16 4 Prcng and vablty wth xed costs 4.1 Admssble prcng rules A contngent clam B to consumpton at the termnal date s a random varable belongng to L 1 (;F;P): A contngent clam B s sad to be attanable (n the model wthout xed cost) f there exsts some frctonless self nancng strategy n W ; such that V = B: Note that the set M of all attanable contngent clams s a lnear space. We shall now dene and characterze prcng rules p (B) on M that are admssble. Denton 4.1 An admssble prcng rule on M s a functonal p dened on M, such that 1. p nduces no arbtrage,.e., t s not possble to nd strateges 1,..., n and one of the two n W ;, for whch P n =1 p V s nonnull., P n =1 V 2. p (B) s (B), where s (B) := nf n V + c, 2 S;, V B o : Part 1 s the usual no-arbtrage condton. Part 2 says that an admssble prce for the contngent clam B must be smaller than ts superreplcaton prce: f t s possble to obtan a payo at least equal to B at a cost s (B), then no ratonal agent (who prefers more to less) wll accept to pay more than s (B) for the contngent clam B: Note that snce B s attanable by a frctonless self nancng strategy, and snce the total tradng costs ncurred by any strategy are bounded, there always exsts at least a self nancng (nclusve of transacton costs) strategy domnatng B;.e. B s also attanable n our model wth xed tradng costs. he followng Proposton characterzes the admssble prcng rules on M through the use of the absolutely contnuous martngale measures obtaned n heorem??. Proposton 4.1 Under Assumpton A and the assumpton of NFL 1, or under the assumpton of NAsFL, any admssble prcng rule p on M can be wrtten as p (B) =E P [B]+c (B) for all B n M where P s any absolutely contnuous martngale measure and c(b)!!1. 16

17 hs means that f B = V then p(b) =V + c(b) snce E P (V )=V for any absolutely contnuous martngale measure P : Moreover, f p (x) [p (x)] for any real number large enough, then the xed cost c s nonnegatve. And f there exsts " >, such that for any large enough, p (x) [p (x), "], then the xed cost c s greater than or equal to ths postve constant ". Notce that under Assumpton A,.e. f the cumulatve xed costs ncurred by any strategy are bounded by a postve real number C, then c (B) := p (B), E P (B) s (B), E P (B) C; for any absolutely contnuous martngale measure P : Also, Proposton?? mples that p(b)!!1 E P [B] for any attanable contngent clam B; where P s any absolutely contnuous martngale measure. hs means that the unt prce of any attanable contngent clam B s equal to E P [B] n the lmt of large quanttes. As usual, we say that the market s complete n the frctonless model f any contngent clam s attanable. If the market s complete, there exsts a unque admssble prcng rule. However, n ncomplete markets (.e., f there are some non attanable contngent clams), even n a frctonless model there s no unversal prcng concept. We can only nd arbtrage bounds and the prcng rules are sublnear 8 lower semcontnuous functonals (see Joun and Kallal (1995a and 1999)). By analogy wth the case of attanable contngent clams, we dene an admssble prcng rule on the set of contngent clams n the followng way. Denton 4.2 A prcng rule on L 1 (;F;P) s admssble f t s of the form p (B) = (B)+c (B) for all B n L 1 (;F;P), where 1. s a sublnear lower semcontnuous functonal and c s such that c(b)!!1 : n o 2. p (B) s (B), where s (B) := nf V + c, 2 S;, V B We then obtan the followng characterzaton of the admssble prcng rules. 8 A functonal s sublnear f (x) = (x) and ^e(x + y) (x) +(y) for all contngent clams x; y and nonnegatve real numbers : 17

18 Proposton 4.2 Under Assumpton A and the assumpton of NFL 1 ; or under the assumpton of NAsFL, any admssble prcng rule p on L 1 (;F;P) can be wrtten as p (B) = sup E P [B]+c(B) P 2K for all B n M where K denotes a convex subset of the set of all absolutely contnuous martngale measures, and c s the xed cost gven n Denton 6. hs means that any admssble sublnear lower semcontnuous functonal can be wrtten as the supremum of a subset of all contnuous lnear functonals ~ l, whch le below, are weakly postve and such that ~ l V = V for all n W ; : It also means that p(b)!!1 sup P 2K E P [B] for any contngent clam B; where K s a convex subset of the set of absolutely contnuous martngale measures. hs means that the unt prce of any attanable contngent clam B must belong to an nterval [, nf P 2K E P [,B] ; sup P 2K E P [B]] n the lmt of large quanttes. Note that snce the absence of free lunch n our model wth xed tradng costs s weaker than the absence of free lunch n a frctonless model, these theorems enable us to prce contngent clams n a wder class of models. We shall now turn to the study of the vablty of such admssble prcng rules. 4.2 Vablty Agents are assumed to be characterzed by ther preferences on the space of net trades R X where X = L 1 (;F;P). A par (r;x) represents r unts of consumpton today and x unts of consumpton tomorrow. Preferences are modeled by complete and transtve bnary relatons on R X. In the usual fashon, denotes the strct preference dened from : We also make Assumpton P: Preferences are assumed to satsfy the followng three requrements: 1. For all (r;x) 2 R X; the set f(r ;x ) 2 R X :(r ;x ) (r;x)g s convex. 2. For all (r;x) 2 R X; the set f(r ;x ) 2 R X :(r ;x ) (r;x)g as well as the set f(r ;x ) 2 R X :(r;x) (r ;x )g are closed. 18

19 3. For all (r;x) 2 R X; r > and x 2 L 1 + such that there exsts a real number "> wth x "; (r + r ;x) (r;x) and (r;x+ x ) (r;x) : he class of such preferences s denoted by A: Part 1 says that agents are rsk averse. Part 2 says that ther preferences are contnuous. Part 3 says that agents prefer more to less. A prce system (M;p) s a subspace M of X and a lnear functonal p on M: In the economy assocated to ths prce system, agents can buy and sell any contngent clam m 2 M at a prce p (m) +c (m) n terms of date consumpton where c(m) s a bounded nonnegatve xed tradng cost satsfyng c()=and c(m) > fm 6= : Denton 4.3 A prce system ( M;p) s sad to be vable f there exst some bnary relaton satsfyng Assumpton P and (r ;m ) n R M such that c (m )+r + p (m ) and (r ;m ) (r;m) for all (r;m) n R M such that c (m)+r + p (m) : hs denton s analogous to the denton n Harrson and Kreps (1979) and Kreps (1981). It means that a prce system s vable f there s some agent wth preferences satsfyng Assumpton P who can nd an optmal net trade subject to hs budget constrant. Note that f we assume that the xed cost functon c s subaddtve,.e. c(m 1 +m 2 ) c(m 1 )+c(m 2 ) for all m 1 ;m 2 2 M; a natural assumpton to make about xed costs, then a prce system s vable f and only f there are some agents wth preferences satsfyng Assumpton P for whom (; ) s an optmal trade, 9.e. who are happy wth ther ntal endowment. hs means that a prce system s vable f and only f t s compatble wth economc equlbrum. 9 Indeed, suppose that there exsts an agent wth preferences satsfyng Assumpton P and such that (r ;m ) s an optmal net trade (.e. c (m )+r + p (m ) and (r ;m ) (r;m) for all ^e (r;m) n R M such that c (m) +r + p (m) ): Dene the preferences ~ by (r1 ;m1) ~ (r2 ;m2) f(r1 + r ;m1 + m ) ~ (r2 + r ;m2 + m ) : hey satsfy Assumpton P: Also note that c () + + p ()=: Now suppose that c (~m)+~r + p (~m) and (~r; ~m) ~ (; ) ;.e. (~r + r ; ~m + m ) (r ;m ) : We have c (~m + m )+~r + r +p (~m + m )=[c (~m + m ),c (~m),c (m )]+c (~m)+~r +p (~m)+c (m )+r +p (m ) c (~m + m ), c (~m), c (m ) by subaddtvty of the xed cost functonal ^ec: 19

20 Denton 4.4 Afree lunch for a prce system (M;p) sasequence (m n ) n2n n M, such that there exst sequences (r n ) n2n, (x n ) n2n n L 1 (;F;P) convergng respectvely to r and x wth r + x 6= ; for whch for all n n N m n x n and c (m n )+r n + p (m n ) : n o We shall now consder the case where M = V ; n W ; ; the set of attanable contngent clams n the frctonless economy, and where the prcng rule s the lnear functonal p dened on M by p V = V for all n W ;. As we have seen n Proposton??, f we want a prce system (M;) to be compatble wth the assumpton of no arbtrage - whch must be the case for vable prce systems as well as for prce systems that admt no free lunch - then we must have = p: We shall now nvestgate the converse,.e. the condtons under whch ths prce system s a vable one and the condtons under whch t admts no free lunch. But rst let us have: Denton 4.5 A free lunch from tme n the frctonless securtes market model s a sequence ( n ) n2n of smple strateges such that there exst sequences (~x n ) n2n of random varables belongng to L 1 (;F;P) and (~r n ) n2n n R N convergng respectvely to x n L 1 (;F;P) and r > n R for whch for all n, n s n W ;, V n,~r n and V n ~x n : We then have heorem 4.1 he followng condtons are equvalent : 1. (M;p) s vable. 2. (M;p) admts no free lunch. 3. here exsts a weakly postve contnuous lnear functonal on L 1 (;F;P) such that j M = p and such that for all f n A = ff 2 L 1 ; 9" > such that f " g ; we have (f) > : 4. here s no free lunch from tme. 2

21 For each date and each event B n F ; we shall dene a prce system M ;B ;p ;B where M ;B s a subspace of X and p ;B a lnear functonal on M ;B. he nterpretaton s that n ths economy, at that date and n that event B; agents are able to buy and sell some contngent clams m n M ;B at a cost p ;B (m) +c (m) n date, event B consumpton. We consder n o M ;B = V ; n W ;B and p ;B dened on M ;B by p ;B V obtan heorem 4.2 he followng condtons are equvalent : 1. For all n ; for all B n F ; M ;B ;p ;B s vable. = V 2. For all n ; for all B n F ; M ;B ;p ;B admts no free lunch. and we 3. here s no free lunch n our securtes markets model wth xed tradng costs. herefore, the prce system we have consdered s vable and admts no free lunch f and only f there s no free lunch n our model wth xed tradng costs. 5 Concluson In ths paper, we have shown that a securtes markets model wth xed tradng costs admts no free lunch f and only f there exsts a famly of absolutely contnuous probablty measures for whch the normalzed (by a numerare) prce processes are martngales, condtonal on any possble future event. he man derence wth the frctonless case s that the martngale measures only need to be absolutely contnuous nstead of equvalent (but we need a whole famly of martngale measures). Snce the absence of arbtrage opportunty or free lunch s a weaker condton n the presence of xed tradng costs than n the frctonless case, ths result wll allow future research to consder a wder class of models. he transacton costs are assumed to be xed n the sense that they are bounded (regardless of the transacton sze). hs s compatble wth xed brokerage fees, brokerage arrangements where margnal fees go to zero beyond a gven volume (a common arrangement n the ndustry), xed nvestment taxes to gan access to a market, operatonal 21

22 and processng costs, xed costs nvolved n settng up an oce and nformaton technology, and the opportunty cost of lookng at a market or of dong a specc trade. We also show that the only arbtrage free prcng rules on the set of attanable contngent clams are those that are equal to the sum of an expected value wth respect to any absolutely contnuous martngale measure and of a bounded xed cost functonal. Moreover, these prcng rules are the only ones to be vable as models of economc equlbrum,.e. such that there exst some ratonal agents who are happy wth ther ntal endowment - and hence for whom supply s equal to demand. 22

23 Appendx Proof of Proposton??, We wll wrte EAO for exstence of an arbtrage opportunty and NAO for no arbtrage opportunty. We wll denote a net gan arbtrage opportunty wth xed costs by AO 3 and a frctonless "- net gan arbtrage opportunty by AO 4. We shall prove that the four notons of NAO are equvalent. We rst treat the case where the xed costs do not depend upon the strategy. 1. NAO 3, NAO 1 : EAO 1 ) EAO 3 s mmedate. EAO 3 ) EAO 1 : we consder the strategy ~ null before and outsde B such that for all t ; ~ t = t +,c, V on B and ~ k t = k t for all k 6= : It s easy to check that ~ s n S ;B ;V ~ + c = and V ~ ; 6= onb. 2. NAO 2, NAO 1 : EAO 1 ) EAO 2 : we consder the strategy ~ null before and outsde B such that ~ = and for all t> ~ t = t, tx j=+1 ~ k t = k t for all k 6=. ( j, j,1 ) Z j on B and hen ~ s n W ;B, V ~ and as c >, we have V ~ < on B. EAO 2 ) EAO 1 : notce that, by consderng some B B, one can replace the condton V < onb by ether the condton \V ; 6= on B" or by the condton \there exsts a postve real number " such that V," on B" because V s F,measurable. So there exsts 1 satsfyng V,C where C = P k= C k and C k = sup!2b c k (!) : We consder the strategy ~ null before and outsde B such that for all t ~ t = t + C, hen ~ s n S ;B and satses V ~ tx j= ~ k t = k t for all k 6= : 23 + c = V c j and + C on B; V ~ :

24 3. NAO 2, NAO 4 : EAO 2 ) EAO 4 s easy wth the techncal remark made for the proof of EAO 2 ) EAO 1. EAO 4 ) EAO 2 : we consder the strategy ~ null before and outsde B and such that for all t ~ t = t, V, "=2 onb and ~ k t = k t for all k 6= : hen ~ s n W ;B and satses V ~ =,("=2) < onb. We have V ~ = outsde B; and V ~ 1 B =(V ~, V ~ )1 B + V ~ 1 B =(V, V )1 B, ("=2)1 B so V ~ "=2 onb and V ~. If the costs depend upon the strategy, then EAO 1 ) EAO 3 s mmedate. For EA 3 ) EAO 4, we easly get the exstence of a strategy 2 W ;B, V, V c on B. hen there exsts a>, such that for B fc > ag, P (B ) > and 1 B s an AO 4. he proof of EAO 4 ) EAO 2 remans the same as above, as well as EAO 2 ) EAO 1, replacng C = P k= C k by C = P k= C k.2 Proof of Proposton?? of Proposton??. We adopt the same notatons as n the proof 1. NFL 3, NFL 1 : We shall treat here the case where the xed cost do not depend upon the strategy. he case where the cost depends upon the strategy s an mmedate extenson, replacng c wth c n each tme t s needed. EFL 1 ) EFL 3 : here exsts a sequence ( n ) n n S ;B for whch V n, V n + c xn +(k n, c ) that converges to x +(k, c ) ; 6=. For EFL 3 ) EFL 1, we consder the sequence ~ of strateges ~ n null before and outsde B such that for all n n N; for all t ( ~ n ) t = ( n ) t +,c, V n on B and ( ~ n ) k t = ( n ) k t for all k 6= It s then easy to check that for all n n N; ~ n s n S ;B, V ~ n + c = and V ~ n = V n, V n + c xn! x ; 6= on B. Notce that n the case where the cost depends upon the strategy, we use the fact that c = c when =. 24

25 2. NFL 2, NFL 4 : EFL 2 ) EFL 4 s mmedate snce we can ndfferently assume r ; 6= or r > or there exsts a postve real number " such that r " by consderng for all n n N the random varables ~r n = r n 1 r> and ^r n = r n 1 r" ; and the followng correspondng strateges ~ n and ^ n such that for all t; ~ n t = n t 1 r> ; ^ n t = n t 1 r". For EFL 4 ) EFL 2 ; we consder the sequence ~ of strateges ~ n null before and outsde B and such that for all n n N; for all t ; ( ~ n ) t = ( n ) t, V n, "=2 onb and ( ~ n ) k t = ( n ) k t for all k 6= : hen for all n n N, ~ n s n W ;B and satses (V ~ n (V n and V ~ n, V n )1 B so V ~ n x n! "=2 on B: As V ~ n = outsde B, ths completes the proof. Proof of Corollary?? mmedate usng Proposton??. Proof of Lemma??, V ~ n )1 B = =,"=2 < on B 1. For NFL 2 ) NFL 1,we prove the mplcaton NFL 2 ) NFL 3, whch s mmedate usng the fact that c > (or that c n c > n the case where the cost depends upon the strategy) and changng a strategy belongng to S ;B nto a strategy belongng to W ;B by procedng lke n the proof of Proposton??. 2. Under Assumpton A, NFL 1 ) NFL 2 : suppose there s a FL 2 ; n the form of a sequence ( n ) n2n of smple strateges lke n Denton??. As we have seen n the proof of Proposton??, we can nderently assume that r ; 6= or r> or there exsts a postve real number " such that r " by consderng for all n n N the random varables ~r n = r n 1 r> and ^r n = r n 1 r" ; and the followng correspondng strateges ~ n and ^ n such that for all t; ~ n t = n t 1 r> ; ^ n t = n t 1 r". So there exsts a real number 1 such that r > C where C denotes the real number n the addtonal Assumpton A: We consder a sequence ~ of strateges ~ n such that ~ n s n S ;B ( ~ n ) k t = ( n ) k t for all k 6= and for all t ( ~ n ) = ( n ) + C, c : 25

26 We then have for all n; V ~ n = V n + C, c so V ~ n +(r n, C + c ) wth (r n, C + c ), c! r, C > : We can choose ~ such that for all n V ~ n V n x n wth x n! x so the sequence ~ consttutes a free lunch wth xed costs. 3. NAsF L, NFL 2 : EFL 2 ) EAsF L: here agan, we can assume that r s (strctly) greater than some postve real number " on B. he xed cost at each date s supposed to be bounded (c t <C t n the case where the xed cost does not depend on the strategy and c t < C t n the case where the xed cost depends upon the strategy). hen for all n, there exsts n such that n " s greater than the cumulatve xed costs of any smple strategy wth the same tradng dates as n so that for all n, there exsts a strategy ~ n n S ;B for whch We get V ~ n + c ~ n = V n n V ~ n n x n! x V ~ n = n V n + n ", C V ~ n n V n n x n. + " + c~n, C,r n + "!,r + "< n EAsF L ) EFL 2 : by nvestng at each date the xed cost n the rskless asset, we obtan a sequence ( ~ n ) n of strateges n W ;B. Lettng for all n, n := n n,we obtan a sequence ( n ) n of strateges n W ;B such that V n," n!," < onb V n x n! x 26

27 Proof of heorem?? Frst notce that the exstence of such a famly of probablty measures s equvalent to the exstence of a famly of random varables denoted by g ;B n L 1 (;F;P) satsfyng g ;B ; 6= on B and such that for all (s; t) wth s t and for all A n F s ; E h g ;B Z t 1 A\B = h E g ;B Z s 1 A\B : the equvalence s easly obtaned by takng g ;B = dp ;B =dp and by denng P ;B by h P ;B (A) = E g ;B 1 A\B E [g ;B 1 B ] for all A n F s : 1) Assume rst the exstence of such a famly of martngale measures and of a sequence ( n ) n2n of strateges such that there exst n and B n F for whch for all n, n s n W ;B. Let = t n ;t n 1; :::; t n N n = denote the tradng dates of the smple strategy n. hen usng the denton of V n, the fact that n s a frctonless self nancng strategy, the martngale property of P ;B and the fact that n s null outsde B, we have for all n; E P ;B h V n so that for all n, j F = E P ;B [ n h Z j F ] = E P ;B n t Z n j F Nn,1 h h = E P ;B n t E P ;B Z n j F t n Nn,1 Nn,1 h = E P ;B V n t j F n Nn,1 j F E P ;B h V n j F = ::: = E P ;B h V n j F = V n on B a:s: P ;B and E P ;B h (V n, V n )1 B j F =: h hen for all A n F ; for all n n N, E g ;B (V n t s mpossble to have V n, V n, V n )1 B\A =. Now x n wth x n! L 1x " on B because ths would lead to = E h g ;B (V n, V n E h g ;B x n 1 B! E h g ;B x1 B > -because g ;B )1 B E h g ;B x n 1 B and s assumed to be bounded-: there exsts no frctonless "-net gan free lunch, whch usng Proposton??, completes the proof of the rst mplcaton. 27

28 2) Conversely, assume there exsts no frctonless strong free lunch. As we have seen n Corollary??, fc ;B = K ;B, L 1 + and A B = n f 2 L 1 +; 9" > such that f " on B o ; the condton of no frctonless strong free lunch s equvalent to the condton that for all n and for all B n F ; C ;B \ A B = ;: For each xed (; B), we apply a strct separaton theorem n L 1 (;F;P) to the closed convex set C ;B and the compact set f1 B g to nd g ;B n L 1 (;F;P) and two real numbers and wth <such that g ;B j C ;B << D 1 B ;g ;BE : he random varable g ;B s bounded from above on C ;B and therefore on L 1,,sog ;B : As belongs to C ;B and C ;B s a convex cone, we can take =: hen D 1 B ;g ;B E > so g ;B 6=on B. As belongs to L 1 +; we have g ;B j K ;B and we even get the equalty because K ;B savector space. For all s, for all A n F s,we consder for all k n f1; :::; ng, the n -dmensonal random varable s;a;k 2 P s;b gven by k s;a;k = 1 A\B l s;a;k = for all l 6= k: As K ;B f; :::; ng, for all (s; t) wth s t and for all A n F s, we have =Ln n s Z t, Z s ; s 2 P s;b ; s t o, we get that for all k n Z k t 1 A\B, Z k s 1 A\B 2 K ;B. hen for all (s; t) wth s t, for all A n F s we obtan or E h g ;B Z t 1 A\B = E h g ;B Z s 1 A\B : E h g ;B (Z t, Z s )1 A\B = Proof of heorem?? 1) ) 2): see heorem??. 28

29 2) ) 3): consder g ;B = dp ;B =dp: 3) ) 4): let n and B n F be xed. We wll wrte g for g ;B and for ;B : We can assume g =outsde B. As g ; 6= on B; the same s true for the random varable E [g j F ] and there exsts a postve real number such that P (E [g j F ] ) > : Let A = fe [g j F ] g : hen A belongs to F ; A B and P (A) 6= : We dene an operator on R ;B by (C) = E [gc j F ] E [g j F ] 1 A for all C 2 R ;B. he lnear operator s lnear, contnuous and takes values n R ;B. If C ; gc so s weakly postve. Only the last condton remans to be checked. Notce rst that for all s t, E [gz t j F s ]1 A = Z s E [g j F s ]1 A. Now, for all n W ;B wth tradng dates denoted by ( = t ;t 1 ; :::; t N = ), we have V E [g Z j F ] = 1 h A E [g j F ] h = E N,1 E gz j F N,1 j F = E h V E [g j F ] N,1 E h g j F N,1 j F E [g j F ] so V = V N,1 = ::: = V 1 A : 4) ) 1): consder a sequence ( n ) n2n of strateges such that there exst n and B n F such that for all n; n s n W ;B : For all n n N; we then have 1 A ;B V n, V n =: Now t s mpossble to have V n, V n x n wth x n! L 1x " on B because ths would lead to = 1 A ;B V n, V n 1 A ;B (x n ) because ;B s lnear and weakly postve; as ;B s contnuous, 1 A ;B (x n )! 1 A ;B (x) 1 A ;B ("1 A ) "1 A because 1 A ;B (1 A ) = 1 A : a contradcton. 1), 5): see Lemma?? Proof of heorem?? 1) ) 2) s mmedate. 2) ) 3): usng a strct separaton theorem exactly lke n the proof of our man theorem, we get that there exsts a random varable Z n L 1 (;F;P) 1 A 1 A 29