Existence of Shadow Prices in Finite Probability Spaces

Transcription

1 Exisence of Shadow Prices in Finie Probabiliy Spaces Jan Kallsen Johannes Muhle-Karbe Absrac A shadow price is a process S lying wihin he bid/ask prices S, S of a marke wih proporional ransacion coss, such ha maximizing expeced uiliy from consumpion in he fricionless marke wih price process S leads o he same maximal uiliy as in he original marke wih ransacion coss. For finie probabiliy spaces, his noe provides an elemenary proof for he exisence of such a shadow price. Key words: ransacions coss, porfolio opimizaion, shadow price Mahemaics Subjec Classificaion (2000: 91B28, 91B16 1 Inroducion When considering problems in Mahemaical Finance, one classically works wih a fricionless marke, i.e., one assumes ha securiies can be purchased and sold for he same price S. This is clearly a srong modeling assumpion, since in realiy one usually has o pay a higher ask price when purchasing securiies, whereas one only receives a lower bid price when selling hem. Pu differenly, one is faced wih proporional ransacion coss. The inroducion of even miniscule ransacion coss ofen fundamenally changes he srucure of he problem a hand (cf., e.g., [5, 7, 3]. Therefore models wih ransacion coss have been exensively sudied in he lieraure (see, e.g., he recen monograph [10] and he references herein. Opimizaion problems involving ransacion coss are usually ackled by one of wo differen approaches. Whereas he firs mehod employs mehods from sochasic conrol heory, he second reformulaes he ask a hand as a similar problem in a fricionless marke. This second approach goes back o he pioneering paper of Jouini and Kallal [8]. They showed ha under suiable condiions, a marke wih bid/ask prices S, S is arbirage free Mahemaisches Seminar, Chrisian-Albrechs-Universiä zu Kiel, Wesring 383, D Kiel, Germany, ( kallsen@mah.uni-kiel.de. Deparemen Mahemaik, ETH Zürich, Rämisrasse 101, CH-8092 Zürich, Swizerland, ( johannes.muhle-karbe@mah.ehz.ch. 1

2 if and only if here exiss a shadow price S lying wihin he bid/ask bounds, such ha he fricionless marke wih price process S is arbirage free. The same idea has since been employed exensively leading o various oher versions of he fundamenal heorem of asse pricing in he presence of ransacion coss (cf., e.g., [17, 7] and he references herein. I has also found is way ino oher branches of Mahemaical Finance. For example, [13] have shown ha bid/ask prices can be replaced by a shadow price in he conex of local riskminimizaion, whereas [2, 4, 14, 12] prove ha he same is rue for porfolio opimizaion in cerain Iô process seings. In hese aricles he dualiy heory for fricionless markes is ypically applied o a shadow price, i.e., shadow prices and he corresponding maringale measures consisen price sysems in he erminology of [17, 7] play he role of maringale measures in fricionless markes in markes wih proporional ransacion coss. In he presen sudy we esablish ha in finie probabiliy spaces, his general principle indeed holds rue lierally for invesmen/consumpion problems, i.e., a shadow price always exiss. We firs inroduce our finie marke model wih proporional ransacion coss in Secion 2. Subsequenly, we sae our main resul concerning he exisence of shadow prices and prove i using elemenary convex analysis. For a vecor x = (x 1,..., x d, we wrie x + = (max{x 1, 0},..., max{x d, 0} and x = (max{ x 1, 0},..., max{ x d, 0}. Likewise, inequaliies and equaliies are undersood o be componenwise in a vecor-valued conex. Moreover, for any sochasic process X we wrie X := X X 1. 2 Uiliy maximizaion wih ransacion coss in finie discree ime We sudy he problem of maximizing expeced uiliy from consumpion in a finie marke model wih proporional ransacion coss. Our general framework is as follows. Le (Ω, F, (F {0,1,...,T }, P be a filered probabiliy space, where Ω = {ω 1,..., ω K } and he ime se {0, 1,..., T } are finie. In order o avoid lenghy noaion, we le F = F T = P(Ω, F 0 = {, Ω}, and assume ha P ({ω k } > 0 for all k {1,..., K}. However, one can show ha all following saemens remain rue wihou hese resricions. The financial marke we consider consiss of a risk-free asse 0 (also called bank accoun wih price process S 0 normalized o S 0 = 1, = 0,..., T, and risky asses 1,..., d whose prices are expressed in muliples of S 0. More specifically, hey are modelled by heir (discouned bid price process S = (S 1,..., S d and heir (discouned ask price process S = (S 1,..., S d, where we naurally assume ha S, S are adaped and saisfy S S > 0. Their meaning should be obvious: if one wans o purchase securiy i a ime, one mus pay he higher price S i whereas one receives only S i for selling i. The connecion o proporional ransacion coss is he following. In fricionless markes, one models he (mid price process S of he asses under consideraion. Transacion coss equal o a fracion ε [0,, ε [0, 1 of he amoun ransaced for purchases and 2

3 sales of socks, respecively, hen lead o an ask price of S := (1 + εs and a bid price of S := (1 εs. However, he mid price S does no maer for he modelling of he marke wih ransacion coss, since shares are only bough and sold a S resp. S. Therefore we work direcly wih he bid and ask price processes. Remark 2.1 Our seup amouns o assuming ha he risk-free asse can be purchased and sold wihou incurring any ransacion coss. This assumpion is commonly made in he lieraure dealing wih opimal porfolios in he presence of ransacion coss (cf., e.g., [5], and seems reasonable when hinking of securiy 0 as a bank accoun. For foreign exchange markes where i appears less plausible, a numeraire free approach has been inroduced by [9]. This approach would, however, require he use of mulidimensional uiliy funcions as in [6, 1] in our conex. Definiion 2.2 A rading sraegy is an R d+1 -valued predicable sochasic process (ϕ 0, ϕ = (ϕ 0, (ϕ 1,..., ϕ d, where ϕ i, i = 0,..., d, = 0,..., T + 1 denoes he number of shares held in securiy i unil ime afer rearranging he porfolio a ime 1. A (discouned consumpion process is an R-valued, adaped sochasic process c, where c, = 0,..., T represens he amoun consumed a ime. A pair ((ϕ 0, ϕ, c of a rading sraegy (ϕ 0, ϕ and a consumpion process c is called porfolio/consumpion pair. To capure he noion of a self-financing sraegy, we use he inuiion ha no funds are added or wihdrawn. More specifically, his means ha he proceeds of selling sock mus be added o he bank accoun while he expenses from consumpion and he purchase of sock have o be deduced from he bank accoun whenever he porfolio is readjused from ϕ o ϕ +1 and an amoun c is consumed a ime {0,..., T }. Defining purchase and sales processes ϕ, ϕ as his leads o he following noion. ϕ := ( ϕ +, ϕ := ( ϕ, (2.1 Definiion 2.3 A porfolio/consumpion pair ((ϕ 0, ϕ, c is called self-financing (or (ϕ 0, ϕ c-financing if ϕ 0 +1 = S ϕ +1 S ϕ +1 c, = 0,..., T. (2.2 Remark 2.4 Define he cumulaed purchases ϕ and sales ϕ as ϕ := (ϕ ϕ s, ϕ := (ϕ 0 + s=1 ϕ s, = 1,..., T + 1. Then he self-financing condiion (2.2 implies ha ( (ϕ 0, ϕ, ϕ, c is self-financing in he usual sense for a fricionless marke wih 2d + 1 securiies (1, S, S. Moreover, noe ha for S = S, we recover he usual self-financing condiion for fricionless markes. 3 s=1

4 We consider an invesor who disposes of an iniial endowmen (η 0, η R d+1 +, referring o he iniial number of securiies of ype i, i = 0,..., d, respecively. Definiion 2.5 A self-financing porfolio/consumpion pair ((ϕ 0, ϕ, c is called admissible if (ϕ 0 0, ϕ 0 = (η 0, η and (ϕ 0 T +1, ϕ T +1 = (0, 0. An admissible porfolio/consumpion pair ((ϕ 0, ϕ, c is called opimal if i maximizes ( κ E u (κ (2.3 =0 over all admissible porfolio/consumpion pairs ((ψ 0, ψ, κ, where he uiliy process u is a mapping u : Ω {0,..., T } R [,, such ha (ω, u (ω, x is predicable for any x R and x u (ω, x is a proper (in he sense of Rockafellar [15], uppersemiconinuous, concave funcion for any (ω, Ω {0,..., T }, which is increasing on is convex effecive domain {x R : u (ω, x > } for (ω,, {0,..., T 1} and sricly increasing for (ω, T. In view of Definiion 2.5, we only deal wih porfolio/consumpion pairs where he enire liquidaion wealh of he porfolio is consumed a ime T. Noe ha his can be done wihou loss of generaliy, because he uiliy process is increasing in consumpion. Remark 2.6 Since we allow he uiliy process o be random, assuming S 0 = 1, = 0,..., T also does no enail a loss of generaliy in he presen seup. More specifically, le S 0 be an arbirary sricly posiive, predicable process. In his undiscouned case a porfolio/consumpion pair (ϕ, c should be called self-financing if ϕ 0 +1S 0 = S ϕ +1 S ϕ +1 c, for = 0,..., T. Admissibiliy is defined as before. By direc calculaions, one easily verifies ha ((ϕ 0, ϕ, c is self-financing resp. admissible if and only if ((ϕ 0, ϕ, ĉ = ((ϕ 0, ϕ, c/s 0 is self-financing resp. admissible relaive o he discouned processes Ŝ0 := S 0 /S 0 = 1, Ŝ := S/S0 and Ŝ := S/S0. In view of ( ( E u (c = E û (ĉ =0 for he uiliy process û (x = u (S 0 x, he problem of maximizing undiscouned uiliy wih respec o u is equivalen o maximizing discouned expeced uiliy wih respec o û. We now menion some well-known specificaions ha are included in our seup. Example Maximizing expeced uiliy from erminal wealh a ime T is included as a special case by seing =0 u (x = {, for x < 0, 0, for x 0, for {0,..., T 1}. 4

5 2. One also obains a uiliy process in he sense of Definiion 2.5 via u(ω,, x := D (ωu(x, where D is some posiive predicable discoun facor (e.g., D = exp( r or D = 1/(1+r for r > 0 and u : R R { } is a uiliy funcion in he usual sense, as, e.g., he logarihmic uiliy funcion u(x = log(x, a power uiliy funcion u(x = x 1 p /(1 p, p R + \{0, 1}, or an exponenial uiliy funcion u(x = e px /p, p > 0. In paricular, one does no have o rule ou negaive consumpion from a mahemaical poin of view, even hough allowing i seems raher dubious from an economical perspecive. 3 Exisence of shadow prices We now inroduce he cenral concep of his paper. Definiion 3.1 We call an adaped process S shadow price process if S S S and if he maximal expeced uiliies in he marke wih bid/ask-prices S, S and in he marke wih price process S wihou ransacion coss coincide. The following heorem shows ha in our finie marke model, shadow price processes always exis, excep in he rivial case where all admissible porfolio/consumpion pairs lead o expeced uiliy. The main idea of he proof is o rea purchases and sales separaely in (2.2. This means ha we effecively consider a problem wih wo ses of asses whose holdings mus be in- resp. decreasing. Maybe surprisingly, he Lagrange mulipliers corresponding o hese consrains merge ino only one process (raher han wo. The laer has a naural inerpreaion as a shadow price process. Theorem 3.2 Suppose an opimal porfolio/consumpion pair ((ϕ 0, ϕ, c exiss for he marke wih bid/ask prices S, S. Then if E( T =0 u (c >, a shadow price process S exiss. PROOF. Sep 1: As he uiliy process is increasing, allowing for sales and purchases a he same ime does no increase he maximal expeced uiliy. More precisely, since x u (x is increasing for fixed, maximizing (2.3 over all admissible porfolio/consumpion pairs yields he same maximal expeced uiliy as maximizing (2.3 over he se of all ((ψ 0, ψ, ψ, κ, where (ψ 0 ( =0,...,T +1 is an R-valued predicable process wih ψ 0 0 = η 0 and ψ 0 T +1 = 0, he increasing, Rd -valued predicable processes (ψ =0,...,T +1, (ψ =0,...,T +1 saisfy ψ 0 = η +, ψ 0 = η, ψ T +1 ψ T +1 = 0 and (κ =0,...,T is a consumpion process such 5

6 ha (2.2 holds for = 0,..., T and ((ψ 0, ψ, κ insead of ((ϕ 0, ϕ, c. Moreover, if we define ϕ and ϕ as in (2.1 above and se ϕ := η + + ϕ, ϕ := η + =1 ϕ, =1 hen ((ϕ 0, ϕ, ϕ, c is an opimal sraegy in his se. Sep 2: We now formulae our opimizaion problem as a finie-dimensional convex minimizaion problem wih convex consrains. To his end, denoe by F 1,..., F m he pariion of Ω ha generaes F, {0,..., T }. Since a mapping is F -measurable if and only if i is consan on he ses F j, j = 1,..., m, we can idenify he se of all processes ((ψ 0, ψ, ψ, κ, where (ψ 0 =0,...,T +1 is R-valued and predicable wih ψ0 0 = η 0, (ψ =0,...,T +1 and (ψ =0,...,T +1 are increasing, R d -valued and predicable wih ψ 0 = η +, ψ 0 = η and (κ =0,...,T is a consumpion process such ha (2.2 holds for = 0,..., T wih R 2dn + R n := (R m 0d +... R m T d + (R m 0d +... R m T d + (R m 0... R m T, and vice versa, namely wih ( ψ, ψ, c := ( ψ,1,1 1,..., ψ,m T,d T +1, ψ,1,1 1,..., ψ,m T,d T +1, c 1 0,..., c m T T, where we use he noaion ψ,j,i := ψ,i (ω for i = 1,..., d, = 0,..., T, j = 1,..., m, and ω F j (and analogously for ψ, c, S, S. Using his idenificaion, we can define mappings f : R 2dn + R n R { }, h j 0 : R 2dn + R n R and h j : R 2dn + R n R d (for j = 1,..., m T by ( f( ψ, ψ, c := E h j 0( ψ, ψ, c := η 0 + =1 =1 T +1 h j ( ψ, ψ, c := η + =1 u (c ( (S j 1 ψ,j ( ψ,j, ψ,j. (S j 1 ψ,j c j, Noe ha h 0 resp. h represen he erminal posiions in bonds resp. socks. Wih his noion, ( ϕ, ϕ, c is opimal if and only if i minimizes f over R 2dn + R n subjec o he consrains h j 0 = 0 and h j = 0 for j = 1,..., m T. Since all mappings are acually convex funcions on R (2d+1n, his is equivalen o ( ϕ, ϕ, c minimizing f over R (2d+1n subjec o he consrains h j 0 = 0, h j = 0 (for j = 1,..., m T and g,j, g,j 0 (for = 0,..., T and j = 1,..., m, where he convex mappings g,j, g,j : R (2d+1n R d are given by =0 g,j ( ψ, ψ, c := ψ+1,,j g,j ( ψ, ψ, c := ψ+1.,j 6

7 In view of [15, Theorems 28.2 and 28.3], ( ϕ, ϕ, c is herefore opimal if and only if here exiss a Lagrange muliplier, i.e., real numbers ν j, µ j,i (for i = 1,..., d and j = 1,..., m T and λ,j,i, λ,j,i (for = 0,..., T, i = 1,..., d and j = 1,..., m such ha he following holds. 1. For = 0,..., T, j = 1,..., m and i = 1,..., d, we have λ,j,i ( ϕ, ϕ, c, g,j,i ( ϕ, ϕ, c 0 and λ,j,i g,j,i g,j,i as λ,j,i g,j,i ( ϕ, ϕ, c = h j 0( ϕ, ϕ, c = 0 and h j ( ϕ, ϕ, c = 0 for j = 1,..., m T., λ,j,i 0 as well as ( ϕ, ϕ, c = 0 as well 3. m T 0 f( ϕ, ϕ, c + ν j h j 0( ϕ, ϕ, c + + =0 d m λ,j,i g,j,i ( ϕ, ϕ, c + =0 d m T µ j,i h j,i ( ϕ, ϕ, c d m λ,j,i g,j,i ( ϕ, ϕ, c. Here, denoes he subdifferenial of a convex mapping (cf. [15] for more deails. Sep 3: We now use he opimaliy condiions for he marke wih ransacion coss o consruc a shadow price process. By [16, Proposiion 10.5] we can spli Saemen 3 ino many similar saemens where he subdifferenials on he righ-hand side are replaced wih parial subdifferenials relaive o ϕ,1,1 1,..., ϕ,m T,d T +1, ϕ,1,1 1,..., ϕ,m T,d T +1, c 1,..., c m T T, respecively. In paricular, for cj T, j {1,..., m T }, we obain 0 c j f( ϕ, ϕ, c ν j, (3.1 T where x denoes he parial subdifferenial of a convex funcion relaive o a vecor x. Hence ν j < 0, j = 1,..., m T, because f is sricly decreasing in c j T. Furhermore, since he mappings g,j,i, g,j,i (for = 0,..., T, j = 1,..., m and i = 1,..., d and h j 0, h j,i (for j = 1,..., m T and i = 0,..., d are differeniable, heir parial subdifferenials coincide wih he respecive parial derivaives by [15, Theorem 25.1]. Hence, aking parial derivaives wih respec o ϕ,j,i +1 resp. ϕ,j,i +1, {0,..., T }, j {1,..., m }, i {0,..., d}, Saemen 3 above implies ha 0 = ( µ k,i and likewise k:ω k F j = k:ω k F j 0 = k:ω k F j k:ω k F j µ k,i k:ω k F j µ k,i k:ω k F j λ,j,i 1 + ( 7 1 λ,j,i k:ω k F j λ,j,i k:ω k F j, (3.2. (3.3

8 In paricular we have, for = 0,..., T, j = 1,..., m, i = 1,..., d, ( 1 + λ,j,i k:ω k F j = 1 λ,j,i k:ω k F j =: S j,i. Since S := ( S 1,..., S d is consan on F j by definiion, his defines an adaped process. Furhermore, we have S S S, since λ,j,i, λ,j,i 0, for i = 1,..., d, = 0,..., T and j = 1,..., m, and because < 0 for k = 1,..., m T. Moreover, by Saemen 1 above, we have λ,j,i = 0 if ϕ,j,i > 0 and λ,j,i = 0 if ϕ,j,i > 0, such ha S i = S i on { ϕ,i > 0}, Si = S i on { ϕ,i > 0}. (3.4 Se µ j,i := µ j,i (for j = 1,..., m T, i = 1,..., d, ν j := ν j (for j = 1,..., m T and λ,j,i,j,i, λ := 0 (for = 0,..., T, j = 1,..., m and i = 1,..., d. Saemens 1, 2 and 3 above, Equaions (3.2, (3.3, (3.4 and he definiion of S hen yield he following. λ,j,i λ,j,i 1. For = 0,..., T, i = 1,..., d and j = 1,..., m we have, 0 as well as g,j,i ( ϕ, ϕ, c, g,j,i ( ϕ, ϕ,j,i, c 0 and λ g,j,i ( ϕ, ϕ, c = 0 as well,j,i as λ g,j,i ( ϕ, ϕ, c = 0, 2. h j 0( ϕ, ϕ, c = 0 and h j ( ϕ, ϕ, c = 0 for j = 1,..., m T, 3. 0 f( ϕ m T, ϕ, c + ν j h j 0( ϕ, ϕ, c + =0 d m λ,j,i g,j,i ( ϕ, ϕ, c =0 d m T µ j,i h j,i ( ϕ, ϕ, c d m λ,j,i g,j,i ( ϕ, ϕ, c, where he mappings f, h j 0, h j, g,j, g,j are defined by seing S = S = S in he definiion of he mappings f, h j 0, h j, g,j, g,j above. In view of [15, Theorem 28.3] and Seps 1 and 2 above, (ϕ, c is herefore no only opimal in he marke wih bid/ask prices S, S, bu in he marke wih bid-ask prices S, S (i.e., in he fricionless marke wih price process S as well. Hence S is a shadow price process and we are done. Remark 3.3 Suppose ha, for any ε > 0 and (ω, Ω {0,..., T }, here exis x 1, x 2 such ha x u (ω, x is differeniable a x 1, x 2 and u (ω, x 1 /u (ω, x 2 < ε. Then i follows from sandard argumens in convex analysis along he lines of [11, Lemma 2.9] ha an opimal porfolio consumpion/consumpion pair exiss if he marke does no allow for arbirage. By he fundamenal heorem of asse pricing wih ransacion coss in finie probabiliy spaces (cf. [17], absence of arbirage in our model is equivalen o he exisence of a 8

9 consisen price sysem. This is a pair consising of an adaped process S evolving wihin he bid-ask spread [S, S] and a corresponding equivalen maringale measure Q. Similarly, he following resul characerizes he opimal consumpion process in erms of a specific consisen price sysem, namely a shadow price and a specific maringale measure for he corresponding fricionless marke. In analogy o he fundamenal heorem of asse pricing, we daringly call i a fundamenal heorem of uiliy maximizaion wih ransacion coss. Corollary 3.4 Le ((ϕ 0, ϕ, c be an admissible porfolio consumpion pair for he marke wih bid/ask prices S, S saisfying E( T =0 u (c >. Then we have equivalence beween: 1. ((ϕ 0, ϕ, c is opimal in he marke wih bid/ask prices S, S. 2. There exiss a consisen price sysem ( S, Q and a number α (0, such ha ( d E Q dp F 1 α u (c, = 0,..., T. PROOF. 1 2: We use he noaion from he proof of Theorem 3.2. In paricular, S and ν, µ denoe he shadow price and he corresponding Lagrange mulipliers inroduced here. Since ν j < 0 for j = 1,..., m T, Q(F j T := νj /α, j = 1,..., m T, wih α := m T k=1 νk, defines a measure on F, which is equivalen o P. Moreover, since he Radon-Nikodým densiy of Q wih respec o P is given by (d Q/dP j = ν j /(αp (F j T, j = 1,..., m T, he densiy process of Q wih respec o P is given by ( Z j d := E Q j dp F = k:ω k F j αp (F j, = 1,..., T, j = 1,..., m. By considering he parial subdifferenials wih respec o c j, = 1,..., T, j = 1,..., m in opimaliy condiion 3 for he process S in he proof of Theorem 3.2, we find ha Z lies in he subdifferenial 1 u α (c for = 1,..., T. I herefore remains o show ha Q is a maringale measure for S, i.e., ha Z S i is a P -maringale for i = 1,..., d. By S j,i T definiion of Z resp. S and (3.3, we have Z j T = νj Sj,i T /(αp (F j T = µj,i /(αp (F j T for i = 1,..., d and j = 1,..., m T. Hence Z S is a maringale, because, for i = 1,..., d, = 0,..., T 1 and j = 1,..., m, we have E( Z T Si T F j = k:ω k F j P ({ω k } P (F j µk,i αp ({ω k } = ( k:ω k F j S j αp (F j = Z j S j, where we have again used (3.3 for he second equaliy. 2 1: We firs show ha Saemen 2 implies ha ((ϕ 0, ϕ, c is opimal in he fricionless marke wih price process S. For any admissible porfolio/consumpion pair 9

10 ((ψ 0, ψ, κ, summing (2.2 over = 0,..., T + 1, insering (ψt 0 +1, ψ T +1 = (0, 0, and using he Q-maringale propery of S yields he budge consrain ( u (κ = η 0 + η S0. (3.5 E Q =0 In paricular, his holds for ((ϕ 0, ϕ, c. Now le ((ψ 0, ψ, κ be any compeing admissible sraegy. Since he uiliy process is concave, we have ( ( ( d E u (κ E u (c + αe Q ( κ c, dp =0 =0 by assumpion and definiion of he subdifferenial. Hence (3.5 implies ha ((ϕ 0, ϕ, c is opimal in he fricionless marke wih price process S. Now le ((ψ 0, ψ, κ be any admissible porfolio consumpion pair in he marke wih bid/ask prices S, S. For = 1,..., T + 1, define ψ := ( ψ +, ψ := ( ψ and le =0 κ( := κ( + ( ψ (S S + ( ψ ( S S. Then κ κ since S S S and ((ψ 0, ψ, κ is a self-financing porfolio/consumpion pair in he fricionless marke wih price process S, i.e., wih bid/ask-prices S, S. Since ((ϕ 0, ϕ, c is opimal in his marke, we have ( ( ( E u (κ E u ( κ E u (c. =0 =0 =0 =0 Therefore ((ϕ 0, ϕ, c is opimal in he marke wih bid/ask prices S, S as well. Remark 3.5 If, for fixed (ω, Ω R +, he mapping x u (ω, x is differeniable on is effecive domain wih derivaive u, hen E( d Q F dp 1 u α (c reduces o ( d E Q dp F = 1 α u (c. Acknowledgemens We hank wo anonymous referees for careful reading of he manuscrip and numerous consrucive commens. References [1] CAMPI, L. and OWEN, M. (2010. Mulivariae uiliy maximizaion wih proporional ransacion coss. Finance Soch., o appear. 10

11 [2] CVITANIĆ, J. and KARATZAS, I. (1996. Hedging and porfolio opimizaion under ransacion coss: a maringale approach. Mah. Finance [3] CVITANIĆ, J., PHAM, H., and TOUZI, N. (1999. A closed-form soluion o he problem of superreplicaion under ransacion coss. Finance Soch [4] CVITANIĆ, J. and WANG, H. (2001. On opimal erminal wealh under ransacion coss. J. Mah. Econom., [5] DAVIS, M. and NORMAN, A. (1990. Porfolio selecion wih ransacion coss. Mah. Oper. Res [6] DEELSTRA, G., PHAM, H., and TOUZI, N. (2001. Dual formulaion of he uiliy maximizaion problem under ransacion coss. Ann. Appl. Probab [7] GUASONI, P., RÁSONYI, M., and SCHACHERMAYER, W. (2008. Consisen price sysems and face-lifing pricing under ransacion coss. Ann. Appl. Probab [8] JOUINI, E. and KALLAL, H. (1995. Maringales and arbirage in securiies markes wih ransacion coss. J. Econom. Theory [9] KABANOV, Y. (1999. Hedging and liquidaion under ransacion coss in currency markes. Finance Soch [10] KABANOV, Y. and SAFARIAN, M. (2009. Markes wih Transacion Coss. Mahemaical Theory. Springer, Berlin. [11] KALLSEN, J. (2002. Uiliy-based derivaive pricing in incomplee markes. In Mahemaical Finance Bachelier Congress 2000 (H. Geman, D. Madan, S. Pliska, and T. Vors, eds, p , Springer, Berlin. [12] KALLSEN, J. and MUHLE-KARBE, J. (2010. On using shadow prices in porfolio opimizaion wih ransacion coss. Ann. Appl. Probab [13] LAMBERTON, D., PHAM, H., and SCHWEIZER, M. (1998. Local risk-minimizaion under ransacion coss. Mah. Oper. Res [14] LOEWENSTEIN, M. (2002. On opimal porfolio rading sraegies for an invesor facing ransacion coss in a coninuous rading marke. J. Mah. Econom., [15] ROCKAFELLAR, R. T. (1997. Convex analysis. Princeon Universiy Press, Princeon. [16] ROCKAFELLAR, T. and WETS, R. (1998. Variaional Analysis. Springer, Berlin. [17] SCHACHERMAYER, W. (2004. The fundamenal heorem of asse pricing under proporional ransacion coss in finie discree ime. Mah. Finance