Risk Arbitrage and Hedging to Acceptability Under Transaction Costs

arXiv:1605.07884v3 [q-fin.MF] 15 Apr 2020 Keywords non possibly and unbounded space. T for Euclidean stronger. decre in results a become sets on to conditions random lead relies arbitrage assets technique the no of the ematical basket assessin and a of prices or possibilities asset hedging any the to general, turn respect in framework with However, our multiv asset, setting. fixed when deals single case good a co special into t arbitrage converted the are describe In (risk) sitions We them. no measure. between several risk connections formulate dynamic explore ac multivariate prices, a of superhedging to family allowing a respect of po and directions, set with the two solvency e.g. a of in of positions, increments sum solven setting the that to has this assuming belong cess process and extend costs portfolio We transaction feasible step. convex a each that at assumption ments the on relies -al [email protected] E-mail: Switzerland Bern, Scienc 3012 Actuarial and 22, Statistics Mathematical of Institute Molchanov I. 2092-ElMan University, [email protected] Manar E-mail: Tunis-El GOSAEF, and De France, Lattre Maréchal De du Place University, Paris-Dauphine Lepinette E. Lepinette Emmanuel under Acceptability to Costs Hedging Transaction and Arbitrage Risk Abstract 2020 16, April oddeal good ahmtc ujc lsicto (2010) Classification Subject Mathematics wl eisre yteeditor) the by inserted be (will No. manuscript Noname · h lsia iceetm oe fpootoa rnato cos transaction proportional of model time discrete classical The ovnyset solvency cetneset acceptance · · admset random laMolchanov Ilya · ikarbitrage risk · rnato costs transaction · ikmeasure risk ,Uiest fBr,Alpeneggstr. Bern, of University e, 12,6D5 60G42 60D05, 91G20, asgy 57 ai ee 16, cedex Paris 75775 Tassigny, r Tunisia ar, · superhedging dtosand nditions s fsuper- of ase noteno the into s toi pro- rtfolio raepo- ariate h risk the g emath- he ceptable incre- t esets he -closed · for ts 2 Emmanuel Lepinette, Ilya Molchanov

1 Introduction

Transaction costs in financial markets are often described using solvency sets, which consist of all financial positions (in physical quantities) regarded better than the zero position or at least equivalent to it. In the dynamic discrete time setting, the solvency sets form a set-valued random process (Kt)t=0,...,T adapted to the underlying filtration (Ft)t=0,...,T . The no arbitrage conditions are usually formulated in terms of selections of these solvency sets, that is, for random vectors that a.s. belong to the solvency sets and so correspond to particular choices of solvent portfolios. In many cases, solvency sets are polyhedral cones and the corresponding model is known as Kabanov’s model with proportional transaction costs, see [23,24,32], where the no arbitrage conditions are thoroughly discussed. If ξ is a claim that matures at time T , then the set of initial positions suitable as a starting value for a self-financing portfolio process (Vt)t=0,...,T paying ξ at maturity forms the family of superhedging prices for ξ. In the multivariate setting, the starting values are vectors which are not necessarily comparable to each other, and so, instead of comparing them by a single numerical quantity, it is sensible to look at the whole set of superhedging prices. The self-financing requirement amounts to the fact that the increment Vt−1 − Vt of the portfolio process is solvent at all times, that is, it a.s. belongs to Kt for all t (in other words, the increment is a selection of Kt). In order to reduce these superhedging prices, it is possible to require that the shortfall of the terminal value of the portfolio, in comparison with the claim, is acceptable with respect to a certain risk measure. This approach may provide arbitrage opportunities as Good Deals, i.e. terminal claims attainable from the zero capital and such that the risk of the claim is strictly negative, equivalently, the utility is strictly positive. The No Good Deal condition, first introduced in [11] and then formalised in [4,10], requires that this situation is impossible. Unlike the univariate setting, the existence of a good deal in the multivariate setting does not necessarily mean the existence of a claim whose Rd multivariate utility belongs to +. Indeed, a vector-valued financial position may be acceptable if some acceptable components compensate for the non- acceptable ones. This may result in various types of arbitrage opportunities. Indeed, it is possible to strengthen the no arbitrage requirement by also considering hedging strategies, where the self-financing condition is replaced by the acceptability of all intermediate portfolio changes with respect to a dynamic risk measure, see [7]. The setting of [7,8] involves at least two assets exchangeable without transaction costs and pinpoints a particular asset that is used as the cash equivalent. The portfolio is converted to its cash equivalent, with the acceptability condition imposed on the increments of these cash values for consecutive time moments. The idea of converting portfolios to a single numerical quantity with acceptable increments in view of superhedging one- dimensional claims has been further explored in [5]. However, if there are several currencies (exchangeable with random frictionless rates or with transaction costs), it may well be the case that the Risk arbitrage 3

position expressed in one currency is acceptable, while the position in the other one is not, see [29, Ex. 1.1]. This may lead to regulatory arbitrages, see [34]. If the regulator is prepared to apply a relaxed acceptability criterion for one currency, then it would be logical to expect the same policy with respect to another currency or a basket of currencies. We show how to handle this in a way that treats all components of a portfolio in the same manner. The key idea of this work is to extend the family of self-ﬁnancing portfolio processes by requiring that Vt−1 − Vt equals the sum of a selection of Kt (a solvent position) and another random vector that is not necessarily solvent, but is acceptable with respect to a dynamic multivariate risk measure. It is worth mentioning that Kt is only supposed to be convex, contrarily to the classical literature of linear transaction costs. With this hedging to acceptability approach, all components of the portfolio are treated in the same way. Then, (Vt)t=0,...,T is called an acceptable portfolio process. For example, the classical superhedging setting arises if the componentwise conditional essential inﬁmum is chosen as the risk measure, so that acceptable random vectors necessarily have all a.s. non-negative components. The hedging to acceptability substan- tially increases the choice of possible hedging strategies, but in some cases may lead to arbitrage.

Example 1 Let r be any coherent risk measure. Consider the one period zero interest model with two currencies as the assets. Assume that the exchange rate π (so that π units of the second asset buy one unit of the first one) at time one is log-normally distributed (in the real world) and the exchanges are free from transaction cost. Then, the positions γ′ = (−a,πa) and γ′′ = (a, −πa) for a> 0 are reachable from (0, 0) at zero cost. Their risks are (a,ar(π)) and (−a,ar(−π)). In order to secure the capital reserves for γ′, the agent has to reserve a of the first currency and ar(π) of the second one (note that r(π) < 0). If the exchange rate at time zero is π0, the initial cost expressed in the second currency is π0a + ar(π)= a(π0 + r(π)). ′′ In order to secure γ , the initial cost is a(−π0 + r(−π)). If π0 does not belong to the interval [−r(π), r(−π)], then either π0 + r(π) < 0 or −π0 + r(−π) < 0, and we let a grow to release infinite capital at time zero. Note that this model does not admit financial arbitrage, since there exists a martingale measure. This example can be easily modified by accounting for proportional transaction costs.

It is recognised by now that risks of multivariate positions involving possible exchanges of assets and transaction costs are described as sets, see [2,19]. The multiasset setting naturally makes it possible to oﬀset a risky position using various combinations of assets. In this framework it is also natural to consider the family of all attainable positions as a set-valued portfolio. Treat- ing both arguments and values of a risk measure as random sets leads to law invariant risk measures and makes it possible to iterate the construction, which is essential to handle dynamic risk measures. 4 Emmanuel Lepinette, Ilya Molchanov

One of the aims of this paper is to introduce a geometric characterisation of superhedging prices. On the way, we suggest a constructive definition of dynamic multivariate risks based on the families of acceptable positions and so extend the existing works on dynamic set-valued risk measures [13,14] by letting the arguments of risks and their values be sets of random vectors in Rd. In many instances, these sets may be interpreted as random (possibly, non-closed) sets. The necessary background on random sets is provided in the appendix. In particular, it is shown that the Minkowski (elementwise) sum of two random closed sets is measurable, no matter if the sum is closed or not. Special attention is devoted to the decomposability and infinite decomposability properties, which are the key concepts suitable to relate families of random vectors to selections of random sets. We refer to [12] and [17] for the basics of static risk measures and to [1] for a survey of the dynamic L∞-setting, further extended beyond the L∞-setting by the module approach worked out in [15,16]. Static risk measures are usually defined on Lp(R, F) with p ∈ [1, ∞]. How- ever, in many cases, they are well defined also on larger sets of random variables. For example, r(ξ) = −ess infξ makes sense for all random variables essentially bounded from below by a constant. Similarly, if r(ξ) = −Eξ, then the acceptance set is defined as the family of ξ such that their positive and negative parts satisfy Eξ+ ≥ Eξ− > ∞. The boundedness of Eξ+ is not required. To account for similar effects in relation to multivariate dynamic risk measures, we put forward acceptance sets in place of risk measures. The acceptance sets Ct,s with t ≤ s are subsets of the sum of the family of Fs-measurable random vectors in Rd that admit generalised conditional pth moment with respect Rd to Ft and the family of all Fs-measurable random vectors in +. Section 2 introduces basic conditions on the acceptance sets and several optional ones. 0 d The dynamic selection risk measure Rt,s(Ξ) for a family Ξ ⊂ L (R , FT ) 0 d is introduced as the closure in probability of (Ξ + Ct,s) ∩ L (R , Ft). If Ξ = 0 L (X, FT ) is the family of selections for a random closed set X, then Rt,s(X)= Rt,s(Ξ) itself is an Ft-measurable random closed set. In comparison with [14], this approach explicitly defines a set-valued dynamic risk measure instead of imposing on it some axiomatic properties. This yields a set-valued risk measure with a set-valued argument that can be naturally iterated in the dynamic framework. The conditional convexity of the acceptance sets yields that

Rt,s(λX + (1 − λ)Y ) ⊃ λRt,s(X)+(1 − λ)Rt,s(Y ) a.s.

0 for any λ ∈ L ([0, 1], Ft) and random closed sets X and Y , meaning that the risk measure is also conditionally convex. The static case of this construction was considered in [29], where properties of selection risk measures in the coherent case are obtained, some of them easily extendable for the dynamic convex setting. Comparing to [29], we work with solvency sets instead of portfolios available at price zero and also allow the argument of the risk measure to be a rather general family of random vectors. Risk arbitrage 5

The hedging to acceptability relies on a sequence (Kt)t=0,...,T of solvency sets and the acceptance sets Ct,s for 0 ≤ t ≤ s ≤ T . Note that the solvency sets are not assumed to be conical, since non-conical models naturally ap- pear, e.g. in the order book setting, see [3] and [30]. An acceptable portfolio process (Vt)t=0,...,T introduced in Section 3 satisfies Vt−1 − Vt = kt + ηt for 0 kt ∈ L (Kt, Ft), ηt ∈Ct−1,t, and all t. In other words, the available assets do suffice to pay for the portfolio at the next step up to an amount acceptable with respect to some risk measure. The strongest acceptability condition as- sumes that Ct−1,t consists of random vectors with non-negative components and yields the classical arbitrage theory for markets with transaction costs, see [24]. The weakest acceptability requirement presumes that all ηt from Ct−1,t have non-negative Ft−1-conditional expectations. ξ If ξ is a terminal claim on d assets, then Ξt denotes the set of all initial endowments at time t that ensure the existence of an acceptable portfolio ξ process paying ξ at maturity, that is, VT ∈ ξ + KT a.s. Equivalently, Ξt is the family of Ft-measurable elements of (ξ − At,T ), where At,T is the set of claims ξ attainable at time T starting from zero investment at time t. The family Ξt may be used to assess the risk associated with ξ at time t. The no arbitrage conditions we study are imposed on the set of super- 0 hedging prices Ξt for the zero claim ξ = 0. They may be reformulated as no arbitrage conditions on the set of attainable claims, which are weaker than the usual ones of the literature. These no risk arbitrage conditions are introduced and analysed in Section 4. In difference to [7], we do not rely on the weak compactness of the duals to the acceptance sets and we do not need to pinpoint any reference asset. It should be noted that the risk arbitrage only makes sense in the multiasset setting with some trading opportunities between the assets; Rd if Kt = + (which is always the case on the line), then all no risk arbitrage conditions automatically hold. It is shown that, in some cases, it is possible to represent the families of capital requirements as a set-valued process, and the no risk arbitrage conditions, for linear transaction costs, can be characterised in terms of weakly consistent price systems, so providing a variant of the Fundamental Theorem of Asset Pricing in our framework, see Theorem 4.4. A comparison of our approach with the no good deals setting is provided in Section 5. It is shown that our approach imposes stronger no arbitrage conditions that are more difficult to check, but which result in lower superhedging prices.

Note that the sets Ct,s of acceptable positions always contain the family 0 Rd L ( +, Fs) of random vectors with a.s. non-negative components and, in many cases, Ct,s is a subset of the family of random vectors with non-negative generalised conditional expectation given Ft. Thus, the no risk arbitrage conditions are sandwiched between those for the risk measure based on the conditional essential infimum and on the conditional expectation. The first choice corresponds to the classical financial arbitrage with transaction costs, where our no risk arbitrage conditions become the classical no arbitrage conditions. 6 Emmanuel Lepinette, Ilya Molchanov

Section 6 recovers and extends several results from [24]. In this classical setting, our approach yields a new geometric interpretation of the sets of superhedging prices with possibly non-conical solvency sets; it is formulated using the concept of the conditional core of a random set elaborated in [26]. The result applies also in some cases when the classical consistent price systems characterisation fails. In Section 7 we characterise no arbitrage conditions arising by adopting the generalised conditional expectation as the acceptability criterion. These are the strongest no arbitrage conditions in our framework; their validity ensures the absence of arbitrage for all acceptance criteria satisfying a dynamic version of the dilatation monotonicity condition from [9]. The results from Sections 6 and 7 are illustrated on a two-asset example in Section 8.

2 Dynamic acceptance sets and selection risk measures

2.1 Deﬁnition and main properties

Let (Ω, (Ft)t=0,...,T , P) be a stochastic basis on a complete probability space such that F0 is the trivial σ-algebra and FT = F. In the following, we endow random vectors and events with a subscript that indicates the σ-algebras they are measurable with respect to. The subscript is often omitted for FT - measurable random vectors. Let Lp(Rd, F) with p ∈ [1, ∞] be the family of p-integrable random vectors (essentially bounded if p = ∞), and let L0(Rd, F) be the family of all random vectors in Rd. The closure in the strong topology in Lp(Rd, F) for p ∈ [1, ∞) is 0 d denoted by clp, and cl0 is the closure in probability in L (R , F). If p = ∞, the closure is considered with respect to the a.s. convergence of uniformly bounded sequences. p Rd For a sub-σ-algebra H⊂F, denote LH( , F) the module of F-measurable random vectors that can be represented as γξ with ξ ∈ Lp(Rd, F) and γ ∈ 0 R 1 Rd L ( , H), see [15, Ex. 2.5]. In particular, LH( , F) is the family of all ξ that admit the generalised conditional expectation Eg(ξ|H) with respect to H, see [26, Lemma B.3]. Following [15, Ex. 2.5], the module norm is deﬁned by

E(kξkp|H)1/p, p ∈ [1, ∞), |||ξ|||p,H = (2.1) (ess supHkξk, p = ∞, where esssupHkξk is the H-measurable essential supremum of kξk, see [17, p Rd p Appendix A.5]. We endow the space LH( , F) with the topology of LH- n n convergence by assuming that ξ converges to ξ if |||ξ − ξ|||p,H → 0 in probability if p ∈ [1, ∞). If p = ∞, we use the Ft-bounded convergence in prob- n n ability, meaning that ess supHkξ k is bounded and |||(ξ − ξ) ∧ 1|||1,H → 0 in probability as n → ∞. Risk arbitrage 7

p p Rd p p Rd Denote shortly L = L ( , F), Lt,s = LFt ( , Fs) for t ≤ s, and let p p Rd Lt,s = LFt ( , Fs) be the family of random vectors ξs that can be decomposed ′ ′′ ′ p ′′ 0 Rd as ξs = ξs + ξs , where ξs ∈ Lt,s and ξs ∈ L ( +, Fs). Following the classical definition of an acceptance set in the theory of risk measures, we introduce the acceptance set Ct,s for t ≤ s as the collection of all Fs-measurable financial positions regarded acceptable at time t. Definition 2.1 Discrete time Lp-dynamic convex acceptance sets are a family p {Ct,s, 0 ≤ t ≤ s ≤ T }, such that Ct,s ⊂ Lt,s and the following properties hold for all 0 ≤ t ≤ s ≤ T . 0 Rd 0 Rd (i) Normalisation: Ct,t = L ( +, Ft), Ct,s ⊃ L ( +, Fs), and 0 Rd Ct,s ∩ L ( −, Ft)= {0}. (ii) Integrability: p 0 Rd Ct,s = (Ct,s ∩ Lt,s)+ L ( +, Fs). (2.2) p p (iii) Closedness: Ct,s ∩ Lt,T is closed in Lt,T . 0 ′ ′′ (iv) Conditional convexity: for all αt ∈ L ([0, 1], Ft), and ηs, ηs ∈Ct,s, ′ ′′ αtηs + (1 − αt)ηs ∈Ct,s. 0 d (v) Weak time consistency: Ct,s∩L (R , Fu)= Ct,u for all 0 ≤ t ≤ u ≤ s ≤ T . p 0 Rd (vi) Compensation: if ξs ∈ Lt,s, then (ξs + Ct,s) ∩ L ( , Ft) 6= ∅.

The integrability property implies that Ct,s is an upper set, that is, ηs ∈ ′ Ct,s and ηs ≤ ηs a.s. (all inequalities are understood coordinatewisely) yield ′ p ηs ∈ Ct,s. The compensation property implies that, for all ξs ∈ Lt,s, there 0 d exists γt ∈ L (R , Ft) such that γt + ξs ∈ Ct,s, i.e. it is possible to make the financial position ξs acceptable by adding the position γt. In the following, we need only the acceptance sets Ct−1,t for t =1,...,T . Example 2 (Static univariate convex risk measures) Consider the one-period setting in one dimension with t = 0, 1. If r is a convex Lp-risk measure with p p ∈ [1, ∞), then its acceptance set C0,1 ∩ L (R, F1) is the family of η1 ∈ p L (R, F1) such that r(η1) ≤ 0. The lower semicontinuity of r is equivalent to the closedness of the acceptance set. The conditional convexity property of the acceptance set is equivalent to the convexity property of the risk measure. The compensation property corresponds to the finiteness of r. The following result refers to the infinite decomposability property (see Definition 9.2), also known as the countable concatenation property [15] or σ-stability [16].

Lemma 2.1 The families Ct,s are inﬁnitely Ft-decomposable for all 0 ≤ t ≤ s ≤ T .

i p i Proof If ηs ∈Ct,s ∩ Lt,s and Bt ∈Ft, i ≥ 1, then n n i 1 η¯ = 1 i η + η 1 n i ∈C s Bt s s Ω\∪i=1Bt t,s i=1 X 8 Emmanuel Lepinette, Ilya Molchanov

p by the conditional convexity property, so that Ct,s ∩ Lt,s is Ft-decomposable. n ∞ i Sinceη ¯ → η¯ = 1 i η in the ||| · ||| -norm if p ∈ [1, ∞) and F - s s i=1 Bt s p,Ft t boundedly in probability if p = ∞, we haveη ¯ ∈C ∩Lp . By the integrability P s t,s t,s property, Ct,s is also inﬁnitely decomposable.

Deﬁnition 2.2 The family of dynamic convex acceptance sets is called

0 (i) coherent if αtηs ∈Ct,s for all t ≤ s, αt ∈ L ([0, ∞), Ft), and ηs ∈Ct,s; n (ii) continuous from below at zero if, for all t ≤ s, and any sequence ξs ∈ p Rd n LFt ( −, Fs), n ≥ 1, with |||ξs |||p,Ft → 0 in probability as n → ∞, there n 0 Rd R n n exists a sequence γt ∈ L ( +, Ft), n ≥ 1 and k ∈ +, such that γt +ξs ∈ n n Ct,s and kγt k≤ k|||ξs |||p,Ft a.s. for all n.

If p = ∞ then the continuity from below always holds and is easily veriﬁed n n by choosing γt with all identical components being |||ξs |||∞,Ft .

Example 3 The acceptance sets can be deﬁned using a univariate convex dy- p p namic L -risk measure (rt)t=0,...,T , so that Ct,s ∩ Lt,s is the dth Cartesian p power of the acceptance set for rt. Equivalently, ξs ∈Ct,s ∩ Lt,s if and only if all components of ξs are acceptable under rt. The continuity from below property (with p ∈ [1, ∞)) holds if rt is lower semicontinuous in the |||· |||p,Ft -norm and continuous from below, which is the case if rt is convex and a.s. ﬁnite, see [33, Th. 4.1.4].

Example 4 (Dual construction of conditional acceptance sets) Let p = ∞, and 1 Rd consider families Zt,s ⊂ LFt ( +, Fs) with 0 ≤ t ≤ s ≤ T such that Zt,u ⊂ Zt,s g for all t ≤ u ≤ s, and E (ζs|Ft)=(1,..., 1) for all ζs ∈ Zt,s. Note that we do not assume that Zt,s is weakly compact. Deﬁne

0 Rd ∞ g Ct,s = L ( +, Fs)+ ηs ∈ Lt,s : E (hζs, ηsi|Ft) ≥ 0 , ζs∈Zt,s \

where hζs, ηsi is the scalar product. It is easily seen that conditions (i), (ii), (iv) and (v) of Deﬁnition 2.1 hold and these acceptance sets are coherent. n n n If ξs → ξs in probability for ξs ∈Ct,s, and all ξs are bounded in the norm 0 g by γt ∈ L (R+, Ft), then E (hζs, ξsi|Ft) ≥ 0 by the dominated convergence theorem for generalised conditional expectations. Thus, condition (iii) also hold. ∞ If ξs ∈ Lt,s, then the components of ξs are bounded in the absolute value 0 Rd by ηt ∈ L ( +, Ft). Then, ηt − ξs is non-negative and so belongs to Ct,s, and 0 d ξs + (ηt − ξs) ∈ L (R , Ft). Thus, (vi) also holds.

2.2 Dynamic selection risk measures

0 d Let ΞT be an upper subset of L (R , FT ), that is, with each ξ ∈ ΞT , the family ′ 0 Rd ΞT also contains all ξ ∈ L (ξ + +, FT ). The most important example of such Risk arbitrage 9

0 1 family is the family of selections L (XT , FT ) for an FT -measurable upper Rd Rd random set XT in , that is, XT + + ⊂ XT a.s. If XT is also closed, then its centrally symmetric version (−XT ) is a set-valued portfolio in the terminology of [29].

0 d Deﬁnition 2.3 Let ΞT ⊂ L (R , FT ) be an upper set. For t ≤ s ≤ T , 0 0 Rd Rt,s(ΞT ) = (ΞT + Ct,s) ∩ L ( , Ft) (2.3)

0 d denotes the set of all γt ∈ L (R , Ft), such that γt − ξ ∈Ct,s for some ξ ∈ ΞT . 0 0 Let Rt,s(ΞT ) denote the closure in probability of Rt,s(ΞT ). If ΞT = L (XT , FT ) 0 is the family of selections of an upper random set XT , we write Rt,s(XT ) and 0 Rt,s(XT ) instead of Rt,s(ΞT ) and Rt,s(ΞT ). In view of this, Rt,s(XT ) (and also Rt,s(ΞT )) is called the dynamic selection risk measure. 0 0 0 0 Rd 0 Note that RT,T (ΞT )= ΞT , Rt,s(ΞT )= Rt,s(ΞT ∩L ( , Fs)), and Rt,u(ΞT ) ⊂ 0 Rt,s(ΞT ) for all 0 ≤ t ≤ u ≤ s ≤ T . If only portfolios from a random set Mt are allowed for compensation at time t, as it is the case in [14], it is easy to 0 modify (2.3) by intersecting (ΞT + Ct,s) with L (Mt, Ft). The empty selection risk measure corresponds to completely unaccept- able positions. The compensation property of acceptance sets guarantees that 0 p Rt,s(ΞT ) is not empty if ΞT ∩ Lt,s 6= ∅. The family ΞT is said to be acceptable 0 for the time horizon s if 0 ∈ Rt,s(ΞT ), equivalently, −ΞT contains an element from Ct,s. The dynamic selection risk measure is conditionally convex, that is, 0 ′ ′′ 0 ′ 0 ′′ Rt,s(αtΞT + (1 − αt)ΞT ) ⊃ αtRt,s(ΞT )+(1 − αt)Rt,s(ΞT ) 0 for all αt ∈ L ([0, 1], Ft), and the same holds for the closures. The next result follows from Lemma 2.1.

0 Lemma 2.2 If ΞT is inﬁnitely Ft-decomposable, then Rt,s(ΞT ) and Rt,s(ΞT ) are also inﬁnitely Ft-decomposable.

Lemma 2.3 Let XT be an FT -measurable random upper closed set.

(i) Rt,s(XT ) is the family of selections of an Ft-measurable random upper d closed set in R , which is denoted by Rt,s(XT ). 0 (ii) If XT is a.s. convex, then Rt,s(XT ) is a.s. convex. If XT is a cone and 0 the acceptance sets are coherent, then Rt,s(XT ) is a cone.

Proof (i) By Lemma 2.2, Rt,s(XT ) is an Ft-decomposable family, and so The- orem 9.1 applies. 1 2 0 i i 1 2 0 (ii) If γt ,γt ∈ Rt,s(XT ), then γt −ξs ∈Ct,s, i =1, 2, for ξs , ξs ∈ L (XT , Fs). 1 For any t ∈ (0, 1), the conditional convexity property yields that tγt + (1 − 2 1 2 0 t)γt − ξs ∈Ct,s with ξs = tξs + (1 − t)ξs ∈ L (XT , Fs). The conical property is trivial. Rd p 0 Example 5 If Xs = ξs + + for ξs ∈ Lt,s, then Rt,s(Xs)= Rt,s(Xs)= rt(−ξs)+ Rd p + for a vector-valued dynamic risk measure rt on Lt,s, see [33]. 1 The (graph) measurability of a random set is deﬁned in the appendix. 10 Emmanuel Lepinette, Ilya Molchanov

3 Hedging to acceptability

3.1 Acceptable portfolio process

Let (Kt)t=0,...,T be a sequence of random closed convex sets, such that, for all Rd t, we have Kt ∩ − = {0}, Kt is an upper set, and Kt is Ft-measurable. The set Kt is understood as the family of all solvent positions at time t expressed in physical units and is called a solvency set, see [24]. If the solvency sets are cones, this model is well studied under the name of Kabanov’s model; it describes the market subject to proportional transaction costs, see [24,32]. If the solvency sets are cones and the acceptance sets are coherent, we talk about the coherent conical setting. 0 Let Kt be the largest Ft-measurable linear subspace contained in Kt, that is, 0 Kt = cKt = cKt, c\6=0 c∈Q\\{0} which is also a random closed set. The solvency sets are said to be proper if 0 ˜ Kt = {0} and strictly proper if Kt = Kt ∩ (−Kt) = {0} for all t = 0,...,T . ˜ 0 0 ˜ ˜ If Kt is a cone, then Kt = Kt , while in general Kt ⊂ Kt. Since Kt is convex and origin symmetric, Kt is proper if and only if K˜t is bounded.

0 d Deﬁnition 3.1 A sequence Vt ∈ L (R , Ft), t =0,...,T , is called an acceptable portfolio process if

0 Vt−1 − Vt ∈ L (Kt, Ft)+ Ct−1,t, t =1,...,T. (3.1) By the deﬁnition of the selection risk measure, (3.1) is equivalent to

0 Vt−1 ∈ Rt−1,t(Vt + Kt), t =1,...,T. (3.2)

Thus, paying transaction costs, it is possible to transform Vt−1 − Vt into an acceptable position for the horizon t. Equivalently, Vt−1 does suﬃces to pur- chase Vt + kt + ηt for some kt ∈ Kt and ηt ∈Ct−1,t. The initial endowment at 0 time t is any Vt− ∈ L (Vt + Kt, Ft), so that it is possible to convert Vt− into Vt paying the transaction costs.

3.2 Attainable positions and superhedging

The family of attainable positions at time s>t is the family of random vectors that may be obtained as Vs for acceptable portfolio processes starting from zero investment at time t. By (3.1), the family of attainable positions is given by s s−1 0 At,s = L (−Ku, Fu) − Cu,u+1. u=t u=t X X 0 d Let ξ ∈ L (R , FT ) be a terminal claim (or payoﬀ). The hedging to acceptability aims to come up with an acceptable portfolio process (Vt)t=0,...,T Risk arbitrage 11

that guarantees paying ξ in the sense that the terminal wealth VT belongs ξ 0 ξ to ΞT = L (XT , FT ), being the family of selections of the random closed set ξ XT = ξ + KT . Deﬁne recursively

ξ 0 0 ξ Ξt = L (Kt, Ft)+ Rt,t+1(Ξt+1), t = T − 1,..., 0, (3.3) which is the set of time t superhedging prices for the claim ξ. The family ξ Ξt consists of the time t superhedging prices for ξ and so may serve as a 0 d dynamic convex risk measure of ξ with values being subsets of L (R , Ft). ′ ′′ ′ p Rd ′′ 0 Rd If ξ = ξ − ξ for ξ ∈ L ( , FT ) and ξ ∈ L ( +, FT ), the compensation ξ property of acceptance sets ensures that Ξt is not empty for all t. ˆξ ξ In order to handle the asymptotic version of the risk measure, let ΞT = ΞT , and further ˆξ 0 ξ Ξt = L (Kt, Ft)+ Rt,t+1(Ξt+1), t = T − 1,..., 0. (3.4) ξ ˆξ ξ ˆξ ξ Note that Ξt ⊂ Ξt ⊂ cl0(Ξt ), whence cl0(Ξt ) = cl0(Ξt ) for all t. The families 0 ˆ0 Ξt and ΞT arise by letting ξ = 0 a.s.

0 ξ ξ ˆξ Lemma 3.1 (i) The families Rt,s(Ξs ), Rt,s(Ξs ), and Ξt are convex and inﬁnitely Ft-decomposable for all 0 ≤ t ≤ s ≤ T . ξ (ii) For each t ≤ T , there exists a (possibly non-closed) random set Xt such ˆξ 0 ξ that Ξt = L (Xt , Ft). (iii) For any t ≤ T , the family of all initial endowments Vt− at time t, allowing to start an acceptable portfolio process (Vs)t≤s≤T such that VT ∈ 0 ξ L (ξ + KT , FT ) a.s., coincides with Ξt and ξ 0 Rd Ξt = (−At,T + ξ) ∩ L ( , Ft). (3.5)

ξ 0 0 (iv) If KT is a cone, then Ξt ⊂ Ξt for any ξ ∈ L (KT , FT ). Proof (i) follows from Lemma 2.2. ξ (ii) The existence of Xt is trivial for t = T . Suppose that it holds at time t. The result for t − 1 follows from the induction assumption and (3.4) by Lemma 9.1. (iii) follows from the fact that (γT + At,T ) ∩ (ξ + KT ) 6= ∅ if and only if γT ∈ (−At,T + ξ). (iv) follows from (iii), since ξ + KT ⊂ KT a.s.

Rd Example 6 If Kt = + a.s. for all t (which is always the case if d = 1), then an acceptable portfolio process satisﬁes Vt−1 − Vt ∈Ct−1,t for all t =1,...,T . Then, T −1 ξ 0 Rd Ξt = (ξ + Cs,s+1) ∩ L ( , Ft). s=t X 0 Rd 0 Rd Since Ct−1,t ∩ L ( , Ft−1) = L ( +, Ft−1) for all t ≥ 1 by the weak time consistency and normalisation properties, the induction argument yields that 12 Emmanuel Lepinette, Ilya Molchanov

0 0 Rd R0 ξ Ξt = L ( +, Ft). If ξ does not a.s. vanish, the set t,s(ΞT ) becomes non- trivial. Its static variant is called a regulator risk measure in [20]; it only takes into account the acceptability requirement and disregards any trading opportunities between the components. In the terminology of [20], R0,1(ξ+K1) (in the static setting with a conical K1) is called the market extension of the regulator risk measure.

4 Risk arbitrage

0 Recall that Ξt is the set of time t super-hedging prices for the zero claim. By (3.5), 0 0 Rd Ξt = (−At,T ) ∩ L ( , Ft). (4.1) For multivariate ﬁnancial models, e.g. models with proportional transaction costs, several no arbitrage conditions have been considered. In Kabanov’s model, there is the NA condition, its robust version NAr, but also the NA2 condition derived using an alternative approach, see [31] and [24]. All these conditions are formulated in terms of the set of all terminal claims At,T attainable from the zero initial endowment. Here, we consider weaker no arbitrage conditions imposed on the super-hedging prices for the zero claim.

Deﬁnition 4.1 The multiperiod model satisﬁes ˆ0 0 0 0 (SNR) (strict no risk arbitrage) if Ξt ∩ L (−Kt, Ft) ⊂ L (Kt , Ft) for all t =0,...,T ; 0 0 Rd (NRA) (no risk arbitrage) if Ξt ∩ L ( −, Ft)= {0} for all t =0,...,T ; 0 0 Rd (NARA) (no asymptotic risk arbitrage) if (cl0Ξt ) ∩ L ( −, Ft)= {0} for all t =0,...,T ; (NRA2) (no risk arbitrage opportunity of the second kind) if, for any t = 0 d 0 0,...,T and ηt ∈ L (R , Ft) such that (ηt + At,T ) ∩ L (KT , FT ) 6= ∅, we 0 have ηt ∈ L (Kt, Ft)+ Ct−1,t; T 0 (SNAR) (strong no risk arbitrage) if t=0(kt + ηt) = 0 for kt ∈ L (Kt, Ft) 0 0 and ηt ∈Ct−1,t for all t implies that kt ∈ L (Kt , Ft) and ηt = 0 a.s. for all t. P

ˆ0 0 Let us comment on the (SNR) condition. If pt ∈ Ξt ∩ L (−Kt, Ft), then, starting from the zero initial endowment at time t expressed as 0 = pt − pt, we obtain the zero claim at time T from the price pt and allowing an immediate possible proﬁt at time t, since the liquidation value of −pt ∈ Kt is non-negative. A similar interpretation applies for the (NRA) condition and its asymptotic version (NARA). The (NRA2) condition may be compared to the (NA2) condition of [31], while (SNAR) is a version of [24, Condition (iii), 0 Rd Section 3.2.2]. Note that (NRA) condition is equivalent to At,T ∩L ( +, Ft)= 0 Rd {0}, while the usual (NA) condition At,T ∩ L ( +, FT )= {0} is stronger, see [24, Section 3.2.1]. Example 1 shows that it may be possible to release inﬁnite capital from zero position without compromising the acceptability criterion, in particular, Risk arbitrage 13

it violates the (SNR) condition. By Lemma 3.1, (SNR) can be written as 0 0 0 Rd Xt ∩(−Kt) ⊂ Kt a.s., and (NARA) as (clXt )∩ − = {0} a.s. It is obvious that (NARA) is stronger than (NRA). By (4.1), (NRA) condition is equivalent to 0 Rd Rd 0 0 Rd At,T ∩L ( +, Ft)= {0} for all t. If Kt = + a.s. for all t, then Ξt = L ( +, Ft) and all no arbitrage conditions are satisﬁed, see Example 6.

Lemma 4.1 (SNR) implies that

0 0 0 0 Rt,t+1(Ξt+1) ∩ L (−Kt, Ft) ⊂ L (Kt , Ft), t =0,...,T − 1. The reverse implication holds if the solvency sets are strictly proper.

0 0 0 0 Proof Denote M = Rt,t+1(Ξt+1), A = L (Kt, Ft), and B = L (Kt , Ft). It is easily seen that M ∩ (−A) ⊂ B if (M + A) ∩ (−A) ⊂ B and only if in case A ∩ (−A)= {0}. For the reverse implication, if x ∈ (M + A) ∩ (−A), then x = m+a1 = −a2, where m ∈ M and a1,a2 ∈ A. Therefore, m/2 ∈ M ∩(−A) ⊂ B. Then x/2 ∈ A ∩ (−A), so that x ∈ B = A ∩ (−A)= {0} if K is strictly proper.

Lemma 4.2 Assume that the acceptance sets are strictly proper, that is, Ct,s ∩ (−Ct,s) consists of all random vectors that equal 0 almost surely. 0 ˜ (i) If Kt = Kt for all t, (SNAR) implies 0 0 0 A0,t ∩ (L (Kt, Ft)+ Ct−1,t) ⊂ L (Kt , Ft), t =0,...,T, (4.2) 0 0 0 At,T ∩ (L (Kt, Ft)+ Ct−1,t) ⊂ L (Kt , Ft), t =0,...,T. (4.3) (ii) If the solvency sets are strictly proper and

0 L (−Kt, Ft) ∩Ct−1,t = {0}, t =0,...,T, (4.4) then either of the conditions (4.2), (4.3) implies (SNAR).

Proof (i) Motivated by [24, Lemma 3.2.7], assume that

0 −k0 −···− kt − η0 −···− ηt = gt + ζt ∈ A0,t ∩ (L (Kt, Ft)+ Ct−1,t)

0 0 with ks ∈ L (Ks, Fs), ηs ∈ Cs−1,s, s = 0,...,t, gt ∈ L (Kt, Ft), and ζt ∈ Ct−1,t. Since (ηt + ζt)/2 ∈Ct−1,t by convexity and

−k0/2 −···− kt−1/2 − (kt + gt)/2 − η0/2 −···− (ηt + ζt)/2=0,

0 we have (kt +gt)/2 ∈ Kt and (ηt +ζt)/2 = 0 by (SNAR). The strict properness 1 1 of the acceptance sets yields that ηt = ζt = 0. Furthermore, 2 gt ∈ − 2 kt + 1 0 0 2 Kt ⊂−Kt, so that gt ∈ Kt , i.e. (4.2) holds. Property (4.3) is similarly derived from (SNAR). (ii) In order to show that (4.2) implies (SNAR), proceed by induction as in [24, Lemma 3.2.13]. Let −k0 −···− kT − η0 −···− ηT = 0. Then

T −1 kT + ηT = (−ks − ηs) ∈ A0,T −1 ⊂ A0,T . s=0 X 14 Emmanuel Lepinette, Ilya Molchanov

0 0 By (4.2), kT + ηT ∈ L (KT , FT ). Since kT + ηT is FT −1-measurable and 0 0 the solvency sets are strictly proper, kT + ηT ∈ L (KT −1, FT −1). Therefore, kT +ηT can be merged with kT −1, and then the induction proceeds with T −1 instead of T . To show that (4.3) implies (SNAR), proceed by induction starting from time zero. Since

T k0 + η0 = (−ks − ηs) ∈ A1,T ⊂ A0,T , s=1 X

(4.3) yields that k0 + η0 = 0, and (4.4) implies k0 = η0 = 0.

Condition (4.4) can be viewed as a consistency between the acceptance sets and solvency sets, namely, that −Kt does not contain any acceptable non-trivial selection. The ﬁrst part of the following result shows that (NRA) is similar to the weak no arbitrage property NAw of Kabanov’s model, see [24, Sec. 3.2.1]. Denote by intA the interior and by ∂A the boundary of A ⊂ Rd.

Rd Proposition 4.1 Suppose that + \{0}⊂ intKt a.s. for all t. Then (NRA) is equivalent to each of the following two conditions.

0 0 0 0 (i) Rt,t+1(Ξt+1) ∩ L (−Kt, Ft) ⊂ L (−∂Kt, Ft) for all t. ˆ0 0 Rd (ii) Ξt ∩ L ( −, Ft)= {0} for all t.

0 0 0 Proof (i) Consider xt = γt +kt for γt ∈ M = Rt,t+1(Ξt+1) and kt ∈ L (Kt, Ft). 0 Rd Assume that xt is non-trivial and xt ∈ L ( −, Ft). Hence, γt/2= xt/2−kt/2 ∈ 0 Rd L (−Kt, Ft) and γt/2 ∈ −intKt on {xt 6= 0}, since intKt contains + \{0}, contrary to the assumption. 0 0 Consider any xt ∈ M ∩L (−Kt, Ft) such that xt = −kt for kt ∈ L (Kt, Ft) such that P(kt ∈ intKt) > 0. By a measurable selection argument, kt + γt ∈ 0 0 Rd L (Kt, Ft) for some γt ∈ L ( −, Ft) \{0}. Thus,

0 0 Rd xt + kt + γt = γt ∈ (M + L (Kt, Ft)) ∩ L ( −, Ft), contrary to (NRA). 0 (ii) It suﬃces to show that (NRA) implies (ii). Assume that kt ∈ L (Kt, Ft) 0 Rd and γt ∈ Rt,t+1(Ξt+1) are such that kt + γt ∈ − a.s. and kt + γt 6= 0 with a Rd positive probability. Since kt/2+ + ⊂ ({kt/2} ∪ intKt) a.s., the set (intKt + 0 Rt,t+1(Ξt+1)) with a positive probability has a non-trivial intersection with Rd 0 0 −. By [26, Prop. 2.10] with X = intKt and Ξ = Rt,t+1(Ξt+1), the set (intKt+ 0 0 Rd Rt,t+1(Ξt+1)) has a non-trivial intersection with − with a positive probability, contrary to (NRA).

Theorem 4.1 If the solvency sets are proper, then (SNR) implies (NARA) ˆ0 and the closedness of Ξt in probability for all t =0,...,T . Risk arbitrage 15

0 ˆ0 0 Proof Denote M = Rt,t+1(Ξt+1). Recall that cl0Ξt = cl0Ξt . Assume that n n 0 Rd n 0 n kt + γt → ζt ∈ L ( −, Ft) a.s. for kt ∈ L (Kt, Ft) and γt ∈ M such that n n ˆ0 0 kt +γt ∈ Ξt , n ≥ 1. Since M is L -closed and convex, we may assume by [24, n 0 n Lemma 2.1.2] that kt → kt ∈ L (Kt, Ft) on the set A = {lim infn kkt k < ∞}. n Hence, γt → γt ∈ M, so that

0 0 0 γt = ζt − kt ∈ M ∩ L (−Kt, Ft) ⊂ L (Kt , Ft). 0 Rd Thus, γt ∈ Kt and ζt/2 = γt/2+ kt/2 ∈ Kt. Hence, ζt/2 ∈ − ∩ Kt = {0} and ζt =0 on A. n n If P(Ω\A) > 0, assume that kt = γt = ζt =0on A by Ft-decomposability, n n and use the standard normalisation procedure, i.e. divide kt ,γt , ζt by (1 + n 0 kkt k). Arguing as previously, we obtain kt ∈ L (Kt, Ft) such that kktk = 1 on Ω \ A. Since 0 ∈ M, we have γt ∈ M by conditional convexity. Moreover, n 0 kt+γt = 0 since ζt/(1+kkt k) → 0. Hence, γt 6= 0 belongs to M∩L (−Kt, Ft)= {0}, which is a contradiction in view of Lemma 4.1. ˆ0 0 This argument also yields the closedness of Ξt = L (Kt, Ft)+ M. p p p Rd p Let At,s denote the closure of At,s = At,s ∩ L ( , FT ) in L for p < ∞ and in the Ft-bounded convergence in probability for p = ∞. The following theorem states that the (NARA) and (SNR) conditions are weak no arbitrage conditions of the no free lunch type. Recall that the usual p 0 Rd NFL condition is At,T ∩ L ( +, FT )= {0}. Theorem 4.2 Assume that the acceptance sets are continuous from below and p ∈ [1, ∞]. (i) (NARA) is equivalent to

p 0 Rd At,T ∩ L ( +, Ft)= {0}, t =0,...,T − 1. (4.5) (ii) If the solvency sets are proper, then (SNR) is equivalent to

p 0 At,T ∩ L (Kt, Ft)= {0}, t =0,...,T − 1. (4.6) Moreover, properties (4.5) and (4.6) are equivalent to the same ones with p =1 A p p Rd and also to those obtained by taking the closure of t,T in Lt,T = LFt ( , FT ). Proof (i) Assume that (4.5) holds for the closure with respect to the condi- p tional norm on Lt,T . Thus, (4.5) also holds if the closure is taken with respect p d to the norm on L (R , FT ). Therefore, given (4.1), it suﬃces to show that (NARA) follows from 0 p Rd 0 Rd clp(Ξt ∩ L ( , Ft)) ∩ L ( −, Ft)= {0}, t =0,...,T − 1. (4.7) 0 0 Rd n Assume (4.7) and consider xt ∈ (cl0Ξt ) ∩ L ( −, Ft). Then xt → xt a.s. for n 0 xt ∈ Ξt , n ≥ 1. Hence,

n n xt 1kxt k≤m+11kxtk≤m → xt1kxtk≤m a.s. as n → ∞ 16 Emmanuel Lepinette, Ilya Molchanov

n n 0 for all m ≥ 1, where xt 1kxt k≤m+11kxtk≤m ∈ Ξt by decomposability and since 0 0 ∈ Ξt . The dominated convergence theorem yields that xt1kxtk≤m belongs to 0 p Rd 0 Rd clp(Ξt ∩L ( , Ft)) ∩L ( −, Ft)= {0}, where the closure may be taken with respect to the conditional norm. Letting m → ∞ yields xt = 0, i.e. (NARA) holds. n p Assume (NARA). Consider a sequence (Vt,T )n≥1 from At,T which converges p + p Rd ˜ n n + + p in Lt,T to zt ∈ L ( +, Ft). Then, Vt,T = Vt,T ∧ zt → zt in Lu,T , where the minimum is taken coordinatewisely and u is any time moment between t and ˜ n T , and Vt,T ∈ At,T , so that we may assume without loss of generality that n + n + p Vt,T ≤ zt . Passing to subsequences, assume that Vt,T → zt in Lu,T and almost surely for each given u ≥ t. n n + n Deﬁne ξT = Vt,T − zt ≤ 0. Then |||ξT |||p,FT −1 → 0 in probability. By the n 0 Rd continuity from below, there exists a sequence γT −1 ∈ L ( +, FT −1), n ≥ 1, such that n n n ηT = ξT + γT −1 ∈CT −1,T n n Rd n and 0 ≤ γT −1 ≤ xT |||ξT |||p,FT −1 for all n and some xT ∈ +. Hence, γT −1 → 0 p n p in LT −2,T if T −2 ≥ t. Since −γT −1 → 0 in LT −2,T , the continuity from below n 0 Rd property yields the existence of a sequence γT −2 ∈ L ( +, FT −2) such that

n n n ηT −1 = −γT −1 + γT −2 ∈CT −2,T −1 Rd for all n, and, for some constant xT −1 ∈ +, we have

n n n 0 ≤ γT −2 ≤ xT −1|||γT −1|||p,FT −1 ≤ xT xT −1|||ξT |||p,FT −2 ,

n p so that γT −2 → 0 in LT −3,T if T − 3 ≥ t. Iterate the construction to ﬁnd n n n n n n γT −3,...,γt such that γt → 0 a.s. Then ηu+1 = −γu+1 + γu ∈ Cu,u+1 if t ≤ u ≤ T − 2. Hence,

T −1 n n n ξT + γt = ηu+1 ∈Ct,t+1 + ··· + CT −1,T . u=t X By convexity, 1 1 1 (−z+ + γn)= − V n + (ξn + γn) ∈ Ξ0. 2 t t 2 t,T 2 T t t 1 + 0 0 Rd + Letting n → ∞ yields that − 2 zt ∈ (cl0Ξt ) ∩ L (− +, Ft), so that zt = 0 by (NARA). Thus, (4.5) holds with respect to the conditional norm and also with respect to the Lp-norm. ˆ0 ˆ0 0 (ii) Recall that Ξt = cl0(Ξt ) = cl0(Ξt ) by Theorem 4.1. Following the arguments from (i), we obtain that (SNR) is equivalent to 0 p Rd p clp(Ξt ∩ L ( , Ft)) ∩ L (−Kt, Ft)= {0}, t =0,...,T − 1. (4.8)

0 p p In view of (4.1), clp(Ξt ∩ L ) ⊂ −At,T . Therefore, (4.6) implies (4.8) and (SNR) holds. Risk arbitrage 17

n Ap Now assume (SNR). Consider a sequence (Vt,T )n≥1 from t,T which con- p 0 verges in L to kt ∈ L (Kt, Ft). Then follow the proof of (i) with kt instead + of zt . n 1 1 Consider a sequence (Vt,T )n≥1 from At,T which converges in L to kt ∈ 0 n L (Kt, Ft). We may assume that (Vt,T )n≥1 converges a.s. Then, for every n A1 1 M > 0, (Vt,T 1kktk≤M )n≥1 is a sequence from t,T which converges in L to p kt1kktk≤M ∈ L (Kt, Ft) so that we may assume without loss of generality n that kktk is bounded by M. Passing to a subsequence, assume that E(kVt,T − n k k |F ) → 0 a.s. Therefore, V 1 n → k almost surely and in t t t,T E(kVt,T k|Ft)≤M+1 t 1 Lt,T . Thus, (4.6) holds with p = 1.

Now consider more general claims ξ. Recall that, if KT is a cone and ξ ∈ KT ξ 0 a.s., then Ξt ⊂ Ξt for all t.

Theorem 4.3 If the solvency sets are proper and the acceptance sets are con- ˆξ tinuous from below, then (SNR) yields that Ξt is closed in probability for all p Rd ˆξ 0 ξ t and any ξ ∈ L ( , FT ), so that Ξt = L (Xt , Ft) for a random closed set ξ Xt , t =0,...,T .

n n 0 Rd n 0 Proof Assume that kt + γt → ζt ∈ L ( , Ft) a.s. for kt ∈ L (Kt, Ft) and n ξ n n ˆ0 0 γt ∈ M = Rt,t+1(Ξt+1) such that kt + γt ∈ Ξt . Since M is L -closed and n 0 convex, we may assume by [24, Lemma 2.1.2] that kt → kt ∈ L (Kt, Ft) on the n n ˆξ set A = {lim infn kkt k < ∞}. Hence, γt → γt ∈ M, so that ζt = kt + γt ∈ Ξt . n n If P(Ω\A) > 0, assume that kt = γt = ζt =0on A by Ft-decomposability, ñ n n and use the normalisation procedure, i.e. obtain kt andγ ˜t by scaling kt and n n n −1 n n γt with ct = (1+ kkt k) . We may assume that kγ˜t k ≤ 2, since ct ζt → 0, ñ n ñ ˜ 0 p so that kt +˜γt → 0. Arguing as previously, kt → kt ∈ L (Kt, Ft) in L , and ˜ n ˜ p ˜ kktk =1on Ω \A. Therefore,γ ˜t → γ˜t = −kt in L , so that kt +˜γt = 0. Notice that 0 d M = cl0 (−At+1,T + ξ) ∩ L (R , Ft) . By convexity, n n 0 Rd γ˜t ∈ cl0(−At+1,T + ct ξ) ∩ L ( , Ft). n n n Since kγ˜t k≤ 2, assume without loss of generality thatγ ˜t ∈ clp(−At+1,T +ct ξ). n mn To see this, it suffices to approximate the sequenceγ ˜t byγ ¯t ∈ (−At+1,T + n mn ct ξ), m ≥ 1, and multiply the latter by 1kγ¯t k≤3. Letting n → ∞, (4.6) yields that p 0 −γ˜t ∈ At,T ∩ L (Kt, Ft)= {0}.

Thus, γt = 0, so that P(Ω \ A) = 0 and the conclusion follows.

Lemma 4.3 (NRA2) is equivalent to

ξ 0 Ξt ⊂ L (Kt, Ft)+ Ct−1,t, t =0,...,T, (4.9)

0 for any ξ ∈ L (KT , FT ). If (4.4) holds, then (NRA2) implies (NARA). 18 Emmanuel Lepinette, Ilya Molchanov

0 Proof Note that (ηt + At,T ) intersects L (KT , FT ) if and only if

0 0 d ηt ∈ (−At,T + L (KT , FT )) ∩ L (R , Ft),

ξ 0 equivalently, ηt ∈ Ξt for some ξ = kT ∈ L (KT , FT ).

∗ Denote by Kt = {x : hx, ui ≥ 0,u ∈ Kt} the positive dual set to Kt and ∗ Rd assume that Kt \{0} is a subset of the interior of + for all t.

Deﬁnition 4.2 For t ≤ T , an adapted process Z = (Zs)s=t,...,T is a q- integrable t-weakly consistent price system if it is a Q-martingale for Q ∼ P ∗ such that Zs is a q-integrable under Q, Fs-measurable selection of Ks for every q,w Q s ≥ t and Zt 6= 0 a.s. We denote by Mt,T ( ) the set all q-integrable t-weakly consistent price systems under Q, where q ∈ [1, ∞].

The following result characterises the prices under (SNR) and (NARA) conditions and so may be viewed as the Fundamental Theorem of Asset Pricing in our framework.

Theorem 4.4 Assume the coherent conical setting and that the solvency sets are continuous from below. Let q be the conjugate of the number p that stems from the deﬁnition of the acceptance sets. (i) (NARA) is equivalent to the fact that, for each t, there exists Z ∈ q,w Mt,T (P) such that

EhZu, ηui≥ 0 for all ηu ∈Cu−1,u, u = t +1,...,T. (4.10)

∗ (ii) If intKt 6= ∅ a.s. for all t, then (SNR) is equivalent to the fact that, q,w P for each t, there exists Z ∈ Mt,T ( ) such that (4.10) holds and Zt ∈ 0 ∗ L (intKt , Ft).

q,w Proof (i) The existence of Z ∈ Mt,T (P) such that EhZt, ηi ≥ 0 for all η ∈ Ct,t+1 + ··· + CT −1,T is a direct consequence of the Hahn–Banach separation theorem and Theorem 4.2(i), since we may take p = 1. To prove the converse implication, for each t, assume the existence of Z ∈ q,w p Mt,T (P) and consider xT ∈ At,T . Then

xT = −kt − (kt+1 + ηt+1) −···− (kT + ηT ),

0 ′ ′′ where ηs ∈ Cs−1,s and ks ∈ L (Ks, Fs) for s ≥ t. Since ηs = ηs + ηs with ′ p ′′ 0 Rd ′′ ηs ∈ Cs−1,s ∩ Ls−1,s and ηs ∈ L ( +, Fs), we may merge ηs and ks and ′ suppose without loss of generality that ηs = ηs. Using the backward induction on t ≤ T , we show that EhZT , xT i ≤ 0. If p xT = −kT −1 − kT − ηT , this is trivial. Since ηt+1, kt ∈ Lt,t+1, there exists a i p d partition (B ) from F such that η 1 i , k 1 i ∈ L (R , F ) for all i ≥ 1. t i≥1 t t+1 Bt t Bt t Then,

i p x = (−k −···− k − η −···− η )1 i ∈ A , i ≥ 1. t+1 t+1 T t+2 T Bt t+1,T Risk arbitrage 19

Moreover, ∞ ∞ i EhZ , x i = EhZ , x i + EhZ , −k 1 i i + EhZ , −η i1 i , T T T t+1 T t Bt t t+1 Bt i=1 i=1 X∞ X∞ i = EhZ , x i + EhZ , −k 1 i i + EhZ , −η 1 i i T t+1 t t Bt t+1 t+1 Bt i=1 i=1 X∞ X i ≤ EhZT , xt+1i. i=1 X i The induction hypothesis yields that EhZT , xt+1i≤ 0. Hence, EhZT , xT i≤ 0. p Therefore, EhZT , xT i ≤ 0 for all xT ∈ At,T . In particular, if xT = xt ∈ 0 Rd Rd L ( +, Ft), then EhZt, xti≤ 0 and ﬁnally EhZt, xti = 0. Since Zt ∈ int +, we have xT = 0, i.e. (NARA) holds by Theorem 4.2(i). (ii) Replicate the proof of (i) using the Hahn-Banach theorem and following q,w the arguments of [25, Th. 4.1] in order to construct Z ∈ Mt,T (P) such that 0 ∗ Zt ∈ L (intKt , Ft). Remark 1 Condition (4.10) can be equivalently written in terms of the conditional expectation as E(hZu, ηui|Fu−1) ≥ 0 a.s., which also corresponds to the duality pairing in modules, see [15]. Indeed, suppose that (4.10) holds. Then

E(hZu, ηu1Au−1 i) ≥ 0 for all Au−1 ∈ Fu−1. Therefore, E(hZu, ηui|Fu−1) ≥ 0. The opposite implication is obvious. In other words, (4.10) means that Zu belongs to the positive dual of Cu−1,u.

If the acceptance sets with p = ∞ are generated by convex families Zt,s (see Example 4), then (4.10) means that Zt belongs to the closure of Zt,s with respect to the Ft-bounded convergence in probability. If Kt are all half-spaces (in the frictionless setting), then (NARA) is equivalent to the existence of a martingale Zu = φuSu, where Su is the price vector, such that (4.10) holds.

5 Good Deals hedging

p d Assume that Ct,s consists of random vectors (ηs, 0,..., 0) ∈ L (R , Fs) with all vanishing components except the first one and such that rt,s(ηs) ≤ 0 for a univariate dynamic risk measure rt,s. This corresponds to the case, when the acceptability at each step is assessed by calculating the risk of a portfolio expressed in the units of the first asset, most importantly, cash. An arbitrage opportunity in this setting is called a Good Deal, see [7]. For simplicity, consider the one period setting with zero interest rate and two assets exchangeable without transaction costs, so that without loss of generality the first asset is assumed to be cash and the second one is a risky asset priced at St for t = 0, 1. Let ξ be the cash value of a terminal claim. If x0 is the initial endowment (in cash), then the terminal value of a portfolio is ′ ′′ V1 = (x0, 0)+(−k0S0, k0) + (−k1S1, k1) − (η , η ), 20 Emmanuel Lepinette, Ilya Molchanov

where η′ and η′′ are acceptable with respect to some static convex risk measure r, meaning that r(η′) ≤ 0 and r(η′′) ≤ 0. Given the choice of the acceptance ′′ set C0,1, we have η = 0, so that V1 suﬃces to pay the claim if

′ x0 + k0(S1 − S0) − η ≥ ξ.

′ The smallest value of x0 that ensures the existence of an acceptable η = −ξ + x0 + k0(S1 − S0), satisfying the above inequality, equals the inﬁmum of r(k0(S1 − S0) − ξ) over all deterministic k0 ∈ R. For instance, the zero claim ξ = 0 can be hedged with a negative initial capital if r(S1 − S0) < 0 or r(S0 − S1) < 0, and this means the existence of a Good Deal arbitrage. If this is the case and the risk measure is coherent, then also

r(k0(S1 − S0) − ξ) ≤ r(k0(S1 − S0)) + r(−ξ) < 0 (5.1)

for suﬃciently large positive k0 if r(S1 −S0) < 0 (or negative k0 if r(S0 −S1) < 0), meaning that any claim with ﬁnite r(−ξ) can be also hedged with a negative initial investment. In other words, the No Good Deals (NGD) Arbitrage condition becomes

r(S1/S0) ≥−1 and r(−S1/S0) ≥ 1.

Our setting is more general as the Good Deals hedging, since it allows for more general acceptance sets and eliminates the prescribed choice of the single asset in order to assess the acceptability. As a result, the no arbitrage conditions become stronger and the superhedging price declines. To illustrate this, consider the above two assets one period setting with the acceptance set ′ ′′ ′ ′′ C0,1 that consists of all (η , η ) such that r(η ) ≤ 0 and r(η ) ≤ 0. By allowing a non-trivial η′′, it is possible to decrease the price of a terminal cash claim ξ. For this, note ξ can be paid if

′ x0 − k0S0 − k1S1 − η ≥ ξ ′′ (k0 + k1 − η ≥ 0

′ ′′ for some deterministic k0, F1-measurable k1, and acceptable η , η . This increases the hedging possibilities and so leads to a decrease of the superhedging price, but also creates extra arbitrage opportunities. In particular, ′ ′′ ′ considering (η .η ) ∈ C0,1 with η = 0, the arbitrage becomes possible if r((k0(S1 − S0)+ x0)/S1) ≤ 0 for some x0 < 0 and K0 ∈ R. By letting x0 increase to zero, we see that the necessary no arbitrage condition in addition to (5.1) yields that

r(S0/S1) ≥−1 and r(−S0/S1) ≥ 1. (5.2)

It corresponds to the fact that in two assets, the position expressed in one of them may be not acceptable, while the position expressed in the other one may be acceptable. The necessary and suﬃcient no arbitrage condition is stronger and should also include all possible combinations of the two assets. Risk arbitrage 21

+ Assume that ξ = (S1 − K) for some K > 0 and that the support of S1 is

the whole half-line (0, ∞). If r(X) = esssupF0 (−X), i.e. when the acceptable positions are non-negative random variables, then the minimal price

x0 = inf r(k0(S1 − S0) − ξ) k0∈R

equals S0. If r is non-trivial, we have x0 ≤ S0 + r(S1 − ξ). Note that S1 − ξ = S1 ∧ K, so that x0 ≤ S0 + r(S1) ∧ K and, ﬁnally, x0 ≤ S0 − r(S1), where r(S1) < 0 given that S1 > 0 and r is non-trivial. This simple example illustrates the decrease of the super-hedging price in presence of a non-trivial risk measure.

Example 7 Assume that the risk measure r is the negative essential inﬁmum, that is, consider the setting of conditional cores from Section 6. Then the NGD arbitrage is not possible if ess infF0 S1 ≤ S0 ≤ ess supF0 S1. With this choice of the risk measure, the NGD condition coincides with the (SNR) condition, see Theorem 8.1.

6 Conditional core as risk measure

0 Rd Assume that p = ∞ and Ct,s = L ( +, Fs) for all 0 ≤ t ≤ s ≤ T , so that 0 0 Rd 0 Rd Rt,s(Ξ)= Ξ ∩ L ( , Ft) for any upper set Ξ ⊂ L ( , Fs). If X is an upper random closed set, then

0 Rt,s(X)= Rt,s(X)= m(X|Ft),

where the latter notation designates the largest Ft-measurable random closed subset of X, called the conditional core of X, see [26, Def. 4.1]. An acceptable portfolio process is characterised by Vt−1 − Vt ∈ Kt a.s. for t =1,...,T . Then 0 At,s becomes the sum of L (−Ku, Fu) for u = t,...,s, exactly like in the ξ classical theory of markets with transaction costs [24]. For claim ξ, the set Ξt deﬁned in Section 3 becomes the set of superhedging prices that was used in [27] to deﬁne a risk measure of ξ. The classical no arbitrage condition (NAs) (no strict arbitrage opportunity at any time, see [24, Sec. 3.1.4]) then becomes (4.2); (SNA) (strong no arbitrage, see Condition (iii) in [24, Sec. 3.2.2]) then becomes the special case of (SNAR).

Theorem 6.1 Suppose that the solvency sets (Kt)t=0,...,T are strictly proper. Then (SNR), (NAs) and (SNA) are all equivalent and are also equivalent to each of the following conditions.

p 0 (i) At,T ∩ L (Kt, Ft)= {0}, for all t ≤ T − 1. 0 0 (ii) At,T is closed in L and At,T ∩ L (Kt, Ft)= {0}, for all t ≤ T − 1. 22 Emmanuel Lepinette, Ilya Molchanov

Proof (SNR) is equivalent to (i) by Theorem 4.2(ii). The equivalence of (NAs) and (SNA) follows from Lemma 4.2 given that (4.4) trivially holds. The implication (i)⇒(ii) is simple to show by induction. First, AT,T is n n n 0 closed. Assume that −kt −···− kT → ξ a.s. for ku ∈ L (Ku, Fu), u ≥ t. On n the set {lim infn kkt k = ∞}, we use the normalisation procedure to arrive at n a contradiction with (i). Otherwise, suppose that −kt →−kt ∈−Kt, so that we may use the induction hypothesis to conclude. In order to derive the closedness of At,r under (SNA), it suﬃces to follow 0 the proof of [24, Lemma 3.2.8]. Indeed, since Kt is a linear space, the recession cone ∞ Rd Kt = αKt = {x ∈ : Kt + αx ⊂ Kt ∀α> 0} α>0 \ 0 ∞ 0 satisﬁes Kt ⊂ Kt , see [30]. Therefore, kt + αx ∈ Kt for all kt ∈ Kt, x ∈ Kt , 0 and all α ∈ R. Furthermore, (SNA) trivially implies At,T ∩ L (Kt, Ft) = {0} for all t. In order to show that (ii) implies (NAs), assume

0 −k0 −···− kt = k˜t ∈ A0,t ∩ L (Kt, Ft).

0 Then k0 ∈ A0,T ∩L (K0, F0), i.e. k0 = 0 by (ii). Similarly, k1 = ··· = kt−1 = 0, s so that −kt = k˜t = 0, since Kt is strictly proper. Thus, (NA ) holds. s 0 At last, (NA ) yields (SNA), so that At,T is closed in L . Finally, (SNA) 0 yields (4.3) and so At,T ∩ L (Kt, Ft)= {0}, that is, (i) holds.

(NA2) (no arbitrage opportunity of the second kind) from [31] and [24, p. 135] has the same formulation as (NRA2).

Lemma 6.1 Assume that the solvency sets are cones.

0 0 (i) (NA2) is equivalent to Ξt = L (Kt, Ft) for all t ≤ T . (ii) (NA2) is equivalent to

m(Kt|Ft−1) ⊂ Kt−1, t =1,...,T. (6.1)

(iii) If the solvency sets are strictly proper, then (NA2) implies (SNR).

0 0 Proof (i) By Lemma 4.3 and Lemma 3.1(iv), Ξt = L (Kt, Ft) yields (4.9), and 0 0 so implies (NA2). In the other direction, (NA2) yields that Ξt ⊂ L (Kt, Ft)+ 0 0 Ct−1,t, while (3.3) and the choice of Ct−1,t yields that Ξt ⊃ L (Kt, Ft). 0 0 (ii) If Ξt ⊂ L (Kt, Ft), then 0 0 Rd 0 Ξt+1 ∩ L ( , Ft) ⊂ L (Kt, Ft)

0 0 for all t by (3.3). Since L (Kt+1, Ft+1) ⊂ Ξt+1, we obtain (6.1). If (6.1) holds, then

0 0 0 ΞT −1 = L (KT −1, FT −1)+ m(KT |FT −1) ⊂ L (KT −1, FT −1). Risk arbitrage 23

0 0 Assume that Ξs ⊂ L (Ks, Fs) for s = t +1,...,T . Then

0 0 0 0 Rd Ξt = L (Kt, Ft) + (Ξt+1 ∩ L ( , Ft)) 0 0 ⊂ L (Kt, Ft)+ m(Kt+1|Ft) ⊂ L (Kt, Ft).

The proof is ﬁnished by the induction argument. 0 0 ˆ0 (iii) Since Ξt is closed in probability under (SNR), we have Ξt = Ξt , and 0 0 (SNR) yields that L (Kt, Ft) ∩ L (−Kt, Ft) = {0}, which is the case if the solvency sets are strictly proper.

0 0 For conical solvency sets satisfying m(Kt |Ft−1) ⊂ Kt−1, t ≤ T , in particular, for strictly proper ones, (NAs) is equivalent to the existence of a Q- ∗ martingale evolving in the relative interiors of (Kt )t=0,...,T for a probability measure Q equivalent to P, see [24, Th. 3.2.2]. Such a martingale is called a ∗ strictly consistent price system. If intKt 6= ∅ for all t, this result follows from Theorem 4.4(ii). ξ 0 ξ ξ 0 ξ Note that ΞT = L (XT , FT ) for XT = ξ+KT , and ΞT −1 = L (XT −1, FT −1) ξ is the family of selections for a (possibly non-closed) random set XT −1 = ξ KT −1 + m(XT |FT −1). One needs additional assumptions of the no arbitrage ξ type in order to extend this interpretation for Ξt with t ≤ T −2. Precisely the ξ sum above should be closed, so that m(Xt |Ft−1) exists for t ≤ T − 1, which makes it possible to apply Lemma 9.1.

Theorem 6.2 Assume that the solvency sets are strictly proper and (NAs) ξ 0 ξ ξ (equivalently, (SNR) or (SNA)) holds. Then Ξt = L (Xt , Ft), where Xt is an ξ Ft-measurable random closed convex set, t =0,...,T , such that XT = ξ +KT , and ξ ξ Xt = Kt + m(Xt+1|Ft), t = T − 1,..., 0. (6.2)

Proof It suﬃces to conﬁrm the statement for t = T − 1 and then use the induction. Indeed, by Theorem 6.1, (NAs) is equivalent to (SNR) so that Theo- ξ rem 4.3 applies. Since Ξt is Ft-decomposable, Theorem 9.1 yields the existence ξ ξ 0 ξ of an FT −1-measurable closed set XT −1 such that ΞT −1 = L (XT −1, FT −1). ξ Since XT is closed,

ξ 0 0 ξ ΞT −1 = L (KT −1, FT −1)+ L (m(XT |FT −1), FT −1), 0 ξ = L (KT −1 + m(XT |FT −1), FT −1) 0 ξ = L (XT −1, FT −1),

ξ where XT −1 is a random set by Lemma 9.1.

Proposition 6.1 Suppose that the solvency sets are strictly proper. Then s 0 0 0 0 (NA ) holds if and only if Ξt = L (Xt , Ft) for random closed sets (Xt )t=0,...,T 24 Emmanuel Lepinette, Ilya Molchanov

0 such that Xt ∩ (−Kt)= {0} a.s. for all t. In the conical case, the latter con- 0 ∗ s dition is equivalent to int(Xt ) 6= ∅ for all t, and, under (NA ),

T 0 0 At,T = L (−Xs , Fs), 0 ≤ t ≤ T, (6.3) s=t X 0 where Xt is a strictly proper random closed convex cone for all t.

s 0 0 Proof Assume (NA ), so that Theorem 6.2 applies. Let −gt ∈ L (Xt ∩(−Kt), Ft). 0 Then gt ∈ Kt a.s., and there exist ku ∈ L (Ku, Fu), u = t,...,T , and 0 g˜T ∈ L (KT , FT ), such that −gt − kt − kt+1 −···− kT =g ˜T . Since (SNA) holds, gt + kt = 0, and gt = 0. The reverse implication is trivial. In the conical case, since Kt + Kt = Kt for all t ≤ T , (6.3) follows from the inclusions

0 Kt ⊂ Xt ⊂ Kt + ··· + KT , t ≤ T.

0 Note that XT = KT is strictly proper by assumption. Since 0 0 0 Xt−1 = Kt−1 + m(Xt |Ft−1) ⊂ Kt−1 + Xt ,

0 the induction argument yields that Xt ⊂ Kt + ··· + KT . Since (SNA) holds s 0 under (NA ), Xt is strictly proper for all t. s 0 0 By [24, Lemma 5.1.2], (NA ) holds if and only if Ξt is closed and Ξt = 0 0 L (Xt , Ft) with ∗ 0 ∗ ∗ 0 ∗ intKt ∩ int(Xt ) = intKt ∩ int m(Xt+1|Ft) 6= ∅, t ≤ T. Finally, observe that

∗ 0 ∗ ∗ 0 ∗ 0 ∗ intKt ∩ int m(Xt+1|Ft) = intKt ∩ m(Xt+1|Ft) = int(Xt ) . Equation (6.3) means that, in the superhedging problem, we may replace 0 0 solvency sets Kt with Xt . The solvency sets (Xt )t=0,...,T satisfy (NA2) condition by Lemma 6.1, which is generally required to obtain a dual characterisation of the superhedging prices, see Condition B (equivalent to (NA2)) in [24, Sec. 3.6.3]. Therefore, (NAs) suﬃces for [24, Th. 3.6.3] to hold provided 0 that we consider the consistent price systems associated to (Xt )t=0,...,T . Now consider (NRA) and (NARA) condition for the chosen acceptance ∗ Rd sets. Assume that the solvency sets are conical and satisfy Kt \{0}⊂ int +. ˆ0 ˆ0 0 0 Since Ξt = cl0(Ξt ) = cl0(Ξt ) = Ξt , (NRA) and (NARA) are equivalent by Ap Q Ap Q Proposition 4.1. Denote by t,T ( ) and t,T ( ) for p ∈ [1, ∞] the variants of Ap Ap Q t,T and t,T when the reference probability measure is . Proposition 6.2 The following statements are equivalent. p 0 Rd (i) At,T (Q) ∩ L ( +, Ft)= {0}, for all t ≤ T − 1, p ∈ [1, ∞) and Q ∼ P. q,w (ii) Mt,T (Q) 6= ∅ for every t ≤ T − 1, Q ∼ P and p ∈ [1, ∞). (iii) (NARA). ∞,w (iv) Mt,T (P) 6= ∅ for every t ≤ T − 1. Risk arbitrage 25

1,w P (v) Mt,T ( ) 6= ∅ for every t ≤ T − 1. Proof By Theorem 4.4, (v) and (iii) are equivalent. We deduce the equivalence of (ii) and (iv) by following the proof of [24, Lemma 3.2.4], which makes it possible to construct a (weakly)-consistent price system (see Deﬁnition 4.2) from any consistent price system in L1. In particular, (v) implies (iv) and, clearly, (iv) implies (v). Then (iii) implies (ii), i.e. (NARA) holds for Q in place of P . By Theorem 4.2, (i) holds. At last, (i) implies (NARA) by Theorem 4.2.

7 Arbitrage with acceptable expectations

1 1 Assume that p = 1, and let Ct,s ∩ Lt,s be the set of all ηs ∈ Lt,s such that g E (ηs|Ft) has all non-negative components. In other words, the acceptable positions are those having non-negative generalised conditional expectation. This is the weakest possible acceptability criterion, which is always the case (in the static setting) if the acceptance sets are dilatation monotonic. 1 The generalised conditional expectation is well deﬁned for each γs ∈ Lt,s g g ′ ′′ ′′ by letting E (γs|Ft) = E (γs|Ft)+ E(γs |Ft), where the expectation of γs ∈ 0 Rd L ( +, Fs) may be inﬁnite. If XT is an FT -measurable random upper convex 1 set that admits at least one selection from Lt,s, then let R0 Eg 1 t,s(XT )= (γs|Ft): γs ∈ LFt (XT , Fs) , g and Rt,s(XT ) = E (XT |Ft) is the generalised conditional expectation of the random closed set XT , see [26, Def. 6.3]. 1 Rd ˆξ Let ξ ∈ L ( , FT ). By Lemma 3.1, ΞT −1 is the family of selections of the (possibly, non-closed) random set

ξ XT −1 = KT −1 + E(ξ|FT −1)+ E(KT |FT −1) , which is FT −1-measurable by Lemma 9.1. Note that all solvency sets are integrable and so their generalised conditional expectation coincides with the usual one. Therefore, R ˆξ E ξ T −2,T −1(ΞT −1)= (XT −1|FT −2)

= E(ξ|FT −2)+ E(KT −1 + E(Kt|FT −1)|FT −2),

= E(ξ|FT −2)+ E(KT + KT −1|FT −2). 1 Since kT ∈ LFT −1 (KT , FT ) and

R0 ξ T −2,T −1(ΞT −1) g g 0 = E (kT −1 + E (ξ + kT |FT −1)|FT −2): kT −1 ∈ L (KT −1, FT −1) , ˆξ ξ ξ we deduce that RT −2,T −1(ΞT −1) ⊂ RT −2,T −1(ΞT −1). Since ΞT −1 is a subset ˆξ of ΞT −1, we have ˆξ ξ RT −2,T −1(ΞT −1)= RT −2,T −1(ΞT −1). 26 Emmanuel Lepinette, Ilya Molchanov

Therefore, ξ XT −2 = KT −2 + E(ξ|FT −2)+ E(KT + KT −1|FT −2), ˆξ 0 ξ Continuing recursively, we obtain Ξt = L (Xt , Ft) with a not necessarily closed Ft-measurable random set ξ Xt = Kt + E(ξ|Ft)+ E KT + KT −1 + ··· + Kt+1|Ft . (7.1)

ξ 0 0 Notice that Xt = Xt + E(ξ|Ft), i.e. Xt determines all superhedging prices. Reformulating requirements from Deﬁnition 4.1, we arrive at the following result. Proposition 7.1 For the risk arbitrage conditions formulated for the conditional expectation as the risk measure, the following hold. (i) If the solvency sets are strictly proper, (SNR) is equivalent to

E(Kt+1 + ··· + KT |Ft) ∩ (−Kt)= {0}, a.s., t =0,...,T − 1. (ii) (NARA) is equivalent to Rd Kt + E(Kt+1 + ··· + KT |Ft) ∩ − = {0} a.s., t =0,...,T − 1. Note that statement (ii) above follows from (i). Theorem 4.4 yields the following result.

Proposition 7.2 Assume that the solvency sets (Kt)t=0,...,T are cones. Then (NARA) (resp. (SNR)) is equivalent to the existence of a deterministic point ∗ ∗ z 6=0 that belongs to all Kt (resp. intKt ), t =0,...,T .

Proof Any acceptable position from Cu−1,u is of the form g g + ηu = γu − E (γu|Fu−1) + E (γu|Fu−1)+ ζu g Rd + 0 Rd with E (γu|Fu−1) ∈ + and ζu ∈ L ( +, Fu). Thus, g g g E hZu, ηui|Fu−1 ≥ E (hZu,γu − E γu|Fu−1)|Fu−1 g g = E hZu,γui|Fu−1 − hZu−1, E (γu|Fu−1)i. g ∞,w Hence, E (hZu, ηui|Fu−1) ≥ 0 if there exists Z ∈Mt,T (P), such that g g E hZu,γui|Fu−1 = hZu−1, E (γu|Fu−1)i a.s. 1 Rd for all γu ∈ LFu−1 ( , Fu). The equality follows from (4.10) by taking un- conditional expectation (restricting to a partition if necessary) and applying the same reason with −γu. Given that Zu is essentially bounded, it is possible to let γu be equal to one of the component of Zu multiplied by the corresponding basis vector. Thus, the square of every component of (Zu)u=0,...,T is a martingale, whence Zu equals to deterministic z for all u. The inverse implication follows from Theorem 4.4(i) applied to g ηu = γu − E (γu|Fu−1) ∈Cu−1,u. The proof for (SNR) follows from the same argument and Theorem 4.4(ii). Risk arbitrage 27

Remark 2 It is possible to derive the result of Proposition 7.2 from Proposi- tion 7.1 by using the fact that the expectation of the cone Kt is the whole ∗ space unless Kt contains a deterministic point z distinct from the origin and then EKt is a subset of the half-space with outer normal (−z). In order that 0 Rd Xt does not intersect −, all cones Kt should have non-trivial expectation and the sum of these expectations has to be non-trivial. This amounts to the ∗ ∗ existence of a deterministic point z 6= 0 that belongs to K0 ∩···∩ KT .

8 Application to the two-dimensional model

Consider a ﬁnancial market model composed of two assets. The ﬁrst one has constant value 1 and the second one is a risky asset modelled by a bid-ask b a b a spread Yt = [St ,St ] such that 0 < St ≤ St a.s. for all t ≤ T . This is Ka- ′ ′′ banov’s model with the conical solvency set Kt = C(Yt), where C([s ,s ]) is the positive dual to the smallest cone in R2 containing the set {1}× [s′,s′′]. Consider the acceptance sets from Section 6, so that the conditional core 0 0 is the risk measure. Then XT −1 is the sum of KT −1 and m(XT , FT −1) = C(M(YT |FT −1)), where M(YT |FT −1)) is the conditional convex hull of YT , that is, the smallest FT −1-measurable random closed convex set that contains YT −1, see [26, Def. 5.1]. 0 Since XT −1 is a random closed set, iterating this argument yields that 0 ˜ ˜ Xt = C(Yt) for t =0,...,T , where YT = YT and

Y˜t = M(Y˜t+1|Ft) ∩ Yt, t = T − 1,..., 0.

0 Note that we do not make any no arbitrage assumption to obtain Xt . Observe ˜ ˜b ã ã a ˜b b that Yt = [St , St ], where ST = ST , ST = ST , and ã a ã ˜b b ˜b St = St ∧ ess supFt St+1, St = St ∨ ess infFt St+1, (8.1)

˜b ˜a for t = T − 1,..., 1. Since 0 < St ≤ St a.s. for all t, (NRA) always holds. By Deﬁnition 4.1 and Lemma 4.3, we easily deduce the following result.

Theorem 8.1 b ˜a a ˜b (i) (SNR) holds if and only if St ≤ ess supFt St+1 and St ≥ ess infFt St+1 b a a.s. with strict inequalities when St

Remark 3 (NAs) is equivalent to (SNR) in the proper case but also to the existence of a strictly consistent price system, see [24, Th. 3.2.2]. In the two asset case, the Grigoriev theorem, see [24, Th. 3.2.15] and [18], asserts that (NAs) is equivalent to the existence of a (possibly non-strict) consistent price system, i.e. the existence of a martingale Zt with respect to a b a probability measure Q equivalent to P such that St ≤ Zt ≤ St for all t. By Theorem 8.1, the existence of a consistent price system, equivalently, 28 Emmanuel Lepinette, Ilya Molchanov

s ˜a E b (NA ), implies (SNR). Indeed, ess supFt St+1 ≥ Q(Zt+1|Ft) ≥ St and sim- ˜b E a ilarly ess infFt St+1 ≤ Q(Zt+1|Ft) ≤ St , the inequalities being strict when b b St

b b b The condition in the following corollary means that δt = St /St−1 and a a a R δt = St /St−1 admit conditional full supports on + for all t =1,...,T .

b a Corollary 8.2 If P(δt ≤ c|Ft−1)P(δt ≥ c|Ft−1) > 0 a.s. for all t = 1,...,T and all c> 0, then (NA2) holds.

a a a a a a Proof Let γ = esssupFt−1 St . Then γ1δt ≥c ≥ St 1δt ≥c ≥ cSt−11δt ≥c. Taking the conditional expectation yields that

a a a γP(δt ≥ c|Ft−1) ≥ cSt−1P(δt ≥ c|Ft−1). a Then γ ≥ cSt−1, and letting c → ∞ yields that γ = +∞ a.s. Similarly, b ess infFt−1 St = 0 a.s., and Theorem 8.1(ii) applies. Assume now that the acceptability criterion is based on the generalised conditional expectation as described in Section 7. By Proposition 7.2, (NARA) holds if and only if there is deterministic z that belongs to all Yt, t =0,...,T , and (SNR) additionally requires that this point belongs to the interiors of Yt. Example 8 (Limit order book) Consider the two asset setting, where it is allowed to perform transactions up to one cash unit amount. This is a simple limit order book setting with only one break point. Then Kt is the sum R2 a b [0, αt] + [0,βt]+ +, where αt = (1, −St ) and βt = (−1,St ). By Proposi- tion 7.1, (NARA) (with acceptability based on conditional expectation) holds if and only if T Rd E ([0, αs] + [0,βs]) Ft ∩ − = {0} h s=t+1 i X for all t = 0,...,T − 1. The sum of segments [0, αs] and [0,βs] is a random convex compact set called a zonotope. The setting can be easily extended to the case of limit order books with several break points.

9 Appendix: Random sets and their selections

Let Rd be the Euclidean space with norm k·k and the Borel σ-algebra B(Rd). The closure of a set A ⊂ Rd is denoted by clA. A set-valued function ω 7→ X(ω) from a complete probability space (Ω, F, P) to the family of all subsets of Rd is called F-measurable (or graph-measurable) if its graph GrX = (ω, x) ∈ Ω × Rd : x ∈ X(ω) ⊂ Ω × Rd Risk arbitrage 29

belongs to the product σ-algebra F⊗B(Rd). In this case, X is said to be a random set. In the same way the H-measurability of X with respect to a sub- σ-algebra H of F is deﬁned. The random set X is said to be closed (convex, open) if X(ω) is a closed (convex, open) set for almost all ω.

Deﬁnition 9.1 An F-measurable random element ξ in Rd such that ξ(ω) ∈ X(ω) for almost all ω ∈ Ω is said to be an F-measurable selection (selection in short) of X, L0(X, F) denotes the family of all F-measurable selections of X, and Lp(X, F) is the family of p-integrable ones.

It is known that an a.s. non-empty random set has at least one selection, see [21, Th. 4.4]. Let H be a sub-σ-algebra of F.

Definition 9.2 A family Ξ ⊂ L0(Rd, F) is said to be infinitely H-decomposable if n ξn1An ∈ Ξ for all sequences (ξn)n≥1 from Ξ and all H-measurable partitions (An)n≥1 of Ω; Ξ is H-decomposable if this holds for finite partitions. P The decomposable subsets of L0(Rd, F) are called stable and infinitely decomposable ones are called σ-stable in [6]. The following result for H = F is well known in case p = 1 [22], where the decomposability concept was first introduced; see also [28, Th. 2.1.10] for H = F, and [24, Prop. 5.4.3] for p = 0.

Theorem 9.1 (see [26, Th. 2.4] and [28, Th. 2.1.10]) Let Ξ be a non- empty subset of Lp(Rd, F) for p =0 or p ∈ [1, ∞]. Then

Ξ ∩ Lp(Rd, H)= Lp(X, H). for an H-measurable random closed set X if and only if Ξ is H-decomposable and closed.

d For A1, A2 ⊂ R , deﬁne their elementwise (Minkowski) sum as

A1 + A2 = {x1 + x2 : x1 ∈ A1, x2 ∈ A2} .

The same definition applies to the sum of subsets of L0(Rd, F). The set of pairwise differences of points from A1 and A2 is obtained as A1 + (−A2), or shortly A1 − A2, where −A2 = {−x : x ∈ A2} is the centrally symmetric variant of A2. For the sum A + {x} of a set and a singleton we write shortly A + x. Note that the sum of two closed sets is not necessarily closed, unless at least one of the closed summands is compact. The following result differs from [28, Th. 1.3.25] in considering the possibly non-closed sum of two random closed sets.

Lemma 9.1 Let X and Y be two random sets. Then L0(X, F)+ L0(Y, F)= L0(X + Y, F). If both X and Y are random closed sets, then X + Y is measurable. 30 Emmanuel Lepinette, Ilya Molchanov

Proof It is trivial that L0(X, F)+ L0(Y, F) ⊂ L0(X + Y, F). To prove the reverse inclusion, consider ξ ∈ L0(X +Y, F). Since X and Y are F-measurable, the measurable selection theorem [24, Th. 5.4.1] yields that there exist F- measurable selections ξ′ ∈ L0(X, F) and ξ′′ ∈ L0(Y, F) such that ξ = ξ′ + ξ′′. Now assume that X and Y are closed and consider their Castaing represen- ′ ′′ tations (see [28, Def. 1.3.6]) X(ω) = cl{ξi(ω),i ≥ 1} and Y (ω) = cl{ξi (ω),i ≥ 1}. The measurability of X + Y follows from the representation

Gr(X + Y ) 1 = (ω, x): kx − ξ′(ω) − ξ′′(ω)k≤ , kξ′(ω)k≤ k . (9.1) i j m i k[≥1 m\≥1 i,j[≥1 n o

Indeed, if (ω, x) ∈ Gr(X1 + X2), then x = a + b for a ∈ X1(ω) and b ∈ X2(ω). Let k ≥ 1 such that kak +1 ≤ k. Since a ∈ X1, there exists a subsequence ′ ′ (ξnl )l≥1 such that ξnl (ω) → a. We may assume without loss of generality that ′ ′′ ′ kξnl (ω)k ≤ k. Similarly, ξnl (ω) → b. Therefore, if m > 0, then kx − ξi(ω) − ′′ 1 ′ ξj (ω)k≤ m and kξi(ω)k≤ k for some i, j.

Acknowledgements IM was supported by the Swiss National Science Foundation Grant 200021-153597. EM thanks the program Investissements d’Avenir from the French foundation ANR which supports the Bachelier colloquium, Metabief, France.

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