Arbitrage concepts under trading restrictions in discrete-time financial markets
aa r X i v : . [ q -f i n . M F ] J un ARBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS INDISCRETE-TIME FINANCIAL MARKETS
CLAUDIO FONTANA AND WOLFGANG J. RUNGGALDIER
Abstract.
In a discrete-time setting, we study arbitrage concepts in the presence of convextrading constraints. We show that solvability of portfolio optimization problems is equivalentto absence of arbitrage of the first kind, a condition weaker than classical absence of arbitrageopportunities. We center our analysis on this characterization of market viability and deriveversions of the fundamental theorems of asset pricing based on portfolio optimization arguments.By considering specifically a discrete-time setup, we simplify existing results and proofs that relyon semimartingale theory, thus allowing for a clear understanding of the foundational economicconcepts involved. We exemplify these concepts, as well as some unexpected situations, in thecontext of one-period factor models with arbitrage opportunities under borrowing constraints. Introduction
The notions of arbitrage, market viability and state-price deflators are deeply connected andplay a foundational role in financial economics and, starting with the seminal work [HK79], inmathematical finance. In frictionless discrete-time financial markets, if no trading restrictionsare imposed, the appropriate no-arbitrage concept takes the classical form of absence of arbitrageopportunities ( no classical arbitrage ). By the fundamental theorem of asset pricing of [DMW90],this is equivalent to the existence of an equivalent martingale measure, whose density acts as astate-price deflator. Moreover, always in the absence of trading restrictions, the results of [RS06]imply that no classical arbitrage is equivalent to market viability, intended as the solvability ofportfolio optimization problems. No classical arbitrage thus represents the minimal economicallymeaningful no-arbitrage condition to enforce on a frictionless discrete-time financial market.In the presence of trading restrictions, these results continue to hold true as long as the setof constrained strategies is a cone, provided that equivalent martingale measures are replacedby equivalent supermartingale measures (see [FS16, Theorem 9.9] and Theorem 2.11 below).However, many practically relevant trading restrictions, such as borrowing constraints or thepossibility of limited short sales, correspond to convex non-conic constraints. In this case, as willbe shown below, market viability is no longer equivalent to no classical arbitrage, but rather tothe weaker condition of no arbitrage of the first kind (NA ). Under convex trading restrictions,NA represents therefore the minimal economically meaningful concept of no-arbitrage and isequivalent to the existence of a num´eraire portfolio or, more generally, a supermartingale deflator .The NA condition, introduced under this terminology in [Kar10], corresponds to the ab-sence of positive payoffs that can be super-replicated with an arbitrarily small initial capitaland is equivalent to the no unbounded profit with bounded risk condition studied in the seminal Department of Mathematics “Tullio Levi - Civita”, University of Padova, Italy.
E-mail addresses : [email protected]; [email protected] . Date : June 30, 2020.
Key words and phrases.
Trading constraints; market viability; arbitrage of the first kind; num´eraire portfolio. work [KK07] (see also [Fon15] for an analysis of no-arbitrage conditions equivalent to NA ).In continuous-time, a complete theory based on NA has been developed in a general semi-martingale setting starting with [KK07], also allowing for convex (non-conic) constraints. Theconnection between NA and market viability has been characterized in [CDM15] in an uncon-strained semimartingale setting (see also [CCFM17] for further results in this direction).Scarce attention has, however, been specifically paid to NA in discrete-time models, despitetheir widespread use in economic theory. This is also due to the fact that, for discrete-timemarkets with conic constraints, there is no distinction between NA and no classical arbitrage(see Remark 2.3 below). To the best of our knowledge, the only work that specifically addressesdiscrete-time models from the viewpoint of NA is [KS09], which derives the central results of[KK07] in a one-period setting. The present paper intends to fill this gap in the literature, inthe framework of general discrete-time models with convex (not necessarily conic) constraints.Centering our analysis on the equivalence between NA and market viability, we make a system-atic effort to provide direct and self-contained proofs based on portfolio optimization arguments.The simplicity of the discrete-time structure allows for a clear understanding of the economicconcepts involved, avoiding the technicalities of the continuous-time semimartingale setup.The paper is divided into three sections, whose contents and contributions can be outlined asfollows. In Section 2, we present a complete theory based on NA in a general one-period setting,extending the analysis of [KS09]. We prove the equivalence between NA and the solvability ofportfolio optimization problems (market viability), thus establishing the minimality of NA froman economic standpoint. In turn, this enables us to obtain a direct proof of the characterizationof NA in terms of the existence of the num´eraire portfolio or, more generally, a deflator. Weshow that NA leads to a dual representation of super-hedging values and a characterization ofattainable claims, and permits to rely on several well-known hedging approaches in constrainedincomplete markets, even in the presence of arbitrage opportunities. Besides its pedagogicalvalue, the one-period setting introduces several techniques that will be important for the analysisof the multi-period case.Section 3 illustrates the theory in the context of factor models with borrowing constraints.We introduce a general factor model, where a single factor is responsible of potential arbitrageopportunities. In this setting, the NA condition and the set of arbitrage opportunities admitexplicit descriptions in terms of the factor loadings. When NA holds but no classical arbitragedoes not, we show the existence of a maximal arbitrage strategy. These results can be easilyvisualized in a two-dimensional setting, which enables us to provide examples of situationswhere, despite the existence of arbitrage opportunities, it is not necessarily optimal to invest inthem. The analysis of this section clarifies the interplay between the support of the asset returnsdistribution, their dependence structure and the borrowing constraints.Finally, Section 4 generalizes the central results of Section 2 to a multi-period setting withrandom convex constraints. We derive several new characterizations of NA , showing that itholds globally if and only if it holds in each single trading period, and prove its equivalence tomarket viability. The most general result on the solvability of portfolio optimization problems indiscrete-time was obtained in [RS06], relying on no classical arbitrage. Our Theorem 4.4 extendsthis result by introducing trading restrictions and weakening the no-arbitrage requirement tothe minimal condition of NA (in turn, our proofs of Theorems 2.5 and 4.4 are inspired from RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS 3 [RS06]). By generalizing the one-period analysis, we then give an easy proof of the equivalencebetween NA , the existence of the num´eraire portfolio and the existence of a supermartingaledeflator, for general discrete-time models with random convex constraints.We close this introduction by briefly reviewing some related literature, limiting ourselves toselected contributions that are specifically connected with the present discussion. Relying onthe concept of no classical arbitrage, the fundamental theorem of asset pricing with constraintson the amounts invested in the risky assets is proved in [PT99], in the case of conic constraints(see also [KP00, Pha00] for valuation and hedging problems in that setting), and in [Bra97] inthe case of convex constraints. The specific case of short-sale constraints is treated in the earlierwork [JK95]. General forms of conic constraints have been considered in [Nap03], extending theanalysis of [PT99]. In the case of convex constraints on the proportions invested, as consideredin the present work, versions of the fundamental theorem of asset pricing based on the usualnotion of no classical arbitrage are given in [CPT01, EST04, Rok05]. In comparison to thelatter contributions, we choose to work with the weaker concept of NA , due to its equivalenceto market viability. In an unconstrained setting, the connection between no classical arbitrageand market viability is studied in [RS05, RS06]. A theory of valuation based on the num´eraireportfolio in discrete-time is developed in [BP03]. Finally, we mention the recent work [BCL19],where super-hedging has been studied under a weak no-arbitrage condition, called absence ofimmediate profits. However, the latter condition does not suffice to ensure market viability.2. The single-period setting
We consider a general financial market in a one-period economy, where d risky assets aretraded, together with a riskless asset with constant price equal to one. We assume that assetprices are discounted with respect to a baseline security and are represented by the vector S t = ( S t , . . . , S dt ) ⊤ ∈ R d + , for t = 0 ,
1, expressed in terms of returns as S it = S i (1 + R i ) , for all i = 1 , . . . , d, where R = ( R , . . . , R d ) ⊤ is a d -dimensional random vector on a given probability space (Ω , F , P )such that R i ≥ − i = 1 , . . . , d . We denote by S the support of the distribution of R ,namely the smallest closed set A ⊂ R d such that P ( R ∈ A ) = 1 (see [FS16, Proposition 1.45]).We also denote by L the smallest linear subspace of R d containing S and by L ⊥ its orthogonalcomplement in R d . The orthogonal projection of a vector x ∈ R d on L is denoted by p L ( x ).2.1. Trading restrictions.
Trading strategies are denoted by vectors π ∈ R d and representproportions of wealth invested in the d risky assets. We write V πt ( v ) for the wealth at time t generated by strategy π starting from initial capital v >
0, with V π ( v ) = v and V π ( v ) = v (1 + h π, R i ) , where h· , ·i denotes the scalar product in R d . Note that V πt ( v ) = vV πt (1), for all v > t = 0 ,
1. In the following, we shall use the notation V πt := V πt (1). A trading strategy π is saidto be admissible if V π ≥ adm the set of all admissible trading strategies, itholds that (see, e.g., [KS99, Lemma 4.3])Θ adm = { π ∈ R d : h π, z i ≥ − z ∈ S} . C. FONTANA AND W. J. RUNGGALDIER
In the terminology of [KK07, KS09], the set Θ adm corresponds to the natural constraints ensuringnon-negative wealth. Observe that, with the present parametrization, the notion of admissibilitydoes not depend on the initial capital.Besides the natural constraints, we assume that market participants face additional tradingrestrictions, represented by a convex closed set Θ c ⊆ R d . Realistic examples of trading restric-tions include the following situations (see also [CPT01, Section 4] for additional examples):(i) prohibition of short-selling : Θ c = R d + ;(ii) prohibition of short-selling and borrowing : Θ c = ∆ d , where ∆ d := { π ∈ R d + : h π, i ≤ } ;(iii) limits to borrowing : Θ c = { π ∈ R d : h π, i ≤ c } , for some c > limited positions in the risky assets : Θ c = Q di =1 [ − α i , β i ], for some α i , β i > i = 1 , . . . , d .Market participants are therefore restricted to trade according to strategies that belong to theset Θ := Θ adm ∩ Θ c . We refer to such strategies as allowed strategies .In general, the financial market may contain redundant assets , meaning that different combi-nations of assets may generate identical portfolio returns. This happens whenever L ⊥ is strictlybigger than { } . Indeed, ρ ∈ L ⊥ if and only if h π, R i = h π + ρ, R i a.s. for every π ∈ Θ. Inother words, investing according to a strategy ρ ∈ L ⊥ does not produce any loss or profit and,therefore, does not alter the outcome of any other allowed strategy π . For this reason, we shallassume that investors are always allowed to choose trading strategies in the set L ⊥ , meaningthat L ⊥ ⊂ Θ c . In turn, this implies that L ⊥ ⊂ Θ.To the convex closed set Θ, we associate its recession cone b Θ, defined as the set of all vectors y ∈ R d such that π + λy ∈ Θ for every λ ≥ π ∈ Θ (see [Roc70, Chapter 8]). The set b Θ has a clear financial interpretation: it represents the set of all allowed strategies that can bearbitrarily scaled and added to any other strategy π ∈ Θ without violating admissibility andtrading restrictions. The cone b Θ is closed and, by [Roc70, Corollary 8.3.2], it holds that b Θ = (cid:8) π ∈ R d : a − π ∈ Θ for all a > (cid:9) = \ a> a Θ . As a consequence of the fact that L ⊥ is a linear subspace of Θ, it holds that L ⊥ ⊆ b Θ. In turn,the latter property can be easily seen to imply that p L (Θ) ⊆ Θ, i.e., p L ( π ) ∈ Θ for all π ∈ Θ.2.2.
Arbitrage concepts.
We proceed to recall two important notions of arbitrage. First, wedefine the set I arb := (cid:8) π ∈ R d : h π, z i ≥ z ∈ S (cid:9) \ L ⊥ . Under trading restrictions, the set of arbitrage opportunities is given by I arb ∩ Θ and consistsof all allowed strategies π that generate a non-negative and non-null return (see also [KS09,Definition 3.5]). We say that no classical arbitrage holds if I arb ∩ Θ = ∅ .We now recall a second and stronger notion of arbitrage (see [Kar10, Definition 1]). To thiseffect, we define as follows the super-hedging value v ( ξ ) of a non-negative random variable ξ :(2.1) v ( ξ ) := inf (cid:8) v > ∃ π ∈ Θ such that v (1 + h π, R i ) ≥ ξ a.s. (cid:9) . In the next definition, we denote by L the family of non-negative random variables on (Ω , F ). Definition 2.1.
A random variable ξ ∈ L with P ( ξ > > arbitrage of the first kind if v ( ξ ) = 0. We say that no arbitrage of the first kind (NA ) holds if v ( ξ ) = 0 implies ξ = 0 a.s. RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS 5
An arbitrage of the first kind consists of a non-negative non-null payoff that can be super-replicated starting from an arbitrarily small initial capital. Observe that NA is weaker than noclassical arbitrage, as will be explicitly illustrated by the examples considered in Section 3. Thenext proposition provides three equivalent formulations of NA . Proposition 2.2.
The following are equivalent:(i) the NA condition holds;(ii) I arb ∩ b Θ = ∅ ;(iii) b Θ = L ⊥ ;(iv) the set Θ ∩ L is bounded (and, hence, compact).Proof. ( i ) ⇒ ( ii ): by way of contradiction, suppose that NA holds and there exists π ∈ I arb ∩ b Θ.Then ξ := h π, R i ∈ L and P ( ξ > >
0. For every v >
0, it holds that π/v ∈ Θ and v (1 + h π/v, R i ) > ξ a.s. This implies that v ( ξ ) = 0, yielding a contradiction to NA .( ii ) ⇒ ( iii ): we already know that L ⊥ ⊆ b Θ. Conversely, since b Θ ⊆ T a> a Θ adm , every element π ∈ b Θ satisfies h π, R i ≥ (ii) then implies h π, R i = 0 a.s., so that π ∈ L ⊥ .( iii ) ⇒ ( iv ): the set Θ ∩ L is non-empty, closed and convex. Hence, by [Roc70, Theorem 8.4],Θ ∩ L is bounded if and only if its recession cone \ Θ ∩ L consists of the zero vector alone. By[Roc70, Corollary 8.3.3], it holds that \ Θ ∩ L = b Θ ∩ L . Therefore, condition ( iii ) implies that \ Θ ∩ L = { } , thus establishing the boundedness of Θ ∩ L .( iv ) ⇒ ( i ): by way of contradiction, let ξ ∈ L with P ( ξ > > n ∈ N , there exists π n ∈ Θ such that n − (1 + h π n , R i ) ≥ ξ a.s. In this case, it holds that1 + h p L ( π n ) , R i ≥ nξ a.s., for all n ∈ N . Since P ( ξ > > L ( π n ) ∈ Θ ∩ L , for every n ∈ N , this contradicts the boundedness of the set Θ ∩ L . (cid:3) The three conditions given in Proposition 2.2 admit natural and direct interpretations, whichcan be stated as follows: (ii) there do not exist arbitrage opportunities that can be arbitrarily scaled; (iii) all allowed strategies that can be arbitrarily scaled reduce to trivial strategies; (iv) all allowed strategies not containing degeneracies are bounded.As shown in Sections 2.3 and 2.4 below, the compactness property (iv) is fundamental, sinceit allows solving optimal portfolio and hedging problems under NA , even when no classicalarbitrage fails to hold. The condition I arb ∩ b Θ = ∅ appears in [KK07, KS09] under the name no unbounded increasing profit (NUIP), where the unboundedness refers to the fact that thearbitrage profit generated by an element of I arb ∩ b Θ can be scaled to arbitrarily large values.
Remark 2.3.
Under conic trading restrictions, there are no arbitrage opportunities if and onlyif there are no arbitrages of the first kind. This simply follows from the observation that, if Θ c is a cone, then I arb ∩ b Θ = I arb ∩ Θ. This implies that the two arbitrage concepts differ only inthe presence of additional restrictions beyond conic (and, in particular, natural) constraints.
Remark 2.4 (On relative arbitrage) . The arbitrage concepts introduced so far have been im-plicitly defined with respect to the riskless asset with constant price equal to one. More generally,in the spirit of [FK09, Definition 6.1], a strategy π ∈ Θ is said to be an arbitrage opportunity rel-ative to θ ∈ Θ if P ( V π ≥ V θ ) = 1 and P ( V π > V θ ) > π − θ ∈ I arb ∩ (Θ − θ ).If θ ∈ b Θ c , then I arb ∩ (Θ − θ ) = ∅ implies no classical arbitrage (i.e., I arb ∩ Θ = ∅ ). Conversely, if C. FONTANA AND W. J. RUNGGALDIER − θ ∈ b Θ c , then I arb ∩ Θ = ∅ implies I arb ∩ (Θ − θ ) = ∅ . It follows that, for every θ ∈ b Θ c ∩ ( − b Θ c ),no classical arbitrage coincides with absence of arbitrage opportunities relative to θ . However,there is no general implication between the two conditions I arb ∩ Θ = ∅ and I arb ∩ (Θ − θ ) = ∅ .Observe that, unlike arbitrage opportunities, the notion of arbitrage of the first kind is universal,in the sense that it does not depend on a reference strategy θ (see Definition 2.1).2.3. Market viability and fundamental theorems.
The economic relevance of the NA condition is explained by its equivalence with the solvability of optimal portfolio problems, asshown in the next theorem. We denote by U the set of all random utility functions , consistingof all functions U : Ω × R + → R ∪ {−∞} such that U ( · , x ) is F -measurable and bounded frombelow, for every x >
0, and U ( ω, · ) is continuous, strictly increasing and concave, for a.e. ω ∈ Ω. Besides allowing for the possibility of random endowments or state-dependent preferences, theextension to random utility functions will be needed for our proof of Theorem 2.8 as well as forthe solution of certain hedging and valuation problems (see the last part of Section 2.4).
Theorem 2.5.
The following are equivalent:(i) the NA condition holds;(ii) for every U ∈ U such that sup π ∈ Θ E [ U + ( V π )] < + ∞ , there exists an allowed strategy π ∗ ∈ Θ ∩ L such that (2.2) E (cid:2) U ( V π ∗ ) (cid:3) = sup π ∈ Θ E (cid:2) U ( V π ) (cid:3) . Proof. ( i ) ⇒ ( ii ): note first that (2.2) can be equivalently stated by maximizing over Θ ∩ L ,since for every π ∈ Θ it holds that h π, R i = h p L ( π ) , R i a.s. By Proposition 2.2, NA impliesthat \ Θ ∩ L = b Θ ∩ L = { } . Hence, in view of [Roc70, Theorem 27.3], it suffices to show thatthe proper concave function u : Θ ∩ L → R defined by u : Θ ∩ L ∋ π u ( π ) := E [ U (1 + h π, R i )]is upper semi-continuous. To this effect, we adapt some of the arguments of [RS06, Lemma 2.3](see also [Nut16, Lemma 2.8]). Since the set Θ ∩ L is bounded under NA (see Proposition 2.2),there exists a bounded polyhedral set P ⊂ span(Θ ∩ L ) such that Θ ∩ L ⊆ P (see, e.g., [Roc70,Theorem 20.4]). Denote by { p , . . . , p N } the set of extreme points of P . Since a linear functiondefined on a polyhedral set attains its maximum on the set of extreme points, it holds that h π, R i ≤ max j =1 ,...,N h p j , R i , for all π ∈ Θ ∩ L . By monotonicity of U , this implies that U + (1 + h π, R i ) ≤ N X j =1 U + (1 + h p j , R i ) =: ζ, for all π ∈ Θ ∩ L . We proceed to show that E [ ζ ] < + ∞ . Since U (1) is bounded from below, we can assume withoutloss of generality that U (1) ≥
0. We recall from [RS06, Lemma 2.2] the inequality(2.3) U + ( λx ) ≤ λ (cid:0) U + ( x ) + U (2) (cid:1) , for all x > λ ≥ The condition θ ∈ b Θ c ∩ ( − b Θ c ) amounts to saying that arbitrary long and short positions in the portfolio θ arenot precluded by the trading restrictions represented by Θ c . This condition is conceptually equivalent to therequirement appearing in the definition of num´eraire adopted in [KST16] (see Definition 10 therein). For simplicity of notation, we shall omit to denote explicitly the dependence of U on ω in the following. RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS 7
Let φ be an element of the relative interior of Θ ∩L and ε j ∈ (0 ,
1] such that φ + ε j ( p j − φ ) ∈ Θ ∩L ,for all j = 1 , . . . , N . By inequality (2.3) and monotonicity of U , together with the fact that φ ∈ Θ ∩ L ⊆ Θ adm , we obtain(2.4) U + (cid:0) h p j , R i (cid:1) = U + (cid:0) h φ, R i + h p j − φ, R i (cid:1) ≤ ε j (cid:0) U + (cid:0) ε j (1 + h φ, R i ) + ε j h p j − φ, R i (cid:1) + U (2) (cid:1) ≤ ε j (cid:0) U + (cid:0) h φ + ε j ( p j − φ ) , R i (cid:1) + U (2) (cid:1) . Due to the assumption that sup π ∈ Θ E [ U + ( V π )] < + ∞ , the first term on the last line of (2.4) isintegrable, for each j = 1 , . . . , N . The same assumption implies that E [ U (1)] < + ∞ , from which E [ U (2)] < + ∞ follows by concavity of U . This proves that the random variable ζ is integrable.Let now ( π n ) n ∈ N be a sequence in Θ ∩ L converging to some element π ∈ Θ ∩ L . An applicationof Fatou’s lemma, together with the continuity of U , yields thatlim sup n → + ∞ u ( π n ) ≤ E (cid:2) lim sup n → + ∞ U (1 + h π n , R i ) (cid:3) = u ( π ) , thus proving the upper semi-continuity of the function u introduced above.( ii ) ⇒ ( i ): by way of contradiction, let π ∗ ∈ Θ ∩ L be the maximizer in (2.2) and suppose thatNA fails to hold. By Proposition 2.2, there exists θ ∈ I arb ∩ b Θ. It holds that π ∗ + θ ∈ Θ ∩ L and E [ U ( V π ∗ + θ )] > E [ U ( V π ∗ )], thus contradicting the optimality of π ∗ . (cid:3) The above theorem asserts the equivalence between NA and market viability , intended as theexistence of an optimal strategy for every well-posed expected utility maximization problem. Inparticular, the proof makes clear that one of the crucial consequences of NA is the compactnessof the set of non-redundant allowed strategies (see Proposition 2.2). Remark 2.6.
The proof of Theorem 2.5 relies on the fact that, under NA , the set Θ ∩ L and the function u have no common directions of recession. The relevance of this property inexpected utility maximization problems has been first recognized in the early work [Ber74].The NA condition admits an equivalent characterization in terms of the existence of a (su-permartingale) deflator or of a num´eraire portfolio , defined as follows. Definition 2.7.
A random variable Z ∈ L with P ( Z >
0) = 1 is said to be a deflator if(2.5) E [ Z V π ] ≤ , for all π ∈ Θ . The set of all deflators is denoted by D .An allowed trading strategy ρ ∈ Θ is said to be a num´eraire portfolio if 1 /V ρ ∈ D , meaning that(2.6) E [ V π /V ρ ] ≤ π ∈ Θ . It is well-known (see, e.g., [Bec01]) that a num´eraire portfolio is unique in the sense that if ρ and ρ satisfy (2.6), then ρ − ρ ∈ L ⊥ . The num´eraire portfolio is therefore uniquely definedon Θ ∩ L . The next theorem shows that NA is necessary and sufficient for the existence ofthe num´eraire portfolio. In a general semimartingale setting, the corresponding result has been C. FONTANA AND W. J. RUNGGALDIER proved in [KK07]. In the present context, Theorem 2.5 enables us to give a short and simpleproof based on log-utility maximization, thus highlighting the central role of market viability. Theorem 2.8.
The following are equivalent:(i) the NA condition holds;(ii) D 6 = ∅ ;(iii) there exists the num´eraire portfolio.Moreover, ρ ∈ Θ is the num´eraire portfolio if and only if it is relatively log-optimal , in the sensethat it satisfies E [log( V π /V ρ )] ≤ , for all π ∈ Θ .Proof. ( i ) ⇒ ( iii ): as a preliminary, similarly as in [Kar09, KS09], let ( f n ) n be a family offunctions such that f n : R d → (0 ,
1] and E [log(1+ k R k ) f n ( R )] < + ∞ , for each n ∈ N , and f n ր n → + ∞ . A specific choice is for instance given by f n ( x ) = {k x k≤ } + {k x k > } k x k − /n . Foreach n ∈ N , define the function ( ω, x ) U n ( ω, x ) := log( x ) f n ( R ( ω )), for ( ω, x ) ∈ Ω × (0 , + ∞ ),with U n ( ω,
0) := lim x ↓ U n ( ω, x ) = −∞ . For each n ∈ N , it holds that U n ∈ U and(2.7) E (cid:2) U + n (1 + h π, R i ) (cid:3) ≤ k π k + E (cid:2) log(1 + k R k ) f n ( R ) (cid:3) < + ∞ , for all π ∈ Θ . If NA holds, Proposition 2.2 implies that Θ ∩ L is bounded and, therefore, it holds thatsup π ∈ Θ E [ U + n (1 + h π, R i )] < + ∞ . For each n ∈ N , Theorem 2.5 gives then the existence ofan element ρ n ∈ Θ ∩ L which is the maximizer in (2.2) for U = U n . For an arbitrary element π ∈ Θ and ε ∈ (0 , π ε := επ + (1 − ε ) ρ n ∈ Θ. The optimality of ρ n together with theelementary inequality log( x ) ≥ ( x − /x , for x >
0, implies that0 ≥ ε (cid:16) E (cid:2) U n (1 + h π ε , R i ) (cid:3) − E (cid:2) U n (1 + h ρ n , R i ) (cid:3)(cid:17) = 1 ε E (cid:2) log( V π ε /V ρ n ) f n ( R ) (cid:3) ≥ E (cid:20) h π − ρ n , R i h ρ n , R i + ε h π − ρ n , R i f n ( R ) (cid:21) . (2.8)Noting that xy + εx ≥ xy + x/ ≥ −
2, for all ε ∈ (0 , / y > x ≥ − y , we can let ε ց E (cid:20) h π − ρ n , R i h ρ n , R i f n ( R ) (cid:21) ≤ , for all π ∈ Θ and n ∈ N . Since Θ ∩L is compact (see Proposition 2.2), we may assume that the sequence ( ρ n ) n ∈ N convergesto some ρ ∈ Θ ∩ L as n → + ∞ . Therefore, since h π − ρ n , R i / (1 + h ρ n , R i ) ≥ − f n ր n → + ∞ , another application of Fatou’s lemma gives that E (cid:20) h π − ρ, R i h ρ, R i (cid:21) ≤ , for all π ∈ Θ . Equivalently, it holds that E [ V π /V ρ ] ≤
1, for all π ∈ Θ. In view of Definition 2.7, we have thusshown that NA implies the existence of the num´eraire portfolio.( iii ) ⇒ ( ii ): this implication is immediate by Definition 2.7.( ii ) ⇒ ( i ): let Z ∈ D and consider a random variable ξ ∈ L with P ( ξ > > n ∈ N , there exists π n ∈ Θ such that V π n (1 /n ) ≥ ξ a.s. Definition (2.7) implies that E [ Z ξ ] ≤ E (cid:2) Z V π n (1 /n ) (cid:3) = 1 n E (cid:2) Z V π n (cid:3) ≤ n , for all n ∈ N . In particular, the present proof simplifies the techniques employed in Lemma 6.2 and Theorem 6.3 of [KS09].
RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS 9
Since
Z > n → + ∞ yields that ξ = 0 a.s., thus proving the validity of NA .It remains to prove the last assertion of the theorem. If ρ ∈ Θ satisfies (2.6), then its relativelog-optimality is a direct consequence of Jensen’s inequality. Conversely, if ρ ∈ Θ is relativelylog-optimal, then (2.6) follows by the same arguments used in (2.8)-(2.9). (cid:3)
Remark 2.9.
If there exists a log-optimal portfolio , i.e., an allowed strategy ρ ∈ Θ satisfying E [log( V π )] ≤ E [log( V ρ )] < + ∞ , for all π ∈ Θ, then ρ is also relatively log-optimal and, therefore,coincides with the num´eraire portfolio. The num´eraire property of the log-optimal portfolio canalso be directly deduced from the proof of Theorem 2.8. In applications, computing the log-optimal portfolio typically represents a simple way to determine the num´eraire portfolio (see forinstance Examples 2.13 and 3.10). Remark 2.10. NA is equivalent to the existence of a strategy θ ∈ Θ with V θ > θ , in the sense of Remark 2.4. Indeed, supposethere exists θ ∈ Θ with V θ > π ∈ b Θ. Then π + θ is an arbitrage opportunityrelative to θ if and only if π ∈ I arb . Conversely, if NA holds, then there do not exist arbitrageopportunities relative to the num´eraire portfolio ρ , as a consequence of (2.6). However, absenceof arbitrage opportunities relative to a strategy θ ∈ Θ with V θ > θ is the num´eraire portfolio (see Example 3.9 for an explicit counterexample).Theorems 2.5 and 2.8 represent the central results of arbitrage theory based on NA . Forcompleteness, we now state the fundamental theorem of asset pricing based on no classicalarbitrage, in the general version of [Rok05, Theorem 4] for a one-period setting. We give asimple proof inspired by [KaS09, Proposition 2.1.5] and [Kar09, Theorem 3.7], which in turnfollow an original idea of [Rog94]. Similarly to Theorem 2.8, the proof is based on utilitymaximization arguments. For a set A ⊆ R d , we denote by cone A its conic hull. Theorem 2.11.
Suppose that the set cone Θ is closed. Then no classical arbitrage holds if andonly if there exists a probability measure Q ∼ P such that E Q [ V π ] ≤ , for all π ∈ cone Θ .Proof. Observe first that I arb ∩ Θ = ∅ if and only if I arb ∩ (cone Θ) = ∅ . In turn, this impliesthat no classical arbitrage holds if and only if I arb ∩ C = ∅ , where C := (cone Θ) ∩ L . Define theproper convex function f : C ∋ π f ( π ) := E ′ [exp( − − h π, R i )], where E ′ denotes expectationwith respect to the probability measure P ′ defined by d P ′ / d P = e −k R k / E [ e −k R k ]. By Fatou’slemma, the function f is lower semi-continuous. Since C is closed by assumption, [Roc70,Theorem 27.3] implies that the function f admits a minimizer π ∗ ∈ C if it has no directions ofrecession in common with the cone C . By [Roc70, Theorem 8.5], this amounts to verifying that(2.10) ˆ f ( π ) := lim γ → + ∞ f ( γπ ) γ > , for all π ∈ C \ { } . We now show that (2.10) is always satisfied under no classical arbitrage. Arguing by contradic-tion, let π ∈ C \ { } such that ˆ f ( π ) ≤
0. In this case, by Fatou’s lemma, it holds that0 ≥ ˆ f ( π ) ≥ E ′ " lim inf γ → + ∞ e − − γ h π,R i γ ≥ E ′ " lim inf γ → + ∞ e − − γ h π,R i γ {h π,R i < } . This implies that necessarily h π, R i ≥ π ∈ L , this contradicts no classical arbitrage.[Roc70, Theorem 27.3] then yields the existence of an element π ∗ ∈ C such that f ( π ∗ ) ≤ f ( π ), for all π ∈ C . The definition of P ′ implies that differentiation and integration can be interchanged,so that the gradient of the function f at π ∗ is given by ∇ f ( π ∗ ) = − E ′ [exp( − − h π ∗ , R i ) R ].Therefore, since C is a cone and f is finite on C , [Roc70, Theorem 27.4] implies that0 ≥ (cid:10) π, −∇ f ( π ∗ ) (cid:11) = E ′ (cid:2) e − −h π ∗ ,R i h π, R i (cid:3) . Letting d Q/ d P = e − V π ∗ −k R k / E [ e − V π ∗ −k R k ] yields a probability measure Q ∼ P such that E Q [ V π ] ≤
1, for all π ∈ C and, hence, for all π ∈ cone Θ.Conversely, suppose there exists a probability measure Q ∼ P such that E Q [ V π ] ≤
1, for all π ∈ cone Θ. Then, for every π ∈ Θ, it holds that E Q [ h π, R i ] ≤
0. If π ∈ I arb ∩ Θ, this impliesthat h π, R i ≤ Q -a.s. However, since Q ∼ P , this contradicts the fact that π ∈ I arb . (cid:3) Remark 2.12.
Theorem 2.11 does not hold without the assumption of closedness of cone Θ. Indeed, one can construct a counterexample along the lines of [Rok05, Example 1] where no clas-sical arbitrage holds but there does not exist a probability measure Q ∼ P such that E Q [ V π ] ≤ π ∈ cone Θ. Observe that, in comparison to no classical arbitrage, NA has the additionaladvantage of not requiring any extra technical condition on the model.The probability measure Q appearing in Theorem 2.11 represents an equivalent supermartin-gale measure (ESMM). If the num´eraire portfolio ρ satisfies E [1 /V ρ ] = 1, then an ESMM Q canbe defined by setting d Q/ d P = 1 /V ρ . However, this is not always possible, even when cone Θis closed and no classical arbitrage holds, as the following simple example illustrates (see also[Bec01, Example 6] for a related example in an unconstrained setting). Example 2.13.
Let d = 1 and suppose that R = e Y −
1, with Y ∼ N (0 , S = [ − , + ∞ ) and Θ adm = [0 ,
1] (i.e., short-selling and borrowing from the risklessasset are prohibited). Suppose that Θ c = [0 , c ], for some c ∈ [0 , , c ]. Clearly,no classical arbitrage holds and, therefore, there exists an ESMM Q . For instance, it can beeasily checked that d Q/ d P = exp( αY − α /
2) defines an ESMM, for any α ≤ − /
2. However,if c < /
2, the num´eraire portfolio ρ cannot be used to construct an ESMM, since E [1 /V ρ ] < h : [0 , → R defined by h ( π ) := E [log( V π )]is finite-valued, strictly concave and achieves its maximum at 1 /
2, so that h ′ ( π ) > π < /
2. Therefore, if c < /
2, the log-optimal portfolio and, therefore, the num´eraire portfolio ρ (see Remark 2.9) are given by ρ = c and it holds that h ′ ( ρ ) > E [1 /V ρ ] < Hedging and valuation of contingent claims.
We first prove the fundamental super-hedging duality under trading constraints. Recall that for a random variable ξ ∈ L (contingentclaim) its super-hedging value v ( ξ ) is defined as in (2.1), with the usual convention inf ∅ = + ∞ . Theorem 2.14.
Suppose that NA holds and let ξ ∈ L . Then (2.11) v ( ξ ) = sup Z ∈D E [ Zξ ] . Moreover, there exists a pair ( v, π ) ∈ R + × Θ such that ξ = V π ( v ) a.s. and E [ Zξ ] = v , for some Z ∈ D , if and only if there exists an element Z ∗ ∈ D such that E [ Z ∗ ξ ] = sup Z ∈D E [ Zξ ] < + ∞ . The same assumption is required in the fundamental theorem of asset pricing in the formulation of [CPT01].[Rok05, Theorem 4] requires the closedness of p L (cone Θ), the set of all vectors in R d that are projections onto L of elements of cone Θ. In our setting, since L ⊥ ⊆ b Θ, it holds that p L (cone Θ) = (cone Θ) ∩ L . This implies thatp L (cone Θ) is closed if and only if cone Θ is closed. RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS11
Proof.
Let V ( ξ ) := { v > ∃ π ∈ Θ such that vV π ≥ ξ a.s. } and C := { V π : π ∈ Θ ∩ L} − L .If v ∈ V ( ξ ), there exists π ∈ Θ such that vV π ≥ ξ a.s. Then, for every Z ∈ D it holds that E [ Zξ ] ≤ v E [ ZV π ] ≤ v. By taking the supremum over all Z ∈ D and the infimum over all v ∈ V ( ξ ), we obtain that v ( ξ ) ≥ sup Z ∈D E [ Zξ ] =: v ∗ . The converse inequality is trivial if v ∗ = + ∞ . Assuming thereforethat v ∗ < + ∞ , we will show that v ( ξ ) > v ∗ cannot hold. Indeed, if v ( ξ ) > v ∗ , then ξ / ∈ v ∗ C .Let ρ be the num´eraire portfolio (which exists by Theorem 2.8). Being closed in L (see Lemma2.15 below) and bounded in L , the set v ∗ C /V ρ is closed in L . Therefore, by the Hahn-Banachtheorem (see, e.g., [FS16, Theorem A.58]), there exists a bounded random variable α such that(2.12) + ∞ > v ∗ E (cid:20) α ξV ρ (cid:21) > sup X ∈C E (cid:20) α XV ρ (cid:21) =: s. Since − n { α< } ∈ C , for all n ≥
0, inequality (2.12) implies that α ≥ P ( α > > ∈ C , it holds that s >
0. For ε ∈ (0 , Z ε := (cid:16) ε + (1 − ε ) αs (cid:17) V ρ . It holds that P ( Z ε >
0) = 1 and, for every π ∈ Θ, E [ Z ε V π ] = ε E (cid:20) V π V ρ (cid:21) + 1 − εs E (cid:20) α V π V ρ (cid:21) ≤ , thus showing that Z ε ∈ D , for all ε ∈ (0 , ε , (2.12) togetherwith (2.13) implies that E [ Z ε ξ ] > v ∗ = sup Z ∈D E [ Zξ ], which is absurd. Therefore, we must have ξ ∈ v ∗ C , thus proving that v ( ξ ) ≤ v ∗ = sup Z ∈D E [ Zξ ].To prove the last assertion of the theorem, observe that the first part of the proof yields that v ∗ V π ≥ ξ a.s., for some π ∈ Θ. If there exists Z ∗ ∈ D such that v ∗ = E [ Z ∗ ξ ], then we have that v ∗ = E [ Z ∗ ξ ] ≤ v ∗ E [ Z ∗ V π ] ≤ v ∗ . Since Z ∗ > ξ = V π ( v ∗ ) a.s. Conversely, if ξ = V π ( v ) a.s. for some ( v, π ) ∈ R + × Θ with v = E [ Z ∗ ξ ], for some Z ∗ ∈ D , then (2.5) implies that E [ Z ∗ ξ ] = sup Z ∈D E [ Zξ ]. (cid:3) Lemma 2.15. If NA holds, then the set C := { V π : π ∈ Θ ∩ L} − L is closed in L .Proof. Let ( X n ) n ∈ N ⊆ C be a sequence converging in L to a random variable X as n → + ∞ . Foreach n ∈ N , it holds that X n = 1+ h π n , R i− A n , for ( π n , A n ) ∈ (Θ ∩ L ) × L . By Proposition 2.2,NA implies that the set Θ ∩ L is compact and, therefore, there exists a subsequence ( π n m ) m ∈ N converging to an element π ∈ Θ ∩ L . In turn, this implies that the sequence ( A n m ) m ∈ N convergesin probability to a random variable A ∈ L , thus establishing the closedness of C in L . (cid:3) Whenever the quantity sup Z ∈D E [ Zξ ] is finite, it provides the super-hedging value of ξ . Ina general semimartingale setting, the duality relation (2.11) has been stated in [KK07, Section4.7]. We contribute by providing a transparent and self-contained proof in a one-period setting.In addition, Theorem 2.14 gives a necessary and sufficient condition for the attainability of acontingent claim ξ . When perfect hedging is not possible, one may resort to several alternativehedging approaches, which are all feasible under NA even if no classical arbitrage fails to hold. A first possibility is represented by hedging with minimal shortfall risk , corresponding to(2.14) E (cid:2) ℓ ( ξ − vV π ) (cid:3) = min ! over all ( v, π ) ∈ (0 , v ] × Θ , for some initial capital v >
0, where ℓ : R → R is an increasing convex loss function such that ℓ ( x ) = 0, for all x ≤
0, and E [ ℓ ( ξ )] < + ∞ (see [FS16, Section 8.2]). Problem (2.14) can be solvedby first minimizing E [ ℓ ( ξ − Y )] over all random variables Y ∈ L such that sup Z ∈D E [ ZY ] ≤ v and then considering the pair ( v ( Y ∗ ) , π ∗ ) which super-replicates the minimizing random variable Y ∗ (if ℓ is strictly increasing on [0 , + ∞ ), then v ( Y ∗ ) = v ). As long as NA holds, the feasibilityof this approach is ensured by Theorem 2.14.An alternative way to hedge and compute the value of a contingent claim ξ is provided by utility indifference valuation . For a given utility function u and an initial capital v >
0, thiscorresponds to finding the solution p = p ( ξ ) to the equation(2.15) sup π ∈ Θ E (cid:2) u ( vV π ) (cid:3) = sup π ∈ Θ E (cid:2) u (( v − p ) V π + ξ ) (cid:3) . Defining U ηp ( x, ω ) := u (( v − ηp ) x + ηξ ( ω )), for η ∈ { , } , Theorem 2.5 with U = U ηp showsthat NA is sufficient for the solvability of the two maximization problems appearing in (2.15).Whenever it exists, p ( ξ ) represents a (buyer) value for ξ , while the strategy π ∗ that achieves thesupremum on the right-hand side of (2.15) with p = p ( ξ ) provides a hedging strategy for ξ .As a variant of the latter approach, one can consider marginal utility indifference valuation,in the sense of [Dav97]. This corresponds to finding the value p = p ′ ( ξ ) which solveslim η ↓ E [ U ηp ( V π ∗ )] − E [ U p ( V π ∗ )] η = 0 . where U η is defined as above, for η ∈ [0 , π ∗ ∈ Θ is the strategy solving problem (2.2)with U = u . Similarly as in [FR13], if NA holds and u ( x ) = log( x ), it can be shown that(2.16) p ′ ( ξ ) = E [ ξ/V ρ ] , as long as the expectation is finite, where ρ denotes the num´eraire portfolio (see Theorem 2.8).In the context of the Benchmark Approach (see [BP03, PH06]), formula (2.16) corresponds tothe well-known real-world pricing formula , which is applicable as long as NA is satisfied.3. Factor models with arbitrage under borrowing constraints
In this section, we study the arbitrage concepts discussed above in the context of a one-periodfactor model, under constraints on the proportion of wealth that can be borrowed/invested onthe riskless asset. We start from a general model and then consider more specific cases.3.1.
A general factor model.
In the setting of Section 2, we assume that asset returns aregenerated by the factor model(3.1) R = QY, where Q ∈ R d × ℓ and Y = ( Y , . . . , Y ℓ ) ⊤ is an ℓ -dimensional random vector with independentcomponents, for some ℓ ∈ N . A non-diagonal matrix Q permits to introduce general correlationstructures among the d asset returns. Without loss of generality, we assume that rank( Q ) = d .Under this assumption, it holds that L ⊥ = { } . RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS13
For k = 1 , . . . , ℓ , we denote by Y k the support of Y k and let y inf k := inf Y k and y sup k := sup Y k .In this section, we work under the following standing assumption:(3.2) y inf1 = 0 , y sup1 = + ∞ and y inf k < < y sup k , for all k = 2 , . . . , ℓ. As will become clear in the sequel, condition (3.2) corresponds to viewing the first factor Y as the driving force of possible arbitrage opportunities, while the remaining factors cannot beexploited to generate arbitrage. In the context of the factor model (3.1)-(3.2), the followinglemma gives a necessary and sufficient condition to ensure positive asset prices. For i = 1 , . . . , d and k = 1 , . . . , ℓ , we denote by q i,k the element on the i -th row and k -th column of Q . Lemma 3.1.
In the context of the model of this section, for each i = 1 , . . . , d , it holds that R i ≥ − a.s. if and only if the following condition is satisfied: (3.3) q i, ≥ and ℓ X k =2 (cid:0) q + i,k y inf k − q − i,k y sup k (cid:1) ≥ − , with the convention × ( −∞ ) = 0 and × (+ ∞ ) = 0 .Proof. Condition (3.3) is obviously sufficient to ensure that R i ≥ − i = 1 , . . . , d .Conversely, let i ∈ { , . . . , d } and suppose that R i ≥ − n ∈ N and k = 1 , . . . , ℓ , let y inf k ( n ) := (cid:16) y inf k + 1 n (cid:17) ∨ ( − n ) and y sup k ( n ) := (cid:16) y sup k − n (cid:17) ∧ n, With this notation, it holds that P ( Y k ≤ y inf k ( n )) > P ( Y k ≥ y sup k ( n )) >
0, for all n ∈ N and k = 1 , . . . , ℓ . Let K + i := { k ∈ { , . . . , ℓ } : q i,k ≥ } and K − i := { , . . . , ℓ } \ K + i . Since P ℓk =1 q i,k Y k ≥ − { Y , . . . , Y ℓ } , it holds that0 < P (cid:18) Y k ≤ y inf k ( n ) and Y j ≥ y sup j ( n ); ∀ k ∈ K + i , ∀ j ∈ K − i (cid:19) = P X k ∈ K + i q i,k Y k ≥ − − X j ∈ K − i q i,j Y j and Y k ≤ y inf k ( n ) and Y j ≥ y sup j ( n ); ∀ k ∈ K + i , ∀ j ∈ K − i ! . In turn, this necessarily implies that P k ∈ K + i q i,k y inf k ( n ) ≥ − − P j ∈ K − i q i,j y sup j ( n ) , for each n ∈ N . Condition (3.3) follows by letting n → + ∞ and using condition (3.2). (cid:3) In particular, condition (3.3) requires that q i,k ≥ y sup k = + ∞ and q i,k ≤ y inf k = −∞ ,for all i = 1 , . . . , d and k = 1 , . . . , ℓ . Observe that condition (3.3) relates the support of therandom factors to the dependence structure of the asset returns, represented by the off-diagonalelements of Q . Arguing similarly as in Lemma 3.1, it can be shown that the set Θ adm ofadmissible strategies can be represented as follows:(3.4) Θ adm = ( π ∈ R d : π ⊤ Q • , ≥ ℓ X k =2 (cid:16) ( π ⊤ Q • ,k ) + y inf k − ( π ⊤ Q • ,k ) − y sup k (cid:17) ≥ − ) , where Q • ,k denotes the k -th column of the matrix Q , with the same convention as in (3.3).We now introduce additional trading restrictions, as considered in Section 2.1. More specifi-cally, we assume the presence of borrowing constraints :(3.5) Θ c := { π ∈ R d : h π, i ≤ c } , for some fixed c >
0. If c ∈ (0 , − c of the initial wealth is invested in the riskless asset, while, if c ≥
1, at most a proportion c − c is nota cone, the notions of arbitrage opportunity and arbitrage of the first kind differ (see Remark2.3). As in Section 2.1, the set Θ of allowed strategies is defined as Θ := Θ adm ∩ Θ c .The following proposition summarizes the arbitrage properties of the factor model underconsideration, in the presence of borrowing constraints. We denote by R ( Q ⊤ ) the range of thematrix Q ⊤ and by e k the k -th vector of the canonical basis of R ℓ , for k = 1 , . . . , ℓ . Proposition 3.2.
In the context of the model of this section, the following hold:(i) there are arbitrage opportunities if and only if e ∈ R ( Q ⊤ ) . In that case, it holds that (3.6) I arb ∩ Θ = (cid:8) λ ( QQ ⊤ ) − Q • , : λ > and λ h ( QQ ⊤ ) − Q • , , i ≤ c (cid:9) ; (ii) if e ∈ R ( Q ⊤ ) , then NA holds if and only if h ( QQ ⊤ ) − Q • , , i > .Proof. (i) : let π be a vector in R d such that h π, QY i ≥ P ℓ X k =1 ( π ⊤ Q • ,k ) Y k < ! ≥ P (cid:16) ( π ⊤ Q • ,k ) + Y k < π ⊤ Q • ,k ) − Y k > , for some k = 1 , . . . , ℓ (cid:17) , it holds that ( π ⊤ Q • ,k ) + Y k ≥ π ⊤ Q • ,k ) − Y k ≤ k = 1 , . . . , ℓ . Recallingcondition (3.2), this implies that π ⊤ Q • , ≥ π ⊤ Q • ,k = 0, for all k = 2 , . . . , ℓ . It followsthat h π, QY i ≥ Q ⊤ π = λ e , for some λ ≥
0. Since rank( Q ) = d , it holdsthat I arb = { λ ( QQ ⊤ ) − Q e : λ > } , from which representation (3.6) of the set I arb ∩ Θ followsdirectly from the definition of the set Θ c in (3.5). (ii) : by Proposition 2.2, NA holds if and only if I arb ∩ b Θ = ∅ . Representation (3.6) implies that I arb ∩ b Θ = ∅ if and only if h ( QQ ⊤ ) − Q • , , i > (cid:3) Remark 3.3.
The vector ( QQ ⊤ ) − Q • , corresponds to the strategy replicating the factor Y .While exact replication of Y may be precluded by borrowing constraints, (3.6) shows that anyallowed strategy that replicates a positive fraction of Y is an arbitrage opportunity. The factor Y can be (super-)replicated at zero cost if h ( QQ ⊤ ) − Q • , , i ≤
0, in which case NA fails. Remark 3.4.
The proof of Proposition 3.2 shows that a strategy π ∈ I arb ∩ Θ necessarilysatisfies π ⊤ Q • ,k = 0, for all k = 2 , . . . , ℓ . When ℓ = d , this corresponds to a set of d − d variables. This set defines a line in R d , which we call arbitrage line . This conceptwill be illustrated in the two-dimensional model considered in Section 3.3.In view of Theorem 2.5, NA ensures the well-posedness of optimal portfolio problems. In thepresence of arbitrage opportunities, the borrowing constraint (3.5) is binding for every optimalallowed strategy. This is a direct consequence of the following simple result. Lemma 3.5.
In the context of the model of this section, suppose that e ∈ R ( Q ⊤ ) and NA holds. Then, for every π ∈ Θ , there exists an element ˆ π ∈ Θ such that h ˆ π, QY i ≥ h π, QY i a.s. and h ˆ π, i = c. RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS15
Moreover, there exists a strategy π max , explicitly given by (3.7) π max = c h ( QQ ⊤ ) − Q • , , i ( QQ ⊤ ) − Q • , , such that h π max , i = c and h π max , QY i ≥ h π, QY i a.s., for all π ∈ I arb ∩ Θ .Proof. Let π be an arbitrary allowed strategy. Letting λ := ( c − h π, i ) h ( QQ ⊤ ) − Q • , , i − ≥ π := π + λ ( QQ ⊤ ) − Q • , . Clearly, it holds that h ˆ π, i = c and, in addition, h ˆ π, QY i = h π, QY i + λ e ⊤ Q ⊤ ( QQ ⊤ ) − QY = h π, QY i + λY ≥ h π, QY i a.s. The second part ofthe lemma follows as a direct consequence of the characterization (3.6) of the set I arb ∩ Θ. (cid:3) We call maximal arbitrage strategy the strategy π max given in (3.7). Whenever NA fails tohold (i.e., h ( QQ ⊤ ) − Q • , , i ≤ π max is not necessarily the optimal strategy inan expected utility maximization problem of the type (2.2). Similarly, π max does not necessarilycoincide with the num´eraire portfolio ρ . This will be explicitly illustrated in Examples 3.8–3.10. Remark 3.6 (On relative arbitrage) . (1) In the context of the model of this section, let usassume that NA holds. Then, for θ ∈ Θ, there exist arbitrage opportunities relative to θ if andonly if h θ, i < c . Indeed, if h θ, i < c , then the existence of an arbitrage opportunity relative to θ follows from Lemma 3.5. Conversely, suppose that h θ, i = c and let π ∈ R d with π − θ ∈ I arb .In view of Proposition 3.2, this holds if and only if π − θ = η ( QQ ⊤ ) − Q • , , for some η > h π, i = h θ, i + η h ( QQ ⊤ ) − Q • , , i > c, the strategy π is not an allowed tradingstrategy. This shows that there cannot exist arbitrage opportunities relative to θ if h θ, i = c .In particular, there do not exist arbitrage opportunities relative to π max .(2) One can also study the existence of arbitrage opportunities relative to the market portfolio π mkt defined by π mkt i := S i / h S , i , for i = 1 , . . . , d (see [FK09, Section 2]). As a consequence ofpart (1) of this remark, arbitrage opportunities relative to the market exist if and only if c > c >
1, then it is possible to invest the whole initial capital v inthe market portfolio, borrow an amount v ( c −
1) from the riskless asset and invest that amountin the strategy π max , thus improving the performance of the market portfolio. The strategy π ∗ ∈ Θ which best outperforms the market portfolio is given by π ∗ = π mkt + c − c π max .3.2. The case of a unit triangular matrix Q . Let us consider the special case where Q isa ( d × d ) upper triangular matrix with q i,i = 1, for all i = 1 , . . . , d . In this case, some of theresults presented above can be stated explicitly in terms of the elements of Q . First, condition(3.3) ensuring the positivity of asset prices can be rewritten in the following recursive form:(3.8) y inf d ≥ − y inf i ≥ − − d X k = i +1 (cid:0) q + i,k y inf k − q − i,k y sup k (cid:1) , for all i = 1 , . . . , d − . In view of (3.4), the set Θ adm of admissible strategies takes the form(3.9)Θ adm = ( π ∈ R d : π ≥ d X k =2 k − X i =1 π i q i,k + π k ! + y inf k − k − X i =1 π i q i,k + π k ! − y sup k ! ≥ − ) . Since rank( Q ) = d , the condition e ∈ R ( Q ⊤ ) is automatically satisfied and, therefore, thereexist arbitrage opportunities (see Proposition 3.2). More specifically, it holds that(3.10) I arb ∩ Θ = (cid:8) λQ − , • : λ > λ h Q − , • , i ≤ c (cid:9) , where Q − , • denotes the first row of the matrix Q − , written as a column vector. The followinglemma gives an explicit representation of the vector Q − , • , which determines all the arbitrageproperties of the model under consideration. Lemma 3.7.
In the context of the model of this section, suppose that Q is a unit triangularmatrix. Then, for all k = 1 , . . . , d , it holds that Q − ,k = α k , where α k is defined by α := 1 and α k := X J ∈ A ( k ) ( − | J |− | J |− Y l =1 q j l ,j l +1 , for k = 2 , . . . , d, and A ( k ) denotes the family of all subsets J = { j , . . . , j r } ⊆ { , . . . , k } , with r ≤ k , such that j = 1 , j r = k and j l < j l +1 , for all l = 1 , . . . , r − .Proof. The vector Q − , • is the unique solution π ∈ R d to the linear system Q ⊤ π = e . Since Q isa unit triangular matrix, the solution π is characterized by π = 1 and by the recursive relation(3.11) π k = − k − X i =1 π i q i,k , for all k = 2 , . . . , d. To prove the lemma, it suffices to show that the vector α = ( α , . . . , α d ) ⊤ satisfies (3.11). Tothis effect, notice that, for every k = 2 , . . . , d , − k − X i =1 α i q i,k = − q ,k − k − X i =2 X J ∈ A ( i ) ( − | J |− | J |− Y l =1 q j l ,j l +1 q i,k = α k . This shows that α = ( α , . . . , α d ) ⊤ satisfies (3.11) and, therefore, it holds that Q − , • = α . (cid:3) In view of (3.10), the vector α introduced in Lemma 3.7 generates all arbitrage strategies,up to a multiplicative factor depending on the borrowing constraint c . More precisely, everyarbitrage strategy π is necessarily of the form π = λα , with λ > λ h α, i ≤ c , and issuch that V π = 1 + λY . Furthermore, by (3.11), all such strategies π belong to the arbitrageline (see Remark 3.4). As an example, for d = 4, all arbitrage strategies are proportional to α = − q , − q , + q , q , − q , + q , q , + q , q , − q , q , q , . In the model considered in this subsection, the condition characterizing the validity of NA takes the simple form h Q − , • , i > :(3.12) NA holds ⇐⇒ X J ⊆{ ,...,d } ( − | J |− | J |− Y l =1 q j l ,j l +1 > , RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS17 where the summation is taken over all sets J = { j , . . . , j r } , with 2 ≤ r ≤ d , such that j = 1and j l < j l +1 , for all l = 1 , . . . , r −
1. In view of (3.7), the same quantity appearing on the rightof (3.12) represents the denominator of the maximal arbitrage strategy π max .3.3. A two-dimensional example with arbitrage.
We now present a two-dimensional modelthat allows for a geometric visualization of the concepts introduced above. Let d = 2 and considera pair ( Y , Y ) of independent random variables such that Y = [0 , + ∞ ) and y inf2 < < y sup2 .Let Q = γ ! , with γ ∈ R , and suppose that the asset returns ( R , R ) are generated as in (3.1). To ensurepositive asset prices, condition (3.8) needs to be satisfied. In this example, the largest possiblesupport of the distribution of the random factor Y is given by y inf2 = − y sup2 = + ∞ , if γ ∈ [0 , y inf2 = − /γ and y sup2 = + ∞ , if γ ≥ y inf2 = − y sup2 = − /γ, if γ < . In view of (3.9), a strategy π = ( π , π ) is admissible if and only if(3.13) π ≥ − γπ ≤ π ≤ − γπ , if γ ∈ [0 , π ≥ − γπ ≤ π ≤ γ − γπ , if γ ≥ π ≥ γ − γπ ≤ π ≤ − γπ , if γ < . In this two-dimensional setting, the borrowing constraint (3.5) takes the form π + π ≤ c .Together with (3.13), this constraint determines the set Θ of allowed strategies. Regardless ofthe values of γ and c , arbitrage opportunities always exist . More specifically, it holds that(3.14) I arb ∩ Θ = (cid:8) π ∈ R : π > , π = − γπ and π (1 − γ ) ≤ c (cid:9) = ∅ . The arbitrage line (see Remark 3.4) is described by the equation π = − γπ . Figure 1 providesa visualization of the set Θ, with the arbitrage line highlighted in red.The NA condition is satisfied if and only if h Q − , • , i >
0. Therefore, we have thatNA holds ⇐⇒ γ < . Indeed, from (3.14) we have that I arb ∩ b Θ = ∅ if and only if γ <
1. Graphically, this conditioncorresponds to requesting that the arbitrage line intersects the borrowing constraint line (seeFigure 1), i.e., the line of equation π = c − π . Observe also that the set Θ is compact if andonly if such an intersection occurs (compare with condition (iv) in Proposition 2.2).For γ <
1, all arbitrage strategies are contained in the line segment passing through the originand the point ( π max1 , π max2 ) corresponding to the maximal arbitrage strategy and given by(3.15) π max1 = c − γ and π max2 = − cγ − γ , as follows from (3.7). Graphically, the strategy π max corresponds to the point of intersectionbetween the arbitrage line and the borrowing constraint line. If the two lines do not intersect,then every arbitrage opportunity can be arbitrarily scaled (i.e., NA fails to hold). π π π = 1 − γπ π = c − π c ( c − γ , − cγ − γ ) π = − γπ Figure 1.
Geometric illustration of the set Θ (yellow area), for c = 2 . γ = 0 . ρ exists if and only if γ <
1. The num´eraireportfolio may or may not coincide with the maximal arbitrage strategy π max , depending on thedistributional properties of Y and Y . For illustration, we discuss three simple examples. Example 3.8.
Let γ ∈ [0 ,
1) and suppose that E [ Y ] = 0. In this case, it holds that ρ = π max .Indeed, let π = ( π , π ) be an arbitrary strategy satisfying (3.13) and π + π ≤ c . By Lemma3.5, there exists a strategy of the form ˆ π = (ˆ π , c − ˆ π ) such that V ˆ π ≥ V π a.s. Due to (3.13), itnecessarily holds that 0 ≤ ˆ π ≤ c/ (1 − γ ). Therefore, using the independence of Y and Y andthe fact that E [ Y ] = 0, we have that E (cid:20) V π V π max (cid:21) ≤ E (cid:20) V ˆ π V π max (cid:21) = E " π Y c − γ Y ≤ , where the last inequality follows from the fact that Y ≥ ρ coincides with the maximal arbitrage strategy π max given in (3.15). Example 3.9.
Let γ = 1 / c = 1. Suppose that Y ∼ Exp(1) and 1 + Y ∼ Exp( β ), with β >
0. In this case, for suitable values of β , the maximal arbitrage strategy is not the num´eraireportfolio. Indeed, considering the strategy (0 , ∈ Θ, we have that E " V (0 , V π max = E (cid:20) Y Y (cid:21) = 1 β E (cid:20)
11 + 2 Y (cid:21) = √ e β Z + ∞ / e − x x d x ≈ . β . For any sufficiently small value of β , it holds that E [ V (0 , /V π max ] > π max cannot be the num´eraire portfolio in that case. Furthermore, since V π max ≥ V π a.s. forall π ∈ I arb ∩ Θ (Lemma 3.5), the num´eraire portfolio does not belong to the set of arbitrageopportunities. In view of Remark 2.9, this implies that, even in the presence of arbitrage, it isnot necessarily optimal (for logarithmic preferences) to invest in an arbitrage strategy.
Example 3.10.
Let γ < E [ Y ] < + ∞ and E [ Y ] < + ∞ . Under theseassumptions, the log-optimal portfolio π ∗ exists and, therefore, it coincides with the num´eraire RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS19 portfolio ρ . Lemma 3.5 together with (3.13) implies that π ∗ is of the form ( π ∗ , c − π ∗ ), with π ∗ ∈ D ( c, γ ) := [ ( c − + − γ , c − γ − γ ]. Consider the function g : D ( c, γ ) → R defined by g ( π ) := E (cid:2) log (cid:0) V ( π ,c − π )1 (cid:1)(cid:3) = E (cid:2) log (cid:0) π (cid:0) Y + ( γ − Y (cid:1) + cY (cid:1)(cid:3) , for π ∈ D ( c, γ ). Since the function g is concave and π max1 = c/ (1 − γ ) belongs to the interiorof the interval D ( c, γ ), the log-optimal portfolio π ∗ is given by π max if and only if g ′ ( π max1 ) = 0.The latter condition is equivalent to(3.16) E " Y c − γ Y = (1 − γ ) E " Y c − γ Y . In the present example, ρ = π max holds if and only if condition (3.16) is satisfied. In particular,unlike in Example 3.8 above, note that (3.16) cannot be satisfied if E [ Y ] = 0.4. The multi-period setting
In this section, we extend the analysis of Section 2 to the multi-period case. We allow forconvex trading constraints evolving randomly over time and prove that NA holds in a dynamicsetting if and only if it holds in each single trading period. This fundamental fact enables us toaddress the multi-period case by relying on arguments similar to those employed in Section 2. Forbrevity of presentation, we prove multi-period versions of only the central results characterizingmarket viability and NA , the remaining results and remarks admitting analogous extensions.4.1. Setting and trading restrictions.
Let (Ω , F , F , P ) be a filtered probability space, where F = ( F t ) t =0 , ,...,T and F is the trivial σ -field completed by the P -nullsets of F , for a fixedtime horizon T ∈ N . Similarly to Section 2, we consider d risky assets and a riskless asset withconstant price equal to one. The discounted prices of the d risky assets are represented by the d -dimensional adapted process S = ( S t ) t =0 , ,...,T . For each i = 1 , . . . , d , we assume that S it = S it − (1 + R it ) , for all t = 1 , . . . , T where each random variable R it is F t -measurable, satisfies R it ≥ − i on the period [ t − , t ]. For each t = 1 , . . . , T , we denote by S t the F t − -conditionalsupport of the random vector R t = ( R t , . . . , R dt ) ⊤ (i.e., the support of a regular version of the F t − -conditional distribution of R t , see [BCL19, Definition 2.2]). We also denote by L t thesmallest linear subspace of R d containing S t and by L ⊥ t its orthogonal complement. Conditionalexpectations are to be understood in the generalized sense (see, e.g., [HWY92, Section 1.4]).A set-valued process A = ( A t ) t =1 ,...,T is said to be predictable if, for each t = 1 , . . . , T , thecorrespondence (set-valued mapping) A t from Ω to R d is F t − -measurable. The processes S = ( S t ) t =1 ,...,T , L = ( L t ) t =1 ,...,T and L ⊥ = ( L ⊥ t ) t =1 ,...,T are all predictable (see [BCL19, Lemma2.4] and [RW98, Exercise 14.12-(d)]). For each t = 1 , . . . , T , the orthogonal projection of a vector x ∈ R d on L t is denoted by p L t ( x ) and it is F t − -measurable (see [RW98, Exercise 14.17]).We describe trading strategies via predictable processes π = ( π t ) t =1 ,...,T , with π t = ( π t , . . . , π dt ) ⊤ representing proportions of wealth invested in the d risky assets between time t − t . We recall that a correspondence A t from Ω to R d is F t − -measurable if, for every open subset G ⊂ R d , it holdsthat { ω ∈ Ω : A t ( ω ) ∩ G = ∅} ∈ F t − , see [RW98, Definition 14.1]. We denote by V πt ( v ) the wealth at time t generated by strategy π starting from capital v > V π ( v ) = v and V πt ( v ) = v t Y k =1 (1 + h π k , R k i ) , for t = 1 , . . . , T. As in Section 2.1, we define V πt := V πt (1). A strategy π is said to be admissible if V πt ≥ t = 1 , . . . , T . Equivalently, introducing the random setsΘ adm ,t := { π ∈ R d : h π, z i ≥ − z ∈ S t } , for t = 1 , . . . , T, a strategy π is admissible if and only if π t ∈ Θ adm ,t holds a.s. for all t = 1 , . . . , T . Note that,for every ( ω, t ) ∈ Ω × { , . . . , T } , the set Θ adm ,t ( ω ) is a non-empty, closed and convex subset of R d . Arguing similarly as in [RW98, Exercise 14.12-(e)], it can be shown that the predictabilityof S implies that the set-valued process Θ adm = (Θ adm ,t ) t =1 ,...,T is predictable. Trading constraints are modelled through a set-valued predictable process Θ c = (Θ c ,t ) t =1 ,...,T such that Θ c ,t ( ω ) is a convex closed subset of R d , for all ( ω, t ) ∈ Ω × { , . . . , T } . Similarly as inSection 2.1, we assume that L ⊥ t ( ω ) ⊂ Θ c ,t ( ω ), for all ( ω, t ) ∈ Ω ×{ , . . . , T } . The family of allowedstrategies is given by all R d -valued predictable processes π = ( π t ) t =1 ,...,T such that π t ∈ Θ t :=Θ adm ,t ∩ Θ c ,t a.s. for all t = 1 , . . . , T . Note that, as a consequence of [RW98, Proposition 14.11],the set-valued process Θ = (Θ t ) t =1 ,...,T is predictable. For brevity of notation, we shall simplywrite π ∈ Θ to denote that a trading strategy π is allowed. For each ( ω, t ) ∈ Ω × { , . . . , T } , theset b Θ t ( ω ) is defined as the recession cone of Θ t ( ω ). The set-valued process b Θ = ( b Θ t ) t =1 ,...,T ispredictable, as a consequence of the predictability of Θ together with [RW98, Exercise 14.21],and admits the same financial interpretation as the recession cone b Θ introduced in Section 2.1in a single-period setting.4.2.
Arbitrage concepts.
An allowed strategy π ∈ Θ is said to be an arbitrage opportunity if(4.1) P ( V πT ≥
1) = 1 and P ( V πT > > . We say that no classical arbitrage holds if there does not exist a strategy π ∈ Θ satisfying(4.1). For t = 1 , . . . , T , we denote by L ( F t ) the family of non-negative F t -measurable randomvariables. Definition 2.1 can be naturally extended to a multi-period setting as follows. Definition 4.1.
A random variable ξ ∈ L ( F T ) with P ( ξ > > arbitrageof the first kind if v ( ξ ) = 0, where v ( ξ ) := inf { v > ∃ π ∈ Θ such that V πT ( v ) ≥ ξ a.s. } . No arbitrage of the first kind (NA ) holds if, for every ξ ∈ L ( F T ), v ( ξ ) = 0 implies ξ = 0 a.s.As a preliminary to the statement of the next proposition, we define, for each t = 1 , . . . , T , I arb ,t := { π ∈ R d : h π, z i ≥ z ∈ S t } \ L ⊥ t . By [RW98, Exercise 14.12-(e)], the random set I arb ,t is F t − -measurable, for all t = 1 , . . . , T .For a random variable ζ ∈ L ( F t ), we define its super-hedging value at time t − t − ( ζ ) := ess inf (cid:8) x ∈ L ( F t − ) : ∃ h ∈ L ( F t − ; Θ t ) such that x (1 + h h, R t i ) ≥ ζ a.s. (cid:9) , where L ( F t − ; Θ t ) denotes the family of F t − -measurable random vectors h : Ω → R d such that h takes a.s. values in Θ t . As already pointed out in [KK07], the fact that Θ adm is a set-valued stochastic process shows that tradingconstraints evolving randomly over time arise naturally as a consequence of the admissibility requirement.
RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS21
For the usual concept of no classical arbitrage, it is well-known that absence of arbitrage ina multi-period setting is equivalent to absence of arbitrage opportunities in each single tradingperiod (see, e.g., [FS16, Proposition 5.11]). In the next proposition, we prove that an analogousproperty holds for NA and we also provide several equivalent characterizations. Proposition 4.2.
The following are equivalent:(i) the NA condition holds;(ii) there does not exist a strategy π ∈ b Θ satisfying (4.1) ;(iii) for every t = 1 , . . . , T and ζ ∈ L ( F t ) , v t − ( ζ ) = 0 a.s. implies ζ = 0 a.s.;(iv) I arb ,t ∩ b Θ t = ∅ a.s., for all t = 1 , . . . , T ;(v) b Θ t = L ⊥ t a.s., for all t = 1 , . . . , T ;(vi) the set Θ t ∩ L t is a.s. bounded (and, hence, compact), for all t = 1 , . . . , T .Proof. ( i ) ⇒ ( iii ): by way of contradiction, assume that NA holds and suppose that, for some t = 1 , . . . , T , there exists ζ ∈ L ( F t ) such that v t − ( ζ ) = 0 a.s. and P ( ζ > >
0. In this case,for every v >
0, one can find h ∈ L ( F t − ; Θ t ) such that v (1 + h h, R t i ) ≥ ζ a.s. Define then thestrategy π = ( π s ) s =1 ,...,T by π s := h if s = t and π s := 0 otherwise. With this definition, it holdsthat π ∈ Θ and V πT ( v ) = v (1 + h h, R t i ) ≥ ζ a.s., contradicting the validity of NA .( iii ) ⇒ ( iv ): we adapt to the present setting the arguments of [KK07, Section 5]. By way ofcontradiction, assume that ( iii ) holds and let P ( I arb ,t ∩ b Θ t = ∅ ) >
0, for some t = 1 , . . . , T . Foreach n ∈ N , define the F t − -measurable random set I n arb ,t := (cid:26) π ∈ R d : h π, z i ≥ z ∈ S t and E (cid:20) h π, R t i h π, R t i (cid:12)(cid:12)(cid:12)(cid:12) F t − (cid:21) ≥ /n (cid:27) ⊂ I arb ,t . We have that I arb ,t ∩ b Θ t = ∅ if and only if I n arb ,t ∩ b Θ t = ∅ for all large enough n ∈ N (see [KK07,Lemma 5.1]). Hence, there exists a sufficiently large n ∈ N such that P ( I n arb ,t ∩ b Θ t = ∅ ) >
0. Itcan be easily checked that the set I n arb ,t ( ω ) ∩ b Θ t ( ω ) is closed and convex, for all ω ∈ Ω. Therefore,by [RW98, Corollary 14.6], there exists an F t − -measurable random vector π nt : Ω → R d such that π nt ( ω ) ∈ I n arb ,t ( ω ) ∩ b Θ t ( ω ) when I n arb ,t ( ω ) ∩ b Θ t ( ω ) = ∅ and π nt ( ω ) = 0 when I n arb ,t ( ω ) ∩ b Θ t ( ω ) = ∅ .The random variable ζ := h π nt , R t i belongs to L ( F t ) and satisfies P ( ζ > >
0. Moreover,since π nt ∈ b Θ t a.s., it holds that π nt /v ∈ Θ t a.s., for all v >
0. Noting that v (1 + h π nt /v, R t i ) > ζ a.s., this implies that v t − ( ζ ) = 0 a.s., thus contradicting property ( iii ).( ii ) ⇔ ( iv ): this equivalence follows by the same arguments used in [FS16, Proposition 5.11],together with the construction of π nt performed in the previous step of the proof.( iv ) ⇒ ( v ) ⇒ ( vi ): these implications can be proved as in Proposition 2.2.( vi ) ⇒ ( i ): by way of contradiction, let ξ ∈ L ( F T ) with P ( ξ > > n ∈ N , there exists an allowed strategy π n ∈ Θ such that V π n T (1 /n ) ≥ ξ a.s. Then, it holds that1 + Q Tt =1 h p L t ( π nt ) , R t i ≥ nξ a.s., for all n ∈ N . Similarly as in the proof of Proposition 2.2, thefact that P ( ξ > > t ∩ L t , for t = 1 , . . . , T . (cid:3) Proposition 4.2 shows that, in a multi-period setting, NA is equivalent to the absence ofarbitrarily scalable arbitrage opportunities (property ( ii )) as well as to the absence of arbitrageof the first kind in each single trading period (property ( iii )). Properties ( iv )–( vi ) can beinterpreted similarly to the analogous properties discussed in Section 2.2. Note also that NA isequivalent to no classical arbitrage if the constraint process Θ c is cone-valued (see Remark 2.3). Remark 4.3.
Property ( vi ) in Proposition 4.2 implies that, for each t = 1 , . . . , T , there existsan F t − -measurable random variable H t such that k π k ≤ H t a.s., for all π ∈ L ( F t − ; Θ t ∩ L t ).The F t − -measurability of H t follows from the closedness and F t − -measurability of Θ t ∩ L t .4.3. Market viability and fundamental theorems.
We proceed to characterize NA interms of the solvability of portfolio optimization problems, extending Theorem 2.5 to the multi-period setting. In view of Proposition 4.2, the NA condition admits a local description. Byemploying a dynamic programming approach, this allows reducing a portfolio optimization prob-lem to a sequence of one-period problems, to which we can apply techniques analogous to thoseused in the proof of Theorem 2.5. This approach is inspired by [RS06], where the implica-tion ( i ) ⇒ ( ii ) of the following theorem has been proved under no classical arbitrage for anunconstrained market. Similarly as in Section 2.3, we denote by U the set of all functions U : Ω × R + → R ∪ {−∞} such that U ( · , x ) is F T -measurable and bounded from below, for every x >
0, and U ( ω, · ) is continuous, strictly increasing and concave, for a.e. ω ∈ Ω. Theorem 4.4.
The following are equivalent:(i) the NA condition holds;(ii) for every U ∈ U such that sup π ∈ Θ E [ U + ( V πT )] < + ∞ , there exists an allowed strategy π ∗ ∈ Θ ∩ L such that E (cid:2) U ( V π ∗ T ) (cid:3) = sup π ∈ Θ E (cid:2) U ( V πT ) (cid:3) . Proof. ( i ) ⇒ ( ii ): suppose that NA holds and let U ∈ U be such that sup π ∈ Θ E [ U + ( V πT )] < + ∞ .Since U ∈ U , it holds that sup π ∈ Θ E [ U + ( xV πT )] < + ∞ for all x ≥
0. The existence of an optimalstrategy π ∗ ∈ Θ ∩ L will be shown in a constructive way by applying dynamic programming.For all ( ω, x ) ∈ Ω × R + , define U T ( ω, x ) := U ( ω, x ) and, for t = 0 , , . . . , T − U t ( ω, x ) := ess sup π t +1 ∈ L ( F t ;Θ t +1 ∩L t +1 ) E (cid:2) U t +1 (cid:0) ω, x (1 + h π t +1 , R t +1 ( ω ) i ) (cid:1)(cid:12)(cid:12) F t (cid:3) ( ω ) , taking a regular version of the conditional expectation (the existence of the conditional expec-tation will follow from the proof below). Proceeding by backward induction, let t < T andsuppose that U t +1 ∈ U and(4.3) sup π t +1 ∈ L ( F t ;Θ t +1 ∩L t +1 ) E (cid:2) U + t +1 (cid:0) x (1 + h π t +1 , R t +1 i ) (cid:1)(cid:3) < + ∞ , for all x ≥ . These hypotheses are satisfied by assumption for t = T − t < T −
1. Since the family { E [ U t +1 ( x (1 + h π t +1 , R t +1 i )) |F t ]; π t +1 ∈ L ( F t ; Θ t +1 ∩ L t +1 ) } isdirected upward, for all x > π nt +1 ( x )) n ∈ N with values in Θ t +1 ∩ L t +1 such that(4.4) lim n → + ∞ E (cid:2) U t +1 (cid:0) x (1 + h π nt +1 ( x ) , R t +1 i ) (cid:1)(cid:12)(cid:12) F t (cid:3) = U t ( x ) a.s.As a consequence of NA , the set Θ t +1 ∩ L t +1 is closed and a.s. bounded (see Proposition 4.2).Therefore, by [FS16, Lemma 1.64], there exists a subsequence ( π n k t +1 ( x )) k ∈ N converging a.s. toan element ˆ π t +1 ( x ) ∈ L ( F t ; Θ t +1 ∩ L t +1 ). By the same arguments used in the proof of theimplication ( i ) ⇒ ( ii ) of Theorem 2.5 (but carried out conditionally on F t , see also [RS06, In the following, for simplicity of notation, we shall omit to denote explicitly the dependence on ω in U t ( ω, x ). RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS23
Lemma 2.3]), the boundedness of Θ t +1 ∩ L t +1 (see Remark 4.3), the properties of U t +1 and (4.3)together imply the existence of an F t +1 -measurable integrable random variable ζ t +1 such that(4.5) U + t +1 (cid:0) x (1 + h π t +1 , R t +1 i ) (cid:1) ≤ ζ t +1 , for all π t +1 ∈ L ( F t ; Θ t +1 ∩ L t +1 ) . Therefore, an application of Fatou’s lemma, together with the continuity of U t +1 , yields thatlim sup k → + ∞ E (cid:2) U t +1 (cid:0) x (1 + h π n k t +1 ( x ) , R t +1 i ) (cid:1)(cid:12)(cid:12) F t (cid:3) ≤ E h lim sup k → + ∞ U t +1 (cid:0) x (1 + h π n k t +1 ( x ) , R t +1 i ) (cid:1)(cid:12)(cid:12)(cid:12) F t i = E (cid:2) U t +1 (cid:0) x (1 + h ˆ π t +1 ( x ) , R t +1 i ) (cid:1)(cid:12)(cid:12) F t (cid:3) . Together with (4.4), this shows that(4.6) U t ( x ) = E (cid:2) U t +1 (cid:0) x (1 + h ˆ π t +1 ( x ) , R t +1 i ) (cid:1)(cid:12)(cid:12) F t (cid:3) . Condition (4.3) implies that U t ( x ) < + ∞ a.s., for all x ≥
0, thus proving the well-posedness of(4.2). Moreover, the same arguments employed in [RS06, Lemma 2.5] allow to show that theoptimizer ˆ π t +1 ( x ) can be chosen F t ⊗B ( R + )-measurable. Since the set Θ t +1 ∩L t +1 is convex andwe assumed that U t +1 ∈ U , the function U t ( ω, · ) inherits the strict increasingness and concavityof U t +1 ( ω, · ), for a.e. ω ∈ Ω. Furthermore, U t ( x ) ≥ E [ U t +1 ( x ) |F t ] and, therefore, U t ( x ) is a.s.bounded from below, for every x >
0. In particular, this implies that U t ( x ) is a.s. finite valuedfor all x > , + ∞ ). To prove continuity at x = 0, note that U t (0) ≤ lim inf n → + ∞ U t (1 /n ). On the other hand, using (4.6), it holds thatlim sup n → + ∞ U t (1 /n ) = lim sup n → + ∞ E (cid:2) U t +1 (cid:0) (1 /n )(1 + h ˆ π t +1 (1 /n ) , R t +1 i ) (cid:1)(cid:12)(cid:12) F t (cid:3) ≤ E h lim sup n → + ∞ U t +1 (cid:0) (1 /n )(1 + h ˆ π t +1 (1 /n ) , R t +1 i ) (cid:1)(cid:12)(cid:12)(cid:12) F t i = E [ U t +1 (0) |F t ] = U t (0) , where, similarly as above, the inequality follows from Fatou’s lemma using (4.5) and the secondequality follows from the continuity of U t +1 together with the a.s. boundedness of Θ t +1 ∩ L t +1 .We have thus shown that U t ∈ U . To complete the proof of the inductive hypothesis, it remainsto show that (4.3) holds true for each t < T −
1. For every x > π t ∈ L ( F t − ; Θ t ∩ L t ),using repeatedly (4.6) and iterated conditioning, we have that(4.7) E (cid:2) U + t (cid:0) x (1 + h π t , R t i ) (cid:1)(cid:3) ≤ E (cid:20) U + (cid:18) x (1 + h π t , R t i ) T − t Y k =1 (1 + h ˆ π t + k ( V t + k − ) , R t + k i ) (cid:19)(cid:21) , with V t := x (1+ h π t , R t i ) and V t + k := V t + k − (1+ h ˆ π t + k ( V t + k − ) , R t + k i ), for k = 1 , . . . , T − t . Sincesup π ∈ Θ E [ U + ( xV πT )] < + ∞ , inequality (4.7) implies the validity of (4.3), for all t = 0 , , . . . , T − π ∗ = ( π ∗ t ) t =1 ,...,T ∈ Θ ∩ L is defined recursively by π ∗ t := ˆ π t ( V π ∗ t − ) , where V π ∗ t = V π ∗ t − (1 + h π ∗ t , R t i ) , for all t = 1 , . . . , T, and V π ∗ = 1 . The optimality of π ∗ follows by noting that, for any strategy π ∈ Θ, E [ U ( V πT )] ≤ E [ U T − ( V πT − )] ≤ . . . ≤ U (1) = E [ U ( V π ∗ )] = . . . = E [ U ( V π ∗ T )] . ( ii ) ⇒ ( i ): in view of Proposition 4.2, this implication follows by the same argument used forproving the implication ( ii ) ⇒ ( i ) in Theorem 2.5. (cid:3) While [RS06] work under no classical arbitrage and do not consider trading constraints, an inspection of theproof of their Lemma 2.5 shows that only the a.s. boundedness of the set of allowed strategies is needed. In ourcontext, the latter property holds under NA as a consequence of Proposition 4.2. Definition 4.5.
An adapted stochastic process Z = ( Z t ) t =0 , ,...,T satisfying Z t > t = 1 , . . . , T and Z = 1 is said to be a supermartingale deflator if ZV π is a supermartingale, forall π ∈ Θ. The set of all supermartingale deflators is denoted by D . An allowed strategy ρ ∈ Θis said to be a num´eraire portfolio if 1 /V ρ ∈ D , i.e., if V π /V ρ is a supermartingale.We now prove a version of the fundamental theorem of asset pricing based on NA in thepresence of convex constraints, extending Theorem 2.8 to the multi-period case. In a continuous-time semimartingale setting, the general version of this result is given in [KK07, Theorem 4.12].By relying on the same approach adopted in the proof of Theorem 2.8, we can give a simple andshort proof in a general discrete-time setting. Theorem 4.6.
The following are equivalent:(i) the NA condition holds;(ii) D 6 = ∅ ;(iii) there exists the num´eraire portfolio.Proof. ( i ) ⇒ ( iii ): let t ∈ { , . . . , T } and consider a family ( f n ) n ∈ N of measurable functionssuch that f n : R d → (0 ,
1] and E [log(1 + k R t k ) f n ( R t )] < + ∞ , for each n ∈ N , and f n ր n → + ∞ (see the proof of Theorem 2.8). For each n ∈ N , let U t,n ( ω, x ) := log( x ) f n ( R t ( ω )),for all ( ω, x ) ∈ Ω × (0 , + ∞ ). For each n ∈ N , it holds that U t,n ∈ U . By Proposition 4.2, NA implies that Θ t ∩ L t is a.s. bounded and, therefore, inequality (2.7) conditionally on F t − impliesthat ess sup π t ∈ L ( F t − ;Θ t ∩L t ) E [ U + t,n (1 + h π t , R t i ) |F t − ] < + ∞ a.s. Using again the boundednessof Θ t ∩ L t , this can be shown to imply the existence of an element ρ nt ∈ L ( F t − ; Θ t ∩ L t ) suchthat E (cid:2) U t,n (1 + h ρ nt , R t i ) (cid:12)(cid:12) F t − (cid:3) = ess sup π t ∈ L ( F t − ;Θ t ∩L t ) E (cid:2) U t,n (1 + h π t , R t i ) (cid:12)(cid:12) F t − (cid:3) a.s.By the same reasoning as in (2.8)-(2.9) (now conditionally on F t − ), we obtain that E (cid:20) h π t − ρ nt , R t i h ρ nt , R t i f n ( R t ) (cid:12)(cid:12)(cid:12)(cid:12) F t − (cid:21) ≤ , for all π t ∈ Θ t and n ∈ N . Since Θ t ∩ L t is bounded and closed, we can assume that ( ρ nt ) n ∈ N converges a.s. to an element ρ t ∈ L ( F t − ; Θ t ∩ L t ) as n → + ∞ (up to passing to a suitable subsequence, see [FS16, Lemma1.64]). Since f n ր n → + ∞ , an application of Fatou’s lemma gives that E (cid:20) h π t − ρ t , R t i h ρ t , R t i (cid:12)(cid:12)(cid:12)(cid:12) F t − (cid:21) ≤ , for all π t ∈ L ( F t − ; Θ t ) . Let π = ( π t ) t =1 ,...,T ∈ Θ. Then, for each t ∈ { , . . . , T − } , the last inequality implies that E (cid:20) V πt V ρt (cid:12)(cid:12)(cid:12)(cid:12) F t − (cid:21) = V πt − V ρt − E (cid:20) h π t , R t i h ρ t , R t i (cid:12)(cid:12)(cid:12)(cid:12) F t − (cid:21) ≤ V πt − V ρt − a.s. , thus proving that the strategy ρ = ( ρ t ) t =1 ,...,T corresponds to the num´eraire portfolio.( iii ) ⇒ ( ii ): this implication is immediate by Definition 4.5.( ii ) ⇒ ( i ): this implication follows by the same argument used in the proof of Theorem 2.8. (cid:3) Finally, we mention that the proof of Theorem 2.11 can be similarly extended to the multi-period case, thus providing a utility maximization proof of the fundamental theorem of assetpricing for no classical arbitrage, in the spirit of [Rog94] (see also [KaS09, Section 2.1.4]). Theo-rem 2.14 also admits a direct extension to the multi-period setting, with an identical statement.
RBITRAGE CONCEPTS UNDER TRADING RESTRICTIONS IN DISCRETE-TIME FINANCIAL MARKETS25
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