On the Existence of Shadow Prices
Giuseppe Benedetti, Luciano Campi, Jan Kallsen, Johannes Muhle-Karbe
aa r X i v : . [ q -f i n . P M ] J a n ON THE EXISTENCE OF SHADOW PRICES
GIUSEPPE BENEDETTI, LUCIANO CAMPI, JAN KALLSEN, AND JOHANNES MUHLE-KARBE
Abstract.
For utility maximization problems under proportional transaction costs, it hasbeen observed that the original market with transaction costs can sometimes be replacedby a frictionless shadow market that yields the same optimal strategy and utility. However,the question of whether or not this indeed holds in generality has remained elusive so far.In this paper we present a counterexample which shows that shadow prices may fail toexist. On the other hand, we prove that short selling constraints are a sufficient conditionto warrant their existence, even in very general multi-currency market models with possiblydiscontinuous bid-ask-spreads. Introduction
Transaction costs have a severe impact on portfolio choice: If securities have to be boughtfor an ask price which is higher than the bid price one receives for selling them, theninvestors are forced to trade off the gains and costs of rebalancing. Consequently, utilitymaximization under transaction costs has been intensely studied in the literature. We referthe reader to Campi and Owen [3] for general existence and duality results, as well as asurvey of the related literature.It has been observed that the original market with transaction costs can sometimes bereplaced by a fictitious frictionless “shadow market”, that yields the same optimal strategyand utility. If such a shadow price exists, then transaction costs do not lead to qualitativelynew effects for portfolio choice, as their impact can be replicated by passing to a suitablymodified frictionless market. Starting from [20], shadow prices have recently also provedto be useful for solving optimization problems in concrete models, see [25, 12, 16, 11].However, unlike in the contexts of local risk minimization [26], no-arbitrage [15, 17, 30],and superhedging [6] (also cf. [18] for an overview) — the question of whether or not sucha least favorable frictionless market extension indeed exists has only been resolved underrather restrictive assumptions so far.More specifically, Cvitani´c and Karatzas [5] answer it in the affirmative in an Itˆo processsetting, however, only under the assumption that the minimizer in a suitable dual problem
Date : June 11, 2012.2010
Mathematics Subject Classification.
JEL Classification : G11.
Key words and phrases.
Transaction costs, shadow prices, short selling constraints.The authors are grateful to Bruno Bouchard, Christoph Czichowsky, Paolo Guasoni, Ioannis Karatzas,Marcel Nutz, Mark P. Owen, and Walter Schachermayer for fruitful discussions, and also acknowledge theconstructive comments of two anonymous referees. The second author thanks the “Chaire Les ParticuliersFace aux Risques”, Fondation du Risque (Groupama-ENSAE-Dauphine), the GIP-ANR “Risk” projectfor their support. The fourth author was partially supported by the National Centre of Competence inResearch Financial Valuation and Risk Management (NCCR FINRISK), Project D1 (Mathematical Methodsin Financial Risk Management), of the Swiss National Science Foundation (SNF). For example, Liu and Loewenstein [27] reckon that “even small transaction costs lead to dramatic changesin the optimal behavior for an investor: from continuous trading to virtually buy-and-hold strategies.” exists and is a martingale. Yet, subsequent work by Cvitani´c and Wang [7] only guaranteesexistence of the minimizer in a class of supermartingales. Hence, this result is hard to applyunless one can solve the dual problem explicitly.A different approach leading closer to an existence result is provided by Loewenstein [28].Here, the existence of shadow prices is established for continuous bid-ask prices in the pres-ence of short selling constraints. In contrast to Cvitani´c and Karatzas [5], Loewenstein [28]constructs his shadow market directly from the primal rather than the dual optimizer. How-ever, his analysis is also based on the assumption that the starting point for his analysis, inthis case his constrained primal optimizer, actually does exist.Finally, Kallsen and Muhle-Karbe [21] show that shadow prices always exist for utilitymaximization problems in finite probability spaces. But, as usual in Mathematical Finance,it is a delicate question to what degree this transfers to more general settings.The present study contributes to this line of research in two ways. On the one hand,we present a counterexample showing that shadow prices do not exist in general withoutfurther assumptions. On the other hand, we establish that Loewenstein’s approach canbe used to come up with a positive result, even in Kabanov’s [19] general multi-currencymarket models with possibly discontinuous bid-ask-spreads:
Theorem 1.1.
In the general multicurrency setting of Section 3, a shadow price alwaysexists, if none of the assets can be sold short. The crucial assumption – which is violated in our counterexample – is the prohibitionof short sales for all assets under consideration. Like Loewenstein [28], we construct ourshadow price from the primal optimizer. Existence of the latter is established by extendingthe argument of Guasoni [14] to the general setting considered here. This is done by makinguse of a compactness result for predictable finite variation processes established in Campiand Schachermayer [4].The paper is organized as follows. In Section 2, we present our counterexample, whichis formulated in simple setting with one safe and one risky asset for better readability.Afterwards, the general multi-currency framework with transaction costs and short sellingconstraints is introduced. In this setting, we then show that shadow prices always exist.Finally, Section 5 concludes. 2.
A counterexample
In this section, we show that even a simple discrete-time market can fail to exhibit ashadow price, if the unconstrained optimal strategy involves short selling. Consider amarket with one safe and one risky asset traded at the discrete times t = 0 , ,
2: The bidand ask prices of the safe asset are equal to 1 and the bid-ask spread [ S, S ] of the riskyasset is defined as follows. The bid prices are deterministically given by S = 3 , S = 2 , S = 1 , while the ask prices satisfy S = 3 and P ( S = 2) = 1 − − n , P ( S = 2 + k ) = 2 − n − k , k = 1 , , , . . . , That is, a consistent price system in the terminology of Schachermayer [30]. Cf. Definition 3.9 below for the formal definition In the general multicurrency notation introduced below, this corresponds to [1 /π , π ], where π ij denotes the number of units of asset i for which the agent can buy one unit of asset j . N THE EXISTENCE OF SHADOW PRICES 3 where n ∈ N is chosen big enough for the subsequent argument to work. Moreover, we set P ( S = 3 + k | S = 2 + k ) = 2 − n − k , P ( S = 1 | S = 2 + k ) = 1 − − n − k for k = 0 , , , . . . The corresponding bid-ask process is illustrated in Figure 1. One readilyverifies that a strictly consistent price system exists in this market. Now, consider themaximization of expected logarithmic utility from terminal wealth, where the maximizationtakes place over all self-financing portfolios that liquidate at t = 2 after starting from aninitial position of − k ... 13 + 03 + 1...3+ k ... − − n − n − − n − k − − n − n − − n − − n − − − n − k − n − k Figure 1.
Illustration of the ask price in the counterexample. The corre-sponding bid price decreases deterministically from 3 to 2 to 1.It is not hard to determine the optimal trading strategy in this setup. Buying a pos-itive amount of stock at time 0 is suboptimal because expected gains would be negative.Consequently, zero holdings are preferable to positive ones. A negative position in the firstperiod, on the other hand, is impossible as well because it may lead to negative terminalwealth. Hence it is optimal to do nothing at time 0, i.e., the optimal numbers b V = ( b V , b V )of safe and risky assets satisfies b V = 0. In the second period, a positive stock holdingwould be again suboptimal because prices are still falling on average. By contrast, buildingup a negative position is worthwhile. The stock can be sold short at time 1 for S = 2 andit can be bought back at time 2 for S = 1 with overwhelming probability and for S = 3resp. 3 + k with very small probability. Consequently, the optimal strategy satisfies b V < G. BENEDETTI, L. CAMPI, J. KALLSEN, AND J. MUHLE-KARBE
If a shadow price process (1 , e S ) for this market exists, then ˜ S must coincide with thebid resp. ask price if a transaction takes place in the optimal strategy b V . Otherwise, onecould achieve strictly higher utility trading ˜ S , by performing the same purchases and salesat sometimes strictly more favorable prices. Consequently, we must have e S = S = 3, e S = S = 2, e S = S . However, (1 , e S ) cannot be a shadow price process. Indeed, e S isdecreasing deterministically by 1 in the first period and would allow for unbounded expectedutility and in fact even for arbitrage. Remark 2.1.
It is important to note that the market discussed in this section does admita shadow price if one imposes short selling constraints. In this case, it is evidently optimalnot to trade at all in the original market with transaction costs: Positive positions are notworthwhile because prices are always falling on average, whereas negative positions are ruledout by the constraints. In this market any supermartingale (1 , e S ) with values in the bid-askspread, i.e., e S = 3, 2 ≤ e S ≤ S and e S = 1, is a shadow price (showing that even if ashadow price exists, it need not be unique). Indeed, Jensen’s inequality yields that positivepositions are suboptimal, and negative holdings are again prohibited by the constraints.Hence it is optimal not to trade at all, as in the original market with transaction costs.This confirms that (1 , e S ) is indeed a shadow price if short selling is ruled out. However,it is clearly not a shadow price in the unconstrained market, as it would allow for obviousarbitrage. 3. The General Multi-Currency Model
Henceforth, we work in the general transaction cost framework of Campi and Schacher-mayer [4], with slight modifications to incorporate short selling constraints. We describehere the main features of the model, but refer to the original paper for further details. Forany vectors x, y in R d , we write x (cid:23) y if x − y ∈ R d + and xy for the Euclidean scalar product.Let (Ω , ( F t ) t ∈ [0 ,T ] , P ) be a filtered probability space satisfying the usual conditions andsupporting all processes appearing in this paper; the initial σ -field is assumed to be trivial.We consider an agent who can trade in d assets according to some bid-ask matrix Π =( π ij ) ≤ i,j ≤ d , where π ij denotes the number of units of asset i for which the agent can buyone unit of asset j . To recapture the notion of currency exchanges, one naturally imposesas in [30] that:(i) π ij ∈ (0 , ∞ ) for every 1 ≤ i, j ≤ d ;(ii) π ii = 1 for every 1 ≤ i ≤ d ;(iii) π ij ≤ π ik π kj for every 1 ≤ i, j, k ≤ d .The first condition means that the bid-ask prices of each asset in terms of the others arepositive and finite, while the interpretation of the second is evident. The third implies thatdirect exchanges are not dominated by several successive trades. In the spirit of [19], theentries of the bid-ask matrix can also be interpreted in terms of the prices S , . . . , S d of theassets and proportional transaction costs λ ij for exchanging asset i into asset j , by setting π ij = (1 + λ ij ) S j S i . We will use both notations in the sequel, one being shorter and the other providing a betterfinancial intuition. Given a bid-ask matrix Π, the solvency cone K (Π) is defined as theconvex polyhedral cone in R d spanned by the canonical basis vectors e i , 1 ≤ i ≤ d , of R d , N THE EXISTENCE OF SHADOW PRICES 5 and the vectors π ij e i − e j , 1 ≤ i, j ≤ d . The convex cone − K (Π) should be interpreted asthose portfolios available at price zero in a market without short selling constraints. Givena cone K , its (positive) polar cone is defined by K ∗ = n w ∈ R d : vw ≥ , ∀ v ∈ K o . We now introduce randomness and time in the model. An adapted, c`adl`ag process(Π t ) t ∈ [0 ,T ] taking values in the set of bid-ask matrices will be called a bid-ask process . Oncea bid-ask process (Π t ) t ∈ [0 ,T ] has been fixed, one can drop it from the notation by writing K τ instead of K (Π τ ) for any stopping time τ , coherently with the framework introducedabove. In accordance with the framework developed in [4] we make the following technicalassumption throughout the paper. It means basically that no price changes take place attime T , which serves only as a date for liquidating the portfolio. This assumption can berelaxed via a slight modification of the model (see [4, Remark 4.2]). For this reason, weshall not explicitly mention it in the following. Assumption 3.1. F T − = F T and Π T − = Π T a.s. In markets with transaction costs, consistent price systems play a role similar to martin-gale measures for the frictionless case (compare, e.g., [30, 15, 18]). For utility maximization,this notion has to be extended, just as it is necessary to pass from martingale measures tosupermartingale densities in the frictionless case [24]:
Definition 3.2.
An adapted, R d + \ { } -valued, c`adl`ag supermartingale Z = ( Z t ) t ∈ [0 ,T ] iscalled a supermartingale consistent price system (supermartingale-CPS) if Z t ∈ K ∗ t a.s.for every t ∈ [0 , T ] . Moreover, Z is called a supermartingale strictly consistent pricesystem (supermartingale-SCPS) if it satisfies the following additional condition: for every [0 , T ] ∪ {∞} -valued stopping time τ , we have Z τ ∈ int( K ∗ τ ) a.s. on { τ < ∞} , and for everypredictable [0 , T ] ∪ {∞} -valued stopping time σ , we have Z σ − ∈ int( K ∗ σ − ) a.s. on { σ < ∞} .The set of all supermartingale-(S)CPS is denoted by Z sup (resp. Z s sup ). As in [4], trading strategies are described by the numbers of physical units of each assetheld at time t : Definition 3.3. An R d -valued process V = ( V t ) t ∈ [0 ,T ] is called a self-financing portfolioprocess for the process K of solvency cones if it satisfies the following properties: (i) It is predictable and a.e. (not necessarily right-continuous) path has finite variation. (ii)
For every pair of stopping times ≤ σ ≤ τ ≤ T , we have V τ − V σ ∈ − K σ,τ , where K s,t ( ω ) denotes the closure of cone { K r ( ω ) , s ≤ r < t } .A self-financing portfolio process V is called admissible if it satisfies the no short sellingconstraint V (cid:23) . We need some more notation related to such processes. For any predictable process offinite variation V , we can define its continuous part V c and its left (resp. right) jump process∆ V t := V t − V t − (resp. ∆ + V t := V t + − V t ), so that V = V c + ∆ V + ∆ + V . The continuouspart, V c , is itself of finite variation, so we can define its Radon-Nykodim derivative ˙ V ct withrespect to its total variation process Var t ( V c ), for all t ∈ [0 , T ] (see [4, Section 2] for details). K (Π) contains precisely the solvent portfolios that can be liquidated to zero by trading according to thebid-ask matrix Π and possibly throwing away positive asset holdings. G. BENEDETTI, L. CAMPI, J. KALLSEN, AND J. MUHLE-KARBE
We will work under the following no-arbitrage assumption, which is the analogue of theexistence of a supermartingale density in frictionless markets.
Assumption 3.4. Z s sup = ∅ . We now turn to the utility maximization problem. Here, we restrict our attention withoutloss of generality to admissible and self-financing portfolio processes that start out with someinitial endowment x ∈ R d + \ { } , and such that V T is nonzero only in the first component(that is, the agent liquidates his wealth to the first asset at the final date). The set of thoseprocesses is denoted by A x,ss , and the set A x,ssT := (cid:8) V T : V ∈ A x,ss (cid:9) consists of all terminal payoffs (in the first asset) attainable at time T from initial endow-ment x . Moreover, the set A x,ssT − := { V T − : V ∈ A x,ss } contains the pre-liquidation values of admissible strategies.The utility maximization problem considered in this paper is the following:(3.1) J ( x ) := sup f ∈A x,ssT E [ U ( f )] . Here, U : (0 , ∞ ) → R is a utility function in the usual sense, i.e., a strictly concave,increasing, differentiable function satisfying(i) the Inada conditions lim x ↓ U ′ ( x ) = ∞ and lim x ↑∞ U ′ ( x ) = 0, and(ii) the condition of reasonable asymptotic elasticity (RAE): lim sup x →∞ xU ′ ( x ) U ( x ) < U ∗ ( y ) = sup x> [ U ( x ) − xy ], y > U , and I := ( U ′ ) − for the inverse function of its derivative. To rule out degeneracies, we assume throughoutthat the maximal expected utility is finite: Assumption 3.5. J ( x ) = sup f ∈A x,ssT E [ U ( f )] < ∞ . A unique maximizer for the utility maximization problem (3.1) indeed exists: Proposition 3.6.
Fix an initial endowment x ∈ R d + \ { } . Under Assumptions 3.4 and3.5, the utility maximization problem (3.1) admits a unique solution b f ∈ A x,ssT .Proof. Step 1 : The compactness result for predictable finite variation processes establishedin [4, Proposition 3.4] also holds in our setting where, in particular, the existence of a strictlyconsistent price system is replaced by the weaker Assumption 3.4. To see this, it sufficesto show that the estimate in [4, Lemma 3.2] is satisfied under Assumption 3.4; then, theproof of [4, Proposition 3.4] can be carried through unchanged. To this end, we can use thearguments in [4, Section 3] with the following minor changes:(i) First, notice that in the proof of [4, Lemma 3.2] the martingale property of Z isonly used to infer that this strictly positive process satisfies inf t ∈ [0 ,T ] k Z t k d > In the absence of constraints, similar existence results have been established for increasingly generalmodels of the bid-ask spread by [5, 8, 2, 14, 3].
N THE EXISTENCE OF SHADOW PRICES 7 (ii) [4, Lemma 3.2] is formulated for strategies starting from a zero initial position,but can evidently be generalized to strategies starting from any initial value x .Moreover, in our case the admissible strategies are all bounded from below by zerodue to the short selling constraints. Consequently, [4, Lemma 3.2] holds under theweaker Assumption 3.4 for V ∈ A x,ss and, in our case, [4, Equation (3.5)] reads as E Q [Var T ( V )] ≤ C k x k for a constant C ≥ Q . Step 2 : Pick a maximizing sequence ( V n ) n ≥ ∈ A x,ss for (3.1) such that E [ U ( V nT )] → J ( x ) as n → ∞ . By Step 1, we can assume (up to a sequence of convex combinations) that V nT → V T a.s. for some V ∈ A x,ss . The rest of the proof now proceeds as in [14, Theorem 5.2]: bymeans of RAE assumption, we can prove that lim n →∞ E [ U ( V nT )] ≤ E [ U ( V T )], implying that V is the optimal solution to (3.1). Uniqueness follows from the strict concavity of U . (cid:3) Consider now a supermartingale-CPS Z ∈ Z sup . By definition, Z lies in the polar K ∗ of the solvency cone (we omit dependence on time for clarity); since Z = 0 this implies inparticular that all components of Z are strictly positive. Moreover, taking any asset, saythe first one, as a numeraire, it means that(3.2) 11 + λ i S i S ≤ Z i Z ≤ (1 + λ i ) S i S , for any i = 1 , . . . , d . In other words, the frictionless price process S Z := Z/Z evolves within the corresponding bid-ask spread. This implies that the terms of trade inthis frictionless economy are at least as favorable for the investor as in the original marketwith transaction costs. For S Z , we use the standard notion of a self-financing strategy: Definition 3.7.
Let S Z := Z/Z for some Z ∈ Z sup . Then, a predictable, R d -valued, S Z -integrable process V = ( V t ) t ∈ [0 ,T ] is called a self-financing portfolio process in the frictionlessmarket with price process S Z , if it satisfies (3.3) V t S Zt = xS Z + Z t V u dS Zu , t ∈ [0 , T ] . It is called admissible if it additionally satisfies the no short selling constraint V (cid:23) . Wesometimes write Z -admissible to stress the dependence on a specific Z ∈ Z sup .The set A x,ssT ( Z ) consists of all payoffs V T S ZT that are attained by a Z -admissible strategy V starting from initial endowment x ∈ R d + \ { } . This notion is indeed compatible with Definition 3.3, in the sense that any payoff in theoriginal market with transaction costs can be dominated in the potentially more favorablefrictionless markets evolving within the bid-ask spread:
Lemma 3.8.
Fix Z ∈ Z sup . For any admissible strategy V in the sense of Definition 3.3there is a strategy ˜ V (cid:23) V which is Z -admissible in the sense of Definition 3.7.Proof. Let V be self-financing in the original market with transaction costs in the sense ofDefinition 3.3. Then, since V is of finite variation, applying the integration by parts formula In particular, we do not restrict ourselves to finite variation strategies here.
G. BENEDETTI, L. CAMPI, J. KALLSEN, AND J. MUHLE-KARBE as in the proof of [4, Lemma 2.8] yields V t S Zt = xS Z + Z t V u dS Zu + Z t S Zu ˙ V cu d Var u ( V c ) + X u ≤ t S Zu − ∆ V u + X u Fix an initial endowment x ∈ R d + \ { } . The process S Z = Z/Z corre-sponding to some Z ∈ Z sup is called a shadow price process , if sup f ∈A x,ssT E [ U ( f )] = sup f ∈A x,ssT ( Z ) E [ U ( f )] . Some remarks are in order here.(i) Even if a shadow price exists it need not be unique, cf. Remark 2.1.(ii) By Lemma 3.8, any payoff that can be attained in the original market with trans-action costs can be dominated in frictionless markets with prices evolving withinthe bid-ask spread. Hence, strict concavity implies that the optimal payoff b f mustbe the same for a shadow price as in the transaction cost market. In order not toyield a strictly higher utility in the shadow market, the optimal strategy b V thatattains b f with transaction costs must therefore also do so in the shadow market.Put differently, a shadow price must match the bid resp. ask prices whenever theoptimal strategy b V entails purchases resp. sales. N THE EXISTENCE OF SHADOW PRICES 9 Existence of Shadow prices under short selling constraints In this section, we prove that a shadow price always exists if short selling is prohibited(cf. Corollary 4.7). To this end, we first derive some sufficient conditions, and then verifythat these indeed hold. Throughout, we assume that Assumptions 3.4 and 3.5 are satisfied.The following result crucially hinges on the presence of short selling constraints. Lemma 4.1. For any supermartingale-CPS Z ∈ Z sup the following holds: (i) The process ZV is a supermartingale for any portfolio process V admissible in thesense of Definition 3.3. (ii) The process ZV = Z V S Z is a supermartingale for any portfolio process V whichis Z -admissible in the sense of Definition 3.7.Proof. (i) Integration by parts gives(4.1) Z t V t − Z x = Z t V u dZ u + Z t Z u ˙ V cu d Var u ( V c ) + X u ≤ t Z u − ∆ V u + X u Proposition 4.2. Let x ∈ R d + \ { } . Suppose there are a supermartingale-CPS Z and ana.s. strictly positive F T -measurable random variable b f ∈ L satisfying: (i) b f ∈ A x,ssT ; (ii) Z T = U ′ ( b f ) ; (iii) S Z,iT = Z iT /Z T = 1 /π i T for i = 1 , . . . , d ; (iv) E [ Z T b f ] = Z x .Then, b f is the optimal payoff both for the frictionless price process S Z and in the originalmarket with transaction costs. Consequently, S Z is a shadow price.Proof. We first prove that b f is the optimal solution to the utility maximization problem (3.1)under transaction costs and short selling constraints. By (i), the payoff b f is attained by some b V ∈ A x,ss after liquidation of b V T − at time T . Now, take any X ∈ A x,ssT − , whose liquidationvalue to the first asset is f = X/π · T − = X/π · T = XZ T /Z T by (iii) and Assumption 3.1.Here 1 /π · is the vector whose components are given by 1 /π i for i = 1 , . . . , d . Then, inview of (iv), we have E [ Z T b f ] = Z x ≥ E [ Z T X ] = E [ Z T f ] . Here, the inequality follows from the supermartingale property of Z and ZV for an admis-sible V leading to X at time T − , since Fatou’s lemma and the positivity of Z, V imply E [ Z T X ] = E [ Z T V T − ] ≤ lim inf t ↑ T E [ Z T V t ] ≤ lim inf t ↑ T E [ Z t V t ] ≤ Z x. Concavity of U together with Property (ii) in turn gives E [ U ( b f ) − U ( f )] ≥ E [ Z T ( b f − f )] ≥ , implying that b f is the optimal solution to the transaction cost problem (3.1).Next, we prove that b f is also optimal in the frictionless market with price process S Z . Tothis end, take a strategy V which is Z -admissible in the sense of Definition 3.7 with V = x ,so that the portfolio value at time t is given by W t := V t S Zt . The process Z t W t = Z t V t isa supermartingale by Lemma 4.1(ii), hence we have E [ Z T W ] ≤ Z x for any W ∈ A x,ssT ( Z ).Now note that the optimization problem in this frictionless market,sup W ∈A x,ssT ( Z ) E [ U ( W )] , is dominated by the static problemsup { E [ U ( W )] : W ∈ L ( R + ) , E [ Z T W ] ≤ Z x } , which by monotonicity of U can be written assup { E [ U ( W )] : W ∈ L ( R + ) , E [ Z T W ] = Z x } . This problem admits a solution which is – recalling that I = ( U ′ ) − – given by c W := I ( Z T ) = b f = b V T − Z T /Z = b V T − S ZT . Indeed, the definition of the conjugate function gives(4.2) E [ U ( W )] ≤ E [ U ∗ ( Z T )] + Z x for all random variables W satisfying E [ Z T W ] = Z x . The random variable c W = I ( Z T )attains the supremum (pointwise) in the definition of U ∗ ( Z T ) and moreover, by assumption, E [ Z T c W ] = Z x . Hence, (4.2) becomes an equality and it follows that c W = b f is also anoptimal payoff in the frictionless market with price process S Z . The latter therefore is ashadow price as claimed. (cid:3) Using the sufficient conditions from Proposition 4.2, we now establish the existence ofa shadow price in our multi-currency market model with transaction costs. We proceedsimilarly as in [28], adapting the arguments to our more general setting, and using thatwe have shown existence of an optimal solution b f to the utility maximization problem inProposition 3.6 above. First, notice that the value function J ( x ) is concave, increasing andfinite for all x in the set R d + \ { } . By [29, Theorem 23.4] it is also superdifferentiable forall x ∈ R d + \ { } . We recall that the superdifferential ∂ϕ ( x ) of any concave function ϕ atsome point x is defined as the set of all y ∈ R d such that ϕ ( z ) ≤ ϕ ( x ) + y ( z − x ) , for all z ∈ R d . Proposition 4.3. Fix x ∈ R d + \ { } , the associated optimal solution b f , and take h =( h , . . . , h d ) in the superdifferential ∂J ( x ) . Then, the following properties hold: (i) h ≥ E [ U ′ ( b f )] ; (ii) h i ≥ E h U ′ ( b f ) /π iT i for i = 2 , . . . , d ; In fact, J can be seen as the restriction to R d + \ { } of some other concave function defined on K thatallows for negative initial endowment (but forces the agent to make an instantaneous trade at time 0 in thatcase). N THE EXISTENCE OF SHADOW PRICES 11 (iii) h ∈ K ∗ ; (iv) hx = E [ U ′ ( b f ) b f ] .In particular, the optimal payoff b f is a.s. strictly positive.Proof. For ǫ > J ( x + e ǫ ) ≥ E [ U ( b f + ǫ )] because one can just hold the extraendowment in asset 1. Hence, the definition of the superdifferential gives h ≥ J ( x + e ǫ ) − J ( x ) ǫ ≥ E " U ( b f + ǫ ) − U ( b f ) ǫ . Since U is concave, the monotone convergence theorem yields h ≥ E [ U ′ ( b f )]. In view of theInada condition lim x ↓ U ′ ( x ) = ∞ , this also shows that b f is a.s. strictly positive.For i = 2 , . . . , d , we have J ( x + e i ǫ ) ≥ E [ U ( b f + ǫ/π iT )] because one can hold the extraendowment in asset i and then liquidate it to asset 1 at time T . Hence, as before, we find h i ≥ E [ U ′ ( b f ) /π iT ].Now notice that, for any i, j = 1 , . . . , d , one can exchange π ij units of asset i for 1 unitof asset j at time zero. Hence, J ( x + e i ǫ ) ≥ J ( x + e j ǫ/π ij ), and the definition of thesuperdifferential yields0 ≤ J ( x + e i ǫ ) − J (cid:16) x + e j ǫ/π ij (cid:17) ≤ ǫ (cid:16) h i − h j /π ij (cid:17) . Together with h i ≥ i = 1 , . . . , d , we obtain h ∈ K ∗ .Finally, hx ( λ − ≥ J ( λx ) − J ( x ) ≥ E [ U ( λ b f ) − U ( b f )]because A λx,ssT = λ A x,ssT . Hence if λ > 1, then hx ≥ E " U ( λ b f ) − U ( b f ) λ − , and the argument of the expectation increases as λ ↓ U . Analogously, for λ < 1, the inequality is reversed and the argument of the expectation decreases as λ ↑ hx = E [ U ′ ( b f ) b f ] as claimed. (cid:3) For any admissible portfolio process V , now define the conditional value process(4.3) J ( V, t ) := ess sup f ∈A V,sst,T E [ U ( f ) | F t ] , where A V,sst,T denotes the terminal values of admissible portfolio processes which agree with V on [0 , t ]. Let b V be the portfolio process in A x,ss leading to the optimal solution b f to(3.1). We can apply [10, Th´eor`eme 1.17] to get the following martingale property for theoptimal value process J ( b V , t ) over the whole time interval [0 , T ]: Lemma 4.4 (Dynamic Programming Principle) . The following equality holds a.s.: J ( b V , s ) = E [ J ( b V , t ) | F s ] , ≤ s ≤ t ≤ T. For i = 1 , . . . , d , now define a process ˜ Z as follows:˜ Z it := lim ǫ ↓ J ( b V + e i ǫ, t ) − J ( b V , t ) ǫ , t ∈ [0 , T ) , ˜ Z iT := U ′ ( b V T ) π i T . Proposition 4.5. ˜ Z is a (not necessarily c`adl`ag) supermartingale satisfying ˜ Z t ∈ K ∗ t a.s.for all t ∈ [0 , T ] .Proof. We adapt the argument of [28, Lemma 4]. Consider ǫ , ǫ > ǫ < ǫ . Usingthe concavity of the utility function U and the definition of the essential supremum yields J ( b V + e i ǫ , t ) = J (cid:18) ǫ ǫ ( b V + e i ǫ ) + (cid:18) − ǫ ǫ (cid:19) b V , t (cid:19) ≥ ǫ ǫ J (cid:16) b V + e i ǫ , t (cid:17) + (cid:18) − ǫ ǫ (cid:19) J (cid:16) b V , t (cid:17) . As a consequence, ˜ Z it is well-defined as the limit of an increasing sequence. For the remainderof the proof, we drop the superscript “ ss ” to ease notation. Since the family { E [ U ( f ) | F t ] : f ∈ A b V + ǫe i t,T } is directed upwards, [23, Theorem A.3] allows to write the essential supremumas a limit which is monotone increasing in n : J ( b V + e i ǫ, t ) = ess sup f ∈A b V + ǫeit,T E [ U ( f ) | F t ] = lim n →∞ E [ U ( f n ) | F t ] , where ( f n ) n ≥ is a sequence of elements of A b V + e i ǫt,T . As A b V + e i ǫt,T ⊆ A b V + e i ǫs,T for 0 ≤ s ≤ t < T , J ( b V + e i ǫ, s ) = ess sup f ∈A b V + eiǫs,T E [ U ( f ) | F s ] ≥ ess sup f ∈A b V + eiǫt,T E [ U ( f ) | F s ] ≥ E [ U ( f n ) | F s ] = E [ E [ U ( f n ) | F t ] | F s ]for all n ≥ 0. But then monotone convergence gives J ( b V + e i ǫ, s ) ≥ lim n →∞ E [ E [ U ( f n ) | F t ] | F s ] = E [ lim n →∞ E [ U ( f n ) | F t ] | F s ]= E [ J ( b V + e i ǫ, t ) | F s ] . Now, Lemma 4.4 implies J ( b V + e i ǫ, s ) − J ( b V , s ) ǫ ≥ E " J ( b V + e i ǫ, t ) − J ( b V , t ) ǫ (cid:12)(cid:12)(cid:12) F s and the supermartingale property of ˜ Z on [0 , T ) follows by monotone convergence for ǫ ↓ T as well notice that, for 0 ≤ t < T , J ( b V + e i ǫ, t ) − J ( b V , t ) ≥ E h U ( b V T + ǫ/π i T ) − U ( b V T ) |F t i because it is admissible to hold the ǫ extra units of asset i before liquidating them into ǫ/π i T − = ǫ/π i T units of asset 1, and J ( b V , t ) = E [ U ( b V T ) |F t ] by Lemma 4.4. Then, monotoneconvergence yields ˜ Z it ≥ E [ U ′ ( b V T ) /π i T |F t ] = E [ ˜ Z iT |F t ] , i = 1 , . . . , d, such that ˜ Z is indeed a supermartingale on [0 , T ]. In particular, it is finite-valued.It remains to show that ˜ Z t ∈ K ∗ t for all t ∈ [0 , T ]. To this end first fix t ∈ [0 , T ) and let( k nl ) l ≥ be a partition of [0 , ∞ ) with mesh size decreasing to zero as n increases. Note that,for all ǫ > 0, on the set { k nl < π ijt ≤ k nl +1 } we have J ( b V + e i ǫ, t ) − J ( b V , t ) ≥ J (cid:16) b V + e j ǫ/k nl +1 , t (cid:17) − J ( b V , t ) N THE EXISTENCE OF SHADOW PRICES 13 because it is admissible to exchange the ǫ extra units of asset i for at least ǫ/k nl +1 units ofasset j immediately after time t . Again using monotone convergence, this in turn implies˜ Z it k nl +1 { k nl <π ijt ≤ k nl +1 } ≥ ˜ Z jt { k nl <π ijt ≤ k nl +1 } , and thus ˜ Z it X l ≥ k nl +1 { k nl <π ijt ≤ k nl +1 } ≥ ˜ Z jt . Then, letting n → ∞ we obtain ˜ Z it π ijt ≥ ˜ Z jt for all i, j = 1 , . . . , d . Hence, ˜ Z t ∈ K ∗ t for t ∈ [0 , T ). For the terminal time T , this follows directly from the definition and property(iii) in the definition of a bid-ask matrix. (cid:3) The process ˜ Z constructed above is a supermartingale but not necessarily c`adl`ag. There-fore, we pass to the regularized c`adl`ag process b Z defined by b Z T = ˜ Z T and b Z it := lim s ↓ t,s ∈ Q ˜ Z is for all i = 1 , . . . , d and t ∈ [0 , T ). Note that the limit exists by [22, Proposition 1.3.14(i)].We can now establish our main result, the existence of shadow prices under short sellingconstraints subject only to the existence of a supermartingale strictly consistent price system(Assumption 3.4) and finiteness of the maximal expected utility (Assumption 3.5). Theorem 4.6. The process b Z belongs to Z sup . Moreover, it satisfies the sufficient conditionsof Proposition 4.2. Consequently, S b Z = b Z/ b Z is a shadow price process.Proof. By [22, Proposition 1.3.14(iii)], the process b Z is a c`adl`ag supermartingale. Moreover,since the bid-ask matrix is right continuous, we have b Z ∈ Z sup . By definition, we have˜ Z T = U ′ ( b f ) and ˜ Z iT / ˜ Z T = 1 /π i T for i = 1 , . . . , d . Since ˜ Z and b Z are equal in T , it thereforeremains to verify condition (iv) in Proposition 4.2. By Proposition 4.3,(4.4) hx = E [ U ′ ( b f ) b f ] = E [ b Z T b V T ] = E [ b Z T b V T ] , for the portfolio process b V attaining b f . The definition of the superdifferential then gives h i ≥ J ( x + e i ǫ ) − J ( x ) ǫ for any ǫ > 0. Hence, h i ≥ ˜ Z i ≥ ˜ Z i = b Z i for all i = 1 , . . . , d by [22, Proposition 1.3.14(ii)].Combined with (4.4) and because x has positive components, we obtain E [ b Z T b V T ] = hx ≥ b Z x Conversely, since b Z ∈ Z sup we can apply the supermartingale property established in Lemma4.1 which gives E [ b Z T b V T ] ≤ b Z x and hence E [ b Z T b V T ] = b Z x . Thus, the sufficient conditionsin Proposition 4.2 are satisfied and the proof is completed. (cid:3) As a result, we can now formulate a precise version of Theorem 1.1 from the introduction: Corollary 4.7. Under short selling constraints and subject to Assumptions 3.4 and 3.5, ashadow price in the sense of Definition 3.9 exists. Conclusion We have shown that shadow prices always exist in the presence of short selling constraints,even in general multi-currency markets with random, time-varying, and possibly discontin-uous bid-ask spreads. On the other hand, we have presented a counterexample showingthat existence generally does not hold beyond finite probability spaces if short selling ispermitted. Yet, in simple concrete models the presence of short selling does not precludethe existence of shadow prices, compare [11]. It is therefore an intriguing question for fu-ture research to identify additional assumptions on the market structure that warrant theirexistence. We also conjecture that shadow prices should always exist for utilities defined onthe whole real line, where there is no solvency constraint that can become binding as in ourcounterexample. Settling this issue, however, will require to resolve the ubiquitous issue ofadmissibility, potentially along the lines of [1], and is therefore left for future research. References [1] Biagini, S., and ˇCern´y, A. Admissible strategies in semimartingale portfolio selection. SIAM J.Control Optim. 49 , 1 (2011), 42–72.[2] Bouchard, B. Utility maximization on the real line under proportional transaction costs. FinanceStoch. 6 , 4 (2002), 495–516.[3] Campi, L., and Owen, M. P. Multivariate utility maximization with proportional transaction costs. Finance Stoch 15 , 3 (2011), 461–499.[4] Campi, L., and Schachermayer, W. A super-replication theorem in Kabanov’s model of transactioncosts. Finance Stoch. 10 , 4 (2006), 579–596.[5] Cvitani´c, J., and Karatzas, I. Hedging and portfolio optimization under transaction costs: a mar-tingale approach. Math. Finance 6 , 2 (1996), 133–165.[6] Cvitani´c, J., Pham, H., and Touzi, N. A closed-form solution to the problem of super-replicationunder transaction costs. Finance Stoch. 3 , 1 (1999), 35–54.[7] Cvitani´c, J., and Wang, H. On optimal terminal wealth under transaction costs. J. Math. Econom.35 , 2 (2001), 223–231.[8] Deelstra, G., Pham, H., and Touzi, N. Dual formulation of the utility maximization problem undertransaction costs. Ann. Appl. Probab. 11 , 4 (2001), 1353–1383.[9] Dellacherie, C., and Meyer, P.-A. Probabilities and potential. B . North-Holland Publishing Co.,Amsterdam, 1982.[10] El Karoui, N. Les aspects probabilistes du contrˆole stochastique . Springer, Berlin, 1979.[11] Gerhold, S., Guasoni, P., Muhle-Karbe, J., and Schachermayer, W. Transaction costs, tradingvolume, and the liquidity premium. Preprint, 2011.[12] Gerhold, S., Muhle-Karbe, J., and Schachermayer, W. The dual optimizer for the growth-optimal portfolio under transaction costs. Finance Stoch. , To appear (2011).[13] Goll, T., and Kallsen, J. Optimal portfolios for logarithmic utility. Stochastic Process. Appl. 89 , 1(2000), 31–48.[14] Guasoni, P. Optimal investment with transaction costs and without semimartingales. Ann. Appl.Probab. 12 , 4 (2002), 1227–1246.[15] Guasoni, P., R´asonyi, M., and Schachermayer, W. Consistent price systems and face-liftingpricing under transaction costs. Ann. Appl. Probab. 18 , 2 (2008), 491–520.[16] Herzegh, A., and Prokaj, V. Shadow price in the power utility case. Preprint, 2011.[17] Jouini, E., and Kallal, H. Martingales and arbitrage in securities markets with transaction costs. J.Econom. Theory 66 , 1 (1995), 178–197.[18] Kabanov, Y., and Safarian, M. Markets with transaction costs . Springer, Berlin, 2009.[19] Kabanov, Y. M. Hedging and liquidation under transaction costs in currency markets. Finance Stoch.3 , 2 (1999), 237–248.[20] Kallsen, J., and Muhle-Karbe, J. On using shadow prices in portfolio optimization with transactioncosts. Ann. Appl. Probab 20 , 4 (2010), 1341–1358. N THE EXISTENCE OF SHADOW PRICES 15 [21] Kallsen, J., and Muhle-Karbe, J. Existence of shadow prices in finite probability spaces. Math.Methods Oper. Res. 73 , 2 (2011), 251–262.[22] Karatzas, I., and Shreve, S. E. Brownian motion and stochastic calculus . Springer, New York, 1988.[23] Karatzas, I., and Shreve, S. E. Methods of mathematical finance . Springer, New York, 1998.[24] Kramkov, D., and Schachermayer, W. The asymptotic elasticity of utility functions and optimalinvestment in incomplete markets. Ann. Appl. Probab. 9 , 3 (1999), 904–950.[25] K¨uhn, C., and Stroh, M. Optimal portfolios of a small investor in a limit order market: a shadowprice approach. Math. Finan. Econ. 3 (2010), 45–72.[26] Lamberton, D., Pham, H., and Schweizer, M. Local risk-minimization under transaction costs. Math. Oper. Res. 23 , 3 (1998), 585–612.[27] Liu, H., and Loewenstein, M. Optimal portfolio selection with transaction costs and finite horizons. Rev. Financ. Stud. 15 , 3 (2002), 805–835.[28] Loewenstein, M. On optimal portfolio trading strategies for an investor facing transactions costs ina continuous trading market. J. Math. Econom. 33 , 2 (2000), 209–228.[29] Rockafellar, R. T. Convex analysis . Princeton University Press, Princeton, N.J., 1970.[30] Schachermayer, W. The fundamental theorem of asset pricing under proportional transaction costsin finite discrete time. Math. Finance 14 , 1 (2004), 19–48. CREST and Universit´e Paris-DauphinePlace du Mar´echal de Lattre de Tassigny, FR-75775 Paris, France E-mail address : [email protected] Universit´e Paris 13, Laboratoire Analyse, G´eom´etrie et Applications, and CREST99, Avenue Jean-Baptiste Cl´ement, FR-93430 Villetaneuse, France E-mail address : [email protected] Christian-Albrechts-Universit¨at zu Kiel, Mathematisches SeminarWestring 383, D-24118 Kiel, Germany E-mail address : [email protected] ETH Z¨urich, Departement f¨ur MathematikR¨amistrasse 101, CH-8092 Z¨urich, Switzerland E-mail address ::