Model Spaces for Risk Measures

Transcription

1 Model Spaces for Risk Measures Felix-Benedikt Liebrich a,1, Gregor Svindland a,2 February 22, 2017 ABSTRACT We show how risk measures originally defined in a model free framework in terms of acceptance sets and reference assets imply a meaningful underlying probability structure. Hereafter we construct a maximal domain of definition of the risk measure respecting the underlying ambiguity profile. We particularly emphasise liquidity effects and discuss the correspondence between properties of the risk measure and the structure of this domain as well as subdifferentiability properties. AUTHORS INFO a Mathematisches Institut der LMU Theresienstr. 39, München 1 liebrich@math.lmu.de 2 svindla@math.lmu.de JEL CLASSIFICATION: D81, G22 KEYWORDS: Model free risk assessment, extension of risk measures, continuity properties of risk measures, subgradients 1. Introduction There is an ongoing debate on the right model space for financial risk measures, i.e. about what an ideal domain of definition for risk measures would be. Typically as risk occurs in face of randomness the risks which are to be measured are identified with random variables on some measurable space (Ω, F). The question which causes debate, however, is which space of random variables one should use as model space. Since risk is often understood as Knightian [21] uncertainty about the underlying probabilistic mechanism, many scholars argue that model spaces should be robust in the sense of not depending too heavily on some specific probabilistic model. We refer to this normative viewpoint as paradigm of minimal model dependence. The literature usually suggests one of the following model spaces: (i) L 0 or L 0 P, the spaces of all random variables or P-almost sure (a.s.) equivalence classes of random variables for some probability measure P on (Ω, F), respectively, see [5, 6]; (ii) L or L P, the spaces of all bounded random variables or P-a.s. equivalence classes of bounded random variables, respectively, see [5, 6, 15, 16, 23] and the references therein; (iii) L p P, p [1, ), the space of P-a.s. equivalence classes of random variables with finite p-th moment, or more generally Orlicz hearts, see e.g. [4, 17, 28]. The spaces in (i) and (ii) satisfy minimal model dependence in that L 0 and L are completely model free, whereas L 0 P and L P in fact only depend on the null sets of the probability measure P. The problem with choosing L 0 or L 0 P, however, is that these spaces are in general too large 1

2 to reasonably define aggregation based risk measures on them. The latter would require some kind of integral to be well-defined. Moreover, if (Ω, F) is not finite, L 0 or L 0 P do not allow for a locally convex topology which make them unapt for optimisation. Important applications of risk measures, however, use them as objective functions or constraints in optimisation problems. Since L and L P are Banach spaces so in particular locally convex spaces and as they satisfy minimal model dependence, these model spaces have become most popular in the literature, and amongst them in particular L P due to nicer analytic properties; see [5, 6, 15, 16, 23] and the references therein. In applications, however, unbounded models for risks are standard, like the log-normal distribution in Black -Scholes market models, etc. Assuming frictionless markets, there is no upper bound on the volumes and thus value of financial positions. Hence unbounded distributions appear quite naturally as limiting objects of bounded distributions, and in statistical modeling of random payoffs, where no upper bound can be assumed a priori. From this point of view model spaces should satisfy the paradigm of maximal domain in that they should at least be sufficiently large to include these standard unbounded models, and the model spaces in (iii) have been proposed to resolve this issue, see e.g. [4, 17, 28]. Problematic though is the strong dependence of L p P, p [1, ), (or in general Orlicz hearts) on the probability measure P in that it is not invariant under equivalent changes of measure anymore. Consequently, maximal domain and minimal model dependence seem to be conflicting paradigms. In the special case of law-invariant risk measures the measured risk is fully determined by the distribution of the risk under a probability measure P on (Ω, F). Thus law-invariance already entails the existence of a meaningful reference probability model P, and the risk measurement is fully depending on P. Hence, the ambiguity structure is such that it is no conceptual problem to define these risk measures on, for instance, L 1 P (see [14]).1 The latter observation shows that the paradigms of minimal model dependence and maximal domain may not be as conflicting as they seem, as long as the underlying probability structure is determined by the considered risk measure. Clearly, a model space like L P is sufficiently robust to carry any kind of risk measure. But given a specific risk measure say de fined on L P and the corresponding ambiguity attitude reflected by it, a model space which respects this ambiguity attitude, which also carries the risk measure, and which is probably larger than L P, is also a reasonable model space for that particular risk measure like in the (unambiguous) case of a law invariant risk measure and the model space L 1 P. Our starting point is an a priori completely model free setting on the model space L and a generalised notion of risk measurement adopted from Farkas, Koch-Medina and Munari in, e.g., [13] and Munari in [25]: all it requires is a notion of acceptability of losses (encoded by an acceptance set A L ), a portfolio of liquidly traded securities allowed for hedging (represented by a subspace S L ), and a set of observable prices for these securities (a linear functional p on S). Using such a risk measurement regime R = (A, S, p), we can define the risk ρ R (X) to be the minimal price one has to pay in order to secure the loss X L with a portfolio in S. This approach has the indisputable advantage of a clear operational interpretation. Section 2 introduces this kind of risk measurement in a unifyingly general framework. In Section 3, we observe that under a standard approximation property of finite risk measures continuity from above they automatically imply a reference probability measure P which allows us to view 1 In fact, law-invariant risk measures are completely unambiguous. 2

3 the risk measure as defined on L P without any loss of information. Next we observe that under some further conditions, e.g., sensitivity and strict monotonicity, we can even find a strong reference probability measure P P such that additionally X L P : ρ R(X) ce P [X] holds for a suitable constant c > 0. We shall see that non-existence of such probability models yields risk arbitrage and market inefficiencies in risk sharing schemes. In Section 4.1, we discuss how these considerations lead to a Banach space L R typically much larger than L P, whose geometry is completely determined by the risk measure ρ R. Beside the latter, L R has a multitude of desirable properties, e.g., being invariant under all strong and weak reference probability models, that is preserving the ambiguity attitude expressed by ρ R, and being a natural maximal domain of definition of the initial risk criterion. In the latter respect we study the canonical extension of the initial ρ R to L R, denoted by ρ R, which preserves the functional form, the dual representation, and acceptability criteria of ρ R. The risk measurement regime R only differs from the original R in that the acceptance set is consistently extended to L R. We also consider the following monotone extensions of ρ R to unbounded loss profiles in L R given by and ξ(x) = lim m lim n ρ R(( n) X m) = sup η(x) = lim n m N lim ρ R(( n) X m) = inf m inf ρ R(( n) X m) n N sup n N m N ρ R (( n) X m) which have been studied in for instance [6, 30]. One would maybe expect that always ρ R = ξ = η, but it turns out that ρ R = ξ always holds, whereas ρ R η is possible, see Example 5.1. We characterise the often desirable regular situation when monotone approximation of risks in the following sense ρ R(X) = η(x) = lim ρ R(( n) X n) (1.1) n is possible, see Theorem 4.7, and show that (1.1) holds if ρ R shows sufficient continuity in the tail of the risk X. For instance, any risk measure to which some kind of monotone or dominated convergence rule can be applied will satisfy (1.1). In Section 4.2, we decompose L R into subsets with a clear interpretation in terms of liquidity risk and show how L R allows to view properties of the risk measure ρ R through a topological lens. Finally, in Section 4.4, we address the issue of subdifferentiability of ρ R on L R based on a brief treatment of the dual of L R in Section 4.3. Subgradients play an important role in risk optimisation and appear as pricing rules in optimal risk sharing schemes, see e.g. [20, 30]. We shall see that the topology on L R being determined by ρ R is fine enough to guarantee a rich class of points where ρ R is subdifferentiable, thereby further illustrating how suited the model space L R is to ρ R. Beside their mere existence, we also aim for reasonable conditions guaranteeing that subgradients correspond to measures on (Ω, F) which means ruling out singular el ements that may exist in the dual space of L R. The motivation for this is the same as in case of L P which in general also admits singular elements in its dual space. It is questionable whether such singular dual elements are reasonable as, for instance, pricing rules, because their effect lies mostly in the tails of the distribution, and 3

4 the lack of countable additivity contradicts the paradigm of diminishing marginal risk. Also, measures show a by far better analytic behavior which may prove to be crucial when solving optimisation problems. Our findings suggest that singular elements do not really matter in a wide range of instances. In particular, we will also see that the local equality (1.1) characterised in Theorem 4.7 is closely related to regular subgradients of ρ R and η. In Section 5 we collect illustrating examples. Two cumbersome proofs are outsourced to the appendices A-B. 2. Some preliminaries Notation and terminology: Given a set M and a function f : M [, ], we define the DOMAIN of f to be the set dom(f) = {m M f(m) < }. f is called PROPER if it does not attain the value and dom(f). For a subset A of a topological space (X, τ), we denote by cl τ (A) and int τ (A) the closure and interior of A, respectively, with respect to the topology τ. If (X, τ) is a topological vector space and τ is generated by a norm on X, we will replace the subscript τ by. A triple (X, τ, ) is called ORDERED TOPOLOGICAL VECTOR SPACE if (X, τ) is a topological vector space and is a partial vector space order compatible with the topology in that the positive cone of X, denoted by X + := {X X 0 X}, is τ-closed. We define X ++ := X + \{0}, and X and X analogously. If (X, τ, ) is a Riesz space and X, Y X, we set X Y := sup{x, Y }, X Y := inf{x, Y }, X + := X 0, and X := ( X) 0. 2 In this section we define risk measurement regimes and risk measures, discuss some properties a risk measure may enjoy, and introduce the building blocks for a duality theory. Definition 2.1. Let (X, τ, ) be an ordered topological vector space. An ACCEPTANCE SET is a non-empty proper and convex subset A of X which is monotone, i.e. A X + A. A SECURITY SPACE is a finite-dimensional linear subspace S X containing a non-null positive element U S X ++. We refer to the elements Z S as security portfolios, or simply securities. A PRICING FUNCTIONAL on S is a positive linear functional p : S R such that p(z) > 0 for all Z S X ++. A triple R := (A, S, p) is a RISK MEASUREMENT REGIME if A is an acceptance set, S is a security space and p is a pricing functional on S such that X X : sup{p(z) Z S, X + Z A} <. (2.1) The RISK MEASURE associated to a risk measurement regime R is the functional ρ R : X (, ], X inf {p(z) Z S, X Z A}. (2.2) Our definition of risk measures is inspired by [13, 25]. Note that: 2 For details concerning ordered vector spaces, we refer to Chapters 5 and 7 of [1]. Since risk measures will appear in this treatment on different domains of definition in all cases spaces of random variables endowed with a pointwise or almost sure order and with varying topologies we define them as functionals on ordered topological vector spaces. However, the reader may th ink of X as a space of random variables and of as a pointwise or almost sure order on the latter. 4

5 (a) The elements X X model losses, not gains. Thus ρ R (X) is the minimal amount which has to be invested in some security portfolio Z S today in order to reduce the loss X tomorrow to an acceptable level. (b) We prescribe convexity of the acceptance set A which means that diversification is not penalised: if X and Y are acceptable so is the diversified λx+(1 λ)y for any λ (0, 1). (c) The notion of a risk measurement regime depends on the interplay of A, S and p by means of (2.1); this condition guarantees that ρ R is a proper function. [13, Propositions 1 and 2] yield criteria for R to be a risk measurement regime in our sense. If S = R U for some U X ++ and p(mu) = m, m R, the setting of [11, 12] with a single liquid eligible asset can be recovered from Definition 2.1. If X is a Riesz space with weak unit 1, S = R 1 and p(m1) = m, m R, the definition covers convex monetary risk measures as comprehensively discussed in [16]. 3 The following is easily verified: Lemma 2.2. Let R = (A, S, p) be a risk measurement regime on an ordered topological vector space X. Then ρ R is convex, MONOTONE, i.e. X Y implies ρ R (X) ρ R (Y ), and S- ADDITIVE, i.e. ρ R (X + Z) = ρ R (X) + p(z) holds for all X V and all Z S. In the same abstract setting we introduce further properties a risk measure can enjoy. Definition 2.3. Let R = (A, S, p) be a risk measurement regime on an ordered topological vector space (X, τ, ) and let ρ R be the associated risk measure. ρ R is called FINITE if it only takes finite values, or equivalently A + S = X. 4 ρ R is NORMALISED if ρ R (0) = 0, or equivalently sup Z A S p(z) = 0. ρ R is COHERENT if ρ R (tx) = tρ R (X) for all t > 0, or equivalently if A is a cone. ρ R is SENSITIVE if it satisfies ρ R (X) > ρ R (0) for all X X ++. ρ R is LOWER SEMICONTINUOUS (l.s.c.) if every lower level set {X X ρ R (X) c}, c R, is τ-closed. ρ R is CONTINUOUS FROM ABOVE if it is finite and for any (X n ) n N X with X n X in order ρ R (X) = lim n ρ R (X n ) holds. Remark 2.4. (i) Normalisation implies that the negative cone X (no losses) will be acceptable, which is economically sound. Every risk measure satisfying ρ R (0) R can be normalised by translating the acceptance set. Indeed, let U S X ++ and define r := ρ R(0) p(u) and à := {X + ru X A}. If R is a risk measurement regime, then so is R := (Ã, S, p). Moreover, ρ R(0) = sup{p(z) Z S, Z ru A} = sup{p(w ) + p(ru) W S A}. 3 In the following, we will refer to this particular case with the term monetary risk measures. 4 [13, Propositions 1-3] give further criteria to decide whether ρ R is finite or not. 5

6 This implies that ρ R(0) = ρ R (0) + ρ R (0) = 0 holds. (ii) Continuity from above is a standard paradigm in risk theory, especially in a very similar but slightly stronger version called Lebesgue property see [16]; essentially, it means that approximating the risk of complex payoffs by the one of potentially easier but worse financial instruments is meaningful as long as the payoffs range in a bounded regime. For further discussion of these properties we refer to [16]. (iii) Lower semicontinuity of ρ R implies that {X X ρ R (X) 0} = cl τ (A + ker(p)). In particular, it is implied by A + ker(p) being closed (see [13, Proposition 4]) and invariant under translations of the acceptance set along S. From an economic perspective this property is not too demanding: security spaces are always finite-dimensional in our setting, hence lower semicontinuity is, e.g., implied by the condition A ker(p) = {0} (cf. [13, Proposition 5]). The latter is sometimes referred to as absence of good deals of the first kind (cf. [19]). (iv) Note that in the case of X being a Banach lattice with norm, every finite risk measure is norm-continuous and therefore also norm-l.s.c. This follows from [29, Proposition 1]: Suppose X is a Banach lattice and f : X (, ] is a proper convex and monotone function. Then f is continuous on int dom(f). We will make frequent use of this fact throughout the paper. For many questions a dual point of view on risk measures is crucial. In our case, its formulation requires the following concepts: Definition 2.5. Assume R = (A, S, p) is a risk measurement regime on an ordered topological vector space (X, τ, ) with topological dual X. We define the SUPPORT FUNCTION of A by σ A : X (, ], l sup l(y ), (2.3) Y A and B(A) := dom(σ A ). Moreover, the EXTENSION SET will refer to the set of positive, continuous extensions of p to X, namely E p := {l X + l S = p}. 3. Model spaces of bounded random variables, and weak and strong reference probability models 3.1. The model space L and weak reference probability measures Fix a measurable space (Ω, F) and let L := L (Ω, F) be the set of bounded measurable realvalued functions. We recall that L is a Banach lattice with norm X := sup ω Ω X(ω) when equipped with the pointwise order, so in particular an ordered topological vector space. On the level of Riesz spaces, Ω ω 1 is an order unit of L. 5 The dual space of L may be identified with ba, the space of all finitely additive set functions µ : F R. As usual ca 5 Recall that e X + is an order unit of a Riesz space (X, ) if {X X λ > 0 : X λe} = X. 6

7 denotes the countably additive set functions in ba, and ca + is the set of finite measures. In the following, the notation will not distinguish between m R and the function Ω ω m. In this section we study risk measures on L. First of all, note that in the L, ba -duality monotonicity of A implies that B(A) ba + has to hold, and an application of the Hahn- Banach Separation Theorem shows cl (A) = {X L µ B(A) : X dµ σ A (µ)}. (3.1) We will mostly assume finiteness of ρ R, which is justified by the domain of definition L that is bounded losses which typically should be hedgeable at potentially large, but finite cost. ρ R is for instance finite whenever the security space S contains some U L ++ being uniformly bounded away from 0, i.e. U δ for some constant δ > 0. In [11, 12], such securities are called NON-DEFAULTABLE. We will show that if the acceptance set is nice enough, then any finite risk measure arising from it in an a priori model-free framework like L indeed implies a probabilistic model, a so-called weak reference model; see Proposition 3.2. In order to prepare this result, let us recall the notion of the DUAL CONJUGATE of ρ R being defined as ρ R : ba (, ], µ sup X dµ ρ R (X). (3.2) X L As a first step towards weak reference probability models, we show now that continuity from above mainly depends on the geometry of the acceptance set A and is thus universal in (S, p) up to finiteness of ρ (A,S,p). Proposition 3.1. (i) Assume R = (A, S, p) is a risk measurement regime such that ρ R is l.s.c. and satisfies ρ R (0) R. Then B(A) E p is non-empty, and for all µ ba it holds that { ρ σ A (µ) if µ B(A) E p, R(µ) = (3.3) otherwise. For all X L we have ρ R (X) = sup µ dom(ρ R ) X dµ ρ R(µ). (3.4) Moreover, if ρ R is coherent, then ρ R(µ) = { 0 if µ B(A) E p, otherwise, (3.5) and ρ R (X) = sup µ dom(ρ R ) X dµ, X L. (3.6) (ii) Assume R = (A, S, p) is a risk measurement regime such that ρ R is finite, then for every c R the lower level set E c := {µ ba : ρ R (µ) c} of ρ R is σ(ba, L )-compact. 7

8 (iii) Given a risk measurement regime R = (A, S, p) such that ρ R is finite, ρ R is continuous from above if and only if B(A) ca. Moreover, given an acceptance set A such that B(A)\ca, no finite risk measure ρ R for any choice of (S, p) is continuous from above. For the special case of a monetary risk measure, parts (i) and (ii) of Proposition 3.1 are wellknown, see e.g. [16, Theorem 4.12 and Proposition 4.17]. However, to our knowledge, so far there is no proof of Proposition 3.1 in this general form in the literature. As the proof is quite technical and thus lengthy we provide it in Appendix A. Note that the representation (3.4) is in terms of pricing rules consistent with (S, p) in that µ dom(ρ R ) only if µ S = p. If S = R and p = id R, these pricing rules can be identified with probability measures. Finally, we remark that continuity from above is indeed a property of the acceptance set as the dichotomy in Proposition 3.1(iii) shows. The following theorem is the already advertised main result of this section. For set functions µ, ν ba, we write µ ν if and only if ν(a) = 0 implies µ(a) = 0, A F. Moreover we define ba ν := {µ ba µ ν}, and ca ν analogously. Theorem 3.2. Let R = (A, S, p) be a risk measurement regime such that ρ R is finite and assume B(A) ca (that is ρ R is continuous from above). (i) There exists a WEAK REFERENCE PROBABILITY MEASURE P, that is a probability measure P on (Ω, F) such ρ R (cp) < for a suitable c > 0 and P dom(ρ R ), i.e. P(N) = 0 µ dom(ρ R) : µ(n) = 0. (3.7) (ii) For P as in (i) we have that dom(ρ R ) (ca P) +. (iii) If ρ R is normalised, then E 0 = {µ ca ρ R (µ) = 0}. Proof. For (i), recall from Proposition 3.1 that the assumption on A implies that any lower level set E n := {µ ca + ρ R (µ) n}, n N, is σ(ca, L )-compact. Together with convexity, this implies countable convexity, i.e. (λ k ) k N [0, 1], k N λ k = 1, (µ k ) k N E n = k N λ k µ k E n. According to [3, Theorem ] E n is σ(ca, ca )-compact and investigating the proof of [3, Theorem ] (i) (ii) for each n N there is a measure ν n E n such that A F : (ν n (A) = 0 µ E n : µ(a) = 0). By (3.4), the sequence (ν n ) n N satisfies ν n (Ω) ν n (Ω) ρ R (ν n) + n ρ R (1) + n. Define ν := n N 2 n ν n, c N := N n=1 2 n, ζ N := c 1 N N 2 n ν n, N N. n=1 8

9 ν ca + follows from the estimate ν(ω) n N 2 n (ρ R (1) + n) = ρ R (1) + 2. Every scalar multiple of ν satisfies (3.7), and moreover, ν = lim N ζ N with respect to σ(ca, L ). Lower semicontinuity and convexity of ρ R and ρ R (ν n) n imply ρ R(ν) lim inf N ρ R(ζ N ) lim N c 1 N N 2 n n = 2. Choosing c := ν(ω), the probability measure P := 1 c ν is a weak reference probability model. (ii) is an immediate consequence of (3.7). In order to prove (iii) note that normalisation implies 0 = ρ R (0) = inf{ρ R (µ) µ dom(ρ R )}. Hence, ρ R 0 and the family of subsets (E k ) k (0,1] of the compact set E 1 has the finite intersection property. Therefore E 0 = k (0,1] E k. Remark 3.3. Note that the converse of Theorem 3.2(i), namely dom(ρ R ) ν for some ν dom(ρ R ) implies continuity from above, is not true. As a counterexample, consider n=1 ess sup(x) := sup{m R P(X m) = 1}, where P is some fixed probability measure on (Ω, F). If we set A := {X L ess sup(x) 0}, the triple R = (A, R, id R ) is a risk measurement regime such that for all X L the estimate ρ R (X) = ess sup(x) X holds, whence dom(ρ R ) = ba P follows. However, ρ R is not continuous from above unless Ω is finite. Whenever a probability measure P satisfies (3.7) and X, Y L are equal P-almost surely (P-a.s.), (3.4) shows that ρ R (X) = ρ R (Y ). Hence, we may view ρ R as a function on the space of equivalence classes L P := L (Ω, F, P) with the corresponding properties. We recall that the least upper bound X := inf{m R P( X m) = 1}, X L P, is a norm on L P, making it into a Banach lattice together with the P-almost sure order, and the equivalence class generated by Ω ω 1 is a strong unit of L P. Its dual may be identified with ba P. Let ι : L L P be the canonical embedding, then it is straightforward to prove the following result. Corollary 3.4. In the situation of Theorem 3.2 define ρ : L P R by ρ( X) = ρ R (X), where X L satisfies ι(x) = X. Then ρ is well-defined and agrees with the risk measure ρ (ι(a),ι(s), p) on L P, where p( Z) = p(z) whenever Z = ι(z). It is norm-continuous and continuous from above. The dual function ρ (µ) := X dµ ρ( X), µ ba P, (3.8) sup X L P where X denotes an arbitrary representative of X, agrees with ρ R ba P. Also ρ( X) = sup µ dom(ρ R ) X dµ ρ R(µ), X L P, where X and X are related as before. 9

10 3.2. The model space L P and strong reference probability measures Supported by our results in Proposition 3.2 and Corollary 3.4 we will from now on consider the model space L P, acceptance sets A L P, security spaces S L P, pricing functionals p : S R and resulting finite risk measures ρ R directly defined on L P, where P is a probability measure on (Ω, F). Moreover, in the following we will stick to the usual convention of identifying equivalence classes of random variables in L P with an arbitrary representative of that class. By similar reasoning as in Proposition 3.1 and Proposition 3.2 we have the following result. Lemma 3.5. Let R = (A, S, p) be a risk measurement regime on L P such that ρ R is finite and normalised. ρ R is continuous from above if and only if { } B(A) := µ ba P sup Y dµ < ca P. Y A In that case where ρ R (X) = ρ R(µ) := sup µ (ca P ) + sup X L P X dµ ρ R(µ), X L P, (3.9) X dµ ρ R (X), µ ba P. (3.10) In particular, dom(ρ R ) = B(A) {ν ca P Z S : Z dν = p(z)}, the lower level sets E c, c R, are σ(ca P, L P )-compact, and E 0. If ρ R is positively homogeneous, then the analogues of (3.5) and (3.6) hold as well. Let us briefly discuss the meaning of the probability measure P in the case of a normalised risk measure. Firstly, the condition P dom(ρ R ) is sensible if one wants to avoid risk arbitrage in risk-sharing schemes. Given two agents operating on the same model space (Ω, F, P), assume agent 1 uses the risk criterion ρ R, agent 2 the simple criterion E P [ ]. If P dom(ρ R ), we can find some A F such that P(A) > 0 = µ(a) for all µ dom(ρ R ). Consequently, we can distribute 0 L P among the agents as follows: in the case of event A, agent 1 pays agent 2 an amount of k > 0 units of currency, which would correspond to individual risks ρ R (k1 A ) = sup ρ R(µ) = 0, E P [ k1 A ] = kp(a), k. µ dom(ρ R ) Leveraging k would lead to a distribution of nothing between two agents operating on the same beliefs such that one of them can increase her utility ad infinitum, whereas the risk profile of the other agent remains neutral, from a regulatory perspective resulting in a market inefficiency. Thus the existence of weak reference models is economically sensible beyond providing a consistent model for an a priori model-free risk criterion, and also beyond continuity from above and Theorem 3.2 and Corollary 3.4. The lower level set E 0 := {µ ca P ρ R (µ) = 0} and its subset P := {µ E 0 µ P} are of particular importance. Note that from (3.9) we directly obtain the following observation: 10

11 Lemma 3.6. For µ ca P it holds that X dµ > ρ R (X) for some X L P µ / E 0. if and only if The lemma can be interpreted as follows: A normative paradigm is that computing the risk of a financial position should be more conservative than pricing this position, i.e. prices should be a lower bound for risk. µ ca P corresponds to a linear pricing rule of the form L P X X dµ, therefore the only linear pricing rules satisfying the paradigm of not pricing beyond the risk level are those corresponding to µ E 0. For these (3.9) implies X L P : X dµ ρ R (X), (3.11) and equality holds at least on S. This interpretation is in line with the standard understanding of ρ R (µ) expressing the subjective belief of an agent in a particular pricing rule by means of penalising the price X dµ with the constant ρ R (µ) 0. Keeping this in mind, assume that the set P is empty and consider the following occurance of risk arbitrage: on the scenario space (Ω, F, P), the mapping ρ 0 : L P R, X sup X dµ, µ E 0 is usually deemed to be the closest coherent approximation of ρ R which is useful if liquidity issues and market frictions are negligible. Consider a second agent who also neglects these effects, but uses the risk criterion ρ 1 = E P [ ], which is in line with the model space agreed upon. Choose A F such that P(A) > 0, but sup µ E0 µ(a) = 0, and consider the following risk sharing scheme for 0 L P : ρ 0 (k1 A ) = 0, ρ 1 ( k1 A ) = E P [ k1 A ] = kp(a), k. We come across the same market inefficiency as above. Given these reflections, we have to ask the question whether there is a STRONG REFERENCE MODEL, this means, is there an element in P := {µ E 0 µ P}? In the following situation this notion is well-known: Proposition 3.7. Let R = (A, S, p) be a risk measurement regime on L P such that ρ R is normalised, and assume ρ R is a monetary risk measure which is also law-invariant with respect to some P P, i.e. ρ R (X) = ρ R (Y ) whenever X and Y share the same law under P. 6 Moreover assume the underlying probability space to be standard and atomless. Then P P. Proof. ρ R is convex, -l.s.c. and non-decreasing with respect to the P-a.s. order, hence it is non-decreasing with respect to second order stochastic dominance ssd (see [31, Theorem 2.1]). 7 By Jensen s inequality E P [X] ssd X, hence E P [X] = ρ R (E P [X]) ρ R (X). We conclude P E 0. 6 Law-invariance is an extremely strong assumption if the securities are random themselves or if the security space is higher-dimensional. 7 ρ R is non-decreasing with respect to ssd if E P [u(x)] E P [u(y )] for all u : R R concave and nondecreasing implies ρ R(X) ρ R(Y ). 11

12 Any ν P corresponds to a pricing rule which the agent fully takes into account and which also determines the underlying probability structure, so that we may replace P by P := 1 ν(ω) ν given that L P = L P. Such a strong reference model will play a crucial role when extending the domain for ρ R in the following section, therefore we devote the remainder of this section to investigating under which conditions they do or do not exist. 8 Clearly, sensitivity (c.f. Definition 2.3) is necessary to have P, but apart from the coherent case it is not sufficient. As an example consider two probability measures Q P such that Q P. Define P β := βq + (1 β)p, ρ : L P X sup E Pβ [X] (1 β) 2. β [0,1] For R = ({X ρ(x) 0}, R, id R ), we have that ρ = ρ R is a sensitive risk measure with E 0 = {Q} and P =. In the following we will use the notation F + := {A F P(A) > 0}. Lemma 3.8. Let R = (A, S, p) be a risk measurement regime such that ρ R is finite, continuous from above, and coherent. Then P if and only if ρ R is sensitive. Proof. We only prove sufficiency. If ρ R is coherent, then dom(ρ R ) = E 0; see the proof of (3.5). Moreover, continuity from above implies E 0 ca +. As ρ R is sensitive, we have that 0 < ρ R (1 A ) = sup µ E0 µ(a) for all A F +. Consequently, there is µ A E 0 such that µ A (A) > 0. In other words, E 0 P. The Halmos-Savage Theorem [16, Theorem 1.61] shows that there is a countable family (µ n ) n N E 0 such that {µ n n N} P. Now convexity and closedness of E 0 ensure that also ν := n N 1 2 n µ n E 0, i.e. P. The following theorems states sufficient conditions under which P without requiring coherence of ρ R. The conditions are all linked to the ability of ρ R to identify arbitrage, see the discussion above. First, we characterise the strong condition E 0 = P. Theorem 3.9. Let R = (A, S, p) be a risk measurement regime, and suppose that ρ R is finite, normalised, continuous from above, and sensitive. The following are equivalent: (i) E 0 = P; (ii) For all A F + we have ρ R ( k1 A ) < 0 for k > 0 sufficiently large; (iii) For all X (L P ) ++ we have ρ R ( X) < 0. Moreover, E 0 (L P ) ++. = P if ρ R is strictly monotone, i.e. ρ R (X) < ρ R (Y ) whenever Y X 8 The subjective interpretation of ρ R above gives another hint why P = should hold. If P is a weak reference probability model, using the convexity of dom(ρ R), E c P holds for any c > 0, so for all A F with P(A) > 0 we can find a µ c E c such that µ c(a) > 0, or in subjective terms, the possibility of the event A is arbitrarily plausible. It would be logical if this possibility is also reflected by the pricing rules in which the agent believes and which are thus fully taken into account without penalisation. 12

13 Proof. (iii) trivially implies (ii). Now assume (i) does not hold, i.e. there is some µ E 0 \P, hence µ(a) = 0 for some A F +. From 0 = ρ R (0) ρ R ( k1 A ) kµ(a) = 0 we infer ρ R ( k1 A ) = 0 for all k > 0, contradicting (ii). This shows that (ii) implies (i). In order to show that (iii) is implied by (i), assume we can find a X 0 in the negative cone with ρ R (X) = 0. As the level sets of ρ R are σ(ca P, L P )-compact, we can find a µ dom(ρ R ) such that 0 = ρ R (X) = X dµ ρ R (µ). This implies ρ R (µ) = 0 = X dµ, a CONTRADICTION to E 0 = P. Finally, strict monotonicity clearly implies (iii) by normalisation. The next aim is a characterisation of P in terms of the components of the risk measurement regime R = (A, S, p). Theorem Suppose that ρ R is sensitive. Let C L P convex cone containing A + ker(p). be the smallest weakly* closed (i) P if and only if C (L P ) ++ =. (ii) P if A + ker(p) satisfies the RULE OF EQUAL SPEED OF CONVERGENCE: Let (X n ) n N A and (Z n ) n N ker(p) be sequences such that X n + Z n 1 for all n. Suppose (t n ) n N is such that t n. If the rescaled vectors V n := t n (X n + Z n ) satisfy V n 0 in probability, then for all sets B F +, it holds that lim sup P(B {V n + n ε}) < P(B). (iii) P = if three sequences (X n ) n N, (Z n ) n N and (t n ) n N violate the rule of equal speed of convergence such that sup t n (X n + Z n ) <. n N A proof is given in Appendix B. 4. The Minkowski domain of a risk measure 4.1. Construction of the Minkowski domain and extension results Throughout Section 4 fix an acceptance set A L P, a security space S L P, and let p : S R be a pricing functional such that ρ R : L P R is a normalised, finite, sensitive risk measure which is continuous from above and satisfies P. By potentially performing an equivalent change of measure we can assume cp P for a suitable constant c. The aim of this section is to lift ρ R to a domain of definition denoted by L R whose structure is completely characterised by ρ R and thus consistent with the initial risk measurement regime, although it is in general strictly bigger than L P. The typical argument for restricting risk measures to bounded random variables namely, that this space is robust and thus not conflicting with the ambiguity 13

14 expressed by ρ R is not valid in this case, since L R will completely reflect the ambiguity as perceived by ρ R. To this end, we remark that ρ( X ) := sup µ dom(ρ R ) X dµ ρ R(µ), (4.1) where ρ R is given in (3.10), is well-defined for all X L0 P := L0 (Ω, F, P), possibly taking the value. In this sense the objects appearing in the following definition are well-defined. Definition 4.1. For c > 0 and X L 0 P let { ( ) } X c,r := inf λ > 0 ρ X λ c (inf := ), and X R := X 1,R. The MINKOWSKI DOMAIN for ρ R is the set L R := {X L 0 P X R < }. Note that we may interpret R as a Minkowski functional given the level set ρ( ) 1 (, 1], and its domain L R is thus called the Minkowski domain. Proposition 4.2. (i) For all c > 0 there exist constants A c, B c > 0 such that A c c,r R B c c,r. In particular L R = {X L 0 P X c,r < } for all c > 0, and ( c,r ) c>0 is a family of equivalent norms on L R. Moreover, X A ρr (1) X R, X L P, and thus L P LR. (ii) (L R, R ) is a Banach lattice. (iii) X Y implies X c,r Y c,r and thus L R is solid. In particular, L R is invariant under rearrangements of profits and losses, i.e. if ϕ L P attaining values in [ 1, 1], then ϕ X L R with ϕx c,r X c,r. Proof. First we set Λ c (X) := {λ > 0 ρ(λ 1 X ) c}, i.e. X c,r = inf Λ c (X). (i): Suppose that c (0, 1) and let X L 0 P. Note that X R = if and only if Λ 1 (X) =, which implies Λ c (X) = or equivalently X c,r =. Now assume X R <, and pick λ Λ 1 (X). As ρ R 0, we have c ρ(c X /λ) = sup µ dom(ρ R ) λ X dµ ρ R(µ) cρ( X /λ) c, which implies X R c X c,r. Trivially, Λ c (X) Λ 1 (X) and therefore X c,r X R. Hence, we may choose A c = c and B c = 1. The case c > 1 is treated similarly. Monotonicity implies that ρ( X / X ) ρ R (1) for all X L P, which yields X X ρr (1),R A ρr (1) X R and L P LR. 14

15 c,r is indeed a norm: The verification of the triangle inequality and homogeneity are straightforward. The definiteness of c,r follows from the fact, that for all λ Λ c (X) and ν P we obtain the estimate X L 1 ν λρ R ( X /λ) λc, i.e. 1 c X L 1 ν X c,r. (4.2) (ii) follows from [26, Proposition 4.10], and (iii) is an immediate consequence of the monotonicity of ρ( ). The proof of Proposition 4.6 will clarify the reason for introducing the norms c,r instead of just R. Remark 4.3. (i) If ρ R is coherent, then X c,r = c 1 ρ( X ). (ii) The Minkowski norm c,r can be interpreted as a generalisation of the so-called Aumann-Serrano economic index of riskiness (see [2] and [7, Example 3]). (iii) The Minkowski domain and similar spaces have appeared in [22, 26, 30]. The definition of L R depends on the nullsets of the probability measure P only, and thus is invariant under any choice of the underlying probability measure P dom(ρ R ). The main purpose for introducing the Minkowski domain L R is to extend ρ R to a larger domain than L P in a robust way in terms of the fundamentals, i.e. the risk measurement regime R = (A, S, p). There is a canonical candidate for this given by R := (Ã, S, p) where à := {X L R µ dom(ρ R) : X dµ ρ R(µ)}, (4.3) so à is given by lifting and thus also preserving the acceptability criteria X dµ ρ R (µ), µ dom(ρ R ), from L P to LR. Indeed the following Theorem 4.4 shows that R is a risk measurement regime, and that the corresponding risk measure ρ R preserves the dual representation of ρ R. Dual approaches to extending convex functions are commonly used in the literature; see, e.g., [14, 26]. Note that ρ R also preserves any functional form ρ R may have, as for instance in the case of the entropic risk measure in Example 5.3 below. Theorem 4.4. R := ( Ã, S, p) is a risk measurement regime on the Banach lattice (L R, R ). ρ R can be expressed as ρ R(X) = sup µ dom(ρ R ) X dµ ρ R(µ), X L R, (4.4) where ρ R is defined as in (3.10). A fortiori, ρ R L P = ρ R. Moreover, ρ R is l.s.c. on (L R, R ), and satisfies ρ R(X) = sup ρ R(X m). (4.5) m N 15

16 Proof. Note that for arbitrary µ dom(ρ R ) and all X 0, we have X dµ = sup X R ε>0 ( X dµ sup ρ X R + ε ε>0 X X R + ε ) + ρ R(µ) 1 + ρ R(µ), hence dµ is a bounded linear functional on L R, and L R L 1 µ. For arbitrary X L R and µ P we have sup{p(z) Z S, X + Z Ã} sup{p(z) Z S, p(z) Xdµ} = Xdµ <. Thus, R satisfies (2.1) and is indeed a risk measurement regime, because à is monotone by dom(ρ R ) (ca P) +, and convex as intersection of convex subsets of L R. It is straightforward to show (4.4), so ρ R is l.s.c. as pointwise supremum of a family of l.s.c. functions. In order to prove (4.5), let µ dom(ρ R ) be arbitrary and note that by the Monotone Convergence Theorem and monotonicity of ρ R, we have X dµ ρ R(µ) = sup m N (X m)dµ ρ R(µ) sup ρ R(X m) ρ R(X). m N Now take the supremum over µ dom(ρ R ) on the left-hand side. Another way to extend ρ R could be considering A := cl R (A). (4.6) and R = (A, S, p). We will discuss this approach in Remark 4.11 where we show that R is no risk measurement regime on L R in general, and that, where ρ R makes sense, it indeed equals ρ R. As announced in the introduction, we also consider the following extensions of ρ R given by monotone approximation procedures: and ξ(x) := sup inf ρ R(( n) X m), X L R, n N m N η(x) := inf sup ρ R (( n) X m), X L R. n N m N The question is under which conditions we have ρ R(X) = ξ(x) = η(x) = lim n ρ R(( n) X n). (4.7) Note that as a byproduct of (4.5), we obtain the estimate ρ R ξ η and X L R : ρ R( X ) = ξ( X ) = η( X ) = ρ( X ). (4.8) The following Theorem 4.5 shows that ρ R possesses some regularity in terms of monotone approximation in that always ρ R = ξ. 16

17 Theorem 4.5. For all X L R and all U L P we have ρ R(X + U) = sup inf ρ R(( n) X m + U). (4.9) n N m N A fortiori, the equality ρ R = ξ holds, and ρ R can equivalently be interpreted as the risk measure associated to the risk measurement regime R ξ := (A ξ, S, p) on (L R, R ), where A ξ := {X L R ξ(x) 0} = {X L R sup inf ρ R(( n) X m) 0}. n N Proof. In the following, for random variables U, V (L P ) + and X L R, we set X U := X ( U) and X V := X V. We show first that ρ R = ξ holds. Let X L R, m N be fixed and n N be arbitrary. Let µ dom(ρ R ) be such that ρ R( X ) 1 ρ R (Xn m ) 1 Xn m dµ ρ R(µ) (X + ) m dµ ρ R(µ). Of course, the first and last inequalities in the latter estimate always hold by monotonicity. For ε > 0 arbitrary we can thus estimate ρ R(µ) 1 (X + ) m dµ ρ R( X ) = 1 (1 + ε)(x + ) m dµ ρ R( X ) 1 + ε ε ρ R((1 + ε)(x + ) m ) ε ρ R(µ) ρ R( X ), m N where we used the positivity of µ. Rearranging this inequality, we obtain ρ R(µ) 1 + ε ε ( ρr ((1 + ε)(x + ) m ) ρ R( X ) + 1 ) =: c, a bound which is independent of n N. Since E c = {µ ca P ρ R (µ) c} is σ(ca P, L P )- compact by Lemma 3.5, we conclude for all n N that ρ R (Xn m ) = max µ Ec f(µ, n), where the function f is given by f : E c N R, f(µ, n) := Xn m dµ ρ R(µ), Our aim is to apply Fan s Minimax Theorem [10, Theorem 2] to the function f in order to infer ξ(x m ) = inf max f(µ, n) = max inf f(µ, n) = max inf X n n m dµ ρ R(µ). (4.10) µ E c µ E c n N µ E c n N To this end we have to check the following conditions: E c is a compact Hausdorff space when endowed with the relative σ(ca P, L P )-topology. This follows from continuity from above. 17

18 f is convex-like on N in that for all n 1, n 2 N and all 0 t 1 there is a n 0 N such that µ E c : f(µ, n 0 ) tf(µ, n 1 ) + (1 t)f(µ, n 2 ). Indeed, choose n 0 := max{n 1, n 2 } and note that tf(µ, n 1 ) + (1 t)f(µ, n 2 ) = t Xn m 1 dµ + (1 t) Xn m 2 dµ ρ R(µ) (t + 1 t) Xn m 0 dµ ρ R(µ) = f(µ, n 0 ). f is concave-like on E c, which is defined analogous to convex-like. Indeed, let µ 1, µ 2 E c and define µ 0 = tµ 1 + (1 t)µ 2 E c (by convexity of E c ). Then for all n N, convexity of ρ R implies tf(µ 1, n) + (1 t)f(µ 2, n) = Xn m dµ 0 tρ R(µ 1 ) (1 t)ρ R(µ 2 ) Xn m dµ 0 ρ R(µ 0 ) = f(µ 0, n). For all n N, the mapping µ f(µ, n) is upper semicontinuous. This follows from the continuity of µ X m n dµ and the lower semicontinuity of ρ R. From (4.10), by the positivity of µ and, e.g., dominated convergence, ξ(x m ) = max X m dµ ρ R(µ) ρ R(X m ), µ E c and ρ R(X m ) = ξ(x m ) holds by (4.8). Taking the limit m, we obtain from the definition of ξ and (4.5) that ρ R(X) = ξ(x). Now, let X L R and U L P be arbitrary and assume m, n u := U. We obtain and in addition (X + U) n = (X + U)1 {X U n} n1 {X< U n} = X1 {X U n} (n + U)1 {X< U n} + U = X U+n + U, (4.11) (X + U) m = (X + U)1 {X m U} + m1 {X>m U} = X1 {X m U} + (m U)1 {X>m U} + U = X m U + U. From these two equations (4.11) and (4.12) we infer This implies that ξ(x + U) = sup inf ρ R((X U+n + U) m ) = sup inf ρ R(X m U n u n u U+n + U). sup inf ρ R(X m m n n m u m u + U) = sup inf ρ R(X m U m n U+n = ξ(x + U) = ρ R(X + U). + U) = sup m inf n ρ R((X + U) m n ) (4.12) (4.9) is proved. ξ = ρ R being S-additive, monotone and proper directly implies R ξ is a risk measurement regime. The equality ρ R = ξ = ρ Rξ obviously holds true. 18

19 Theorem 4.5 appeared as [30, Lemma 2.8] in the context of law-invariant monetary risk measures. Our proof not only serves as an alternative to the one given in [30], relying irreducibly on law-invariance, but also generalises the result to a much wider class of risk measures. In contrast to Theorem 4.5, we demonstrate in Example 5.1 that ρ R η may happen. Before we study conditions under which ρ R displays regularity in the sense of (4.7), we show the following properties of η: Proposition 4.6. Define the acceptance set A η := {X L R inf n N ρ R(( n) X) 0} L R. Then η is the risk measure associated to the risk measurement regime R η := (A η, S, p). Moreover, X L R : η(x) = inf ρ R(( n) X), (4.13) n N and Γ := {X L R ε > 0 : ρ((1 + ε)x + ) < } = int dom(η) int dom(ρ R). Proof. From (4.5) and η L P = ρ R = ρ R L P, we immediately obtain that for all X L R the equality η(x) = inf n N ρ R(( n) X) holds. (4.8) shows that η is proper. In order to prove the theorem, it suffices to check S-additivity, convexity and monotonicity. Let Z S and X L R. From the S-additivity of ρ R and (4.11) we obtain, using the notational conventions introduced in the proof of Theorem 4.5, that η(x) = lim n ρ R(X Z+n + Z) = lim n ρ R(X Z+n ) + p(z) = η(x) + p(z). For each n N, f n (x) := ( n) x is convex and monotone, thus η = lim n ρ R f n is convex and monotone. Next we show that Γ int dom(η). To this end we first show that B := c>0 {Y LR Y c,r < 1} int dom(η). Indeed for any X with X c,r < 1, there is λ < 1 such that ρ( X /λ) c, and thus η(x) η( X ) = ρ R( X ) λρ R( X /λ) = λρ( X /λ) λc <, so B dom(η). Moreover, by definition B is open in (L R, R ). Now, let X Γ, and thus X + B. Hence, there is δ > 0 and a ball B δ (0) := {Y L R Y R < δ} such that {X + } + B δ (0) dom(η). By monotonicity of η it now follows that also {X} + B δ (0) = {X + } + B δ (0) {X } dom(η), so X int dom(η). In order to show Γ int dom(η) let X int dom(η). Then there is ε > 0 such that (1 + 2ε)X dom(η) and thus also (1 + ε)x dom(η), and by (4.13) there must be n N such that (1 + 2ε)(( n) X) dom(ρ R) and (1 + ε)(( n) X) dom(ρ R). Let X n := ( n) X and Y = (1 + ε)(x n) L P, so we have (1 + ε)x + = (1 + ε)x n + Y. If δ > 0 satisfies (1 + δ)(1 + ε) = 1 + 2ε, convexity implies ( ) 1 + δ ρ((1 + ε)x + ) = ρ R((1 + ε)x n + Y ) = ρ R 1 + δ (1 + ε)x δ(1 + δ) n + δ(1 + δ) Y ( ) (1 + δ) δ ρ R((1 + 2ε)X n ) + δ (1 + δ) ρ R Hence, X Γ. int dom(η) int dom(ρ R) follows from ρ R η, see (4.8). δ Y <. (4.14) 19

20 The following Theorem 4.7 states conditions under which (4.7) holds. Theorem 4.7. Let X Γ. Consider the following conditions: (i) there is s > 0 such that for all n N we have ρ R(( n) X) = lim m ρ R(( n) X + sx + 1 {X + m}); (ii) there is s > 0 such that η(x) = lim m η(x + sx + 1 {X + m}); (iii) for all n N we have lim m ρ(nx1 {X m} ) = 0. Any of the conditions (i)-(iii) implies (4.7). The set Γ appears to be a set of reasonable risks in that they can at least be leveraged by a small amount and still remain hedgeable. Risks outside Γ should probably not be considered by any sound agent. Note that the conditions (i)-(iii) are satisfied whenever monotone or dominated convergence results can be applied to ρ R, as is the case for many risk measures used in practice like the entropic risk measure in Example 5.3 or Average Value at Risk based risk measures in Example 5.4. The proof of Theorem 4.7 is based on a study of subgradients of ρ R and η, respectively, and therefore postponed to the end of Section 4.4. It turns out that the regularity condition (4.7) is closely related to the existence of regular subgradients for η and ρ R The structure of the Minkowski domain In this section, we will decompose L R into parts with clear operational meanings. Definition 4.8. We denote the closure of L P HEART of the Minkowski domain to be in LR by M R := cl R (L P ), and define the H R := {X L R ρ(k X ) < for all k > 0}. H R, a concept which clearly adapts the idea of an Orlicz heart, 9 is the set of risky positions which can be hedged at any quantity with finite cost. Proposition 4.9. M R and H R are solid Banach sublattices of L R and M R H R. Moreover, H R Γ, and both ρ R H R and η H R are continuous. Proof. The first assertions are easily verified. Recall the set B from the proof of Proposition 4.6 for which we know that B Γ. We show that H R B. To this end, let 0 X H R, and note that c := ρ(2 X ) γe P [ X ] > 0 is finite, where γ > 0 is a constant such that γp P. Then X c,r 1 2 < 1, so HR B. Finally, as (H R, R ) is a Banach lattice and both η and ρ R are convex, monotone and finite-valued on (H R, R ), ρ R H R and η H R are continuous according to Remark 2.4(iv). From Proposition 4.9 we can derive the following characterisation of M R, a result which can also be found as [26, Lemma 3.3]. 9 For an introduction to Orlicz space theory we refer to [27]. 20

21 Corollary M R = {X L R λ > 0 : lim k ρ(λ X 1 { X k} ) = 0}. Proof. Let X M R and λ, ε > 0 be arbitrary. Let δ > 0 such that Y δ, Y H R, implies ρ( Y ) = ρ R( Y ) ε. This is possible due to Proposition 4.9. Choose now Y L P such that λ(x Y ) R δ 2 and k N such that λy 1 { X k} R δ 2, the latter being due to continuity from above. Then Z := X Y 1 { X k} + Y 1 { X k} satisfies λz R δ, and by monotonicity ρ(λ X 1 { X k} ) ρ R(λZ) ε. The converse inclusion above is obvious. As H R is closed, the set of non-negative directions along which ρ R attains the value infinity is thus norm-open. In particular, we can only approximate such vectors with sequences of vectors along which ρ R behaves equally discontinuous, and limits of well-behaved financial positions are equally well-behaved. Hence shifting to L R yields a structure which conveniently separates regimes of good and bad risk behavior. In that respect consider the set C R := dom(ρ R)\H R L R. C R is the set of less bad positions, and shields H R from the financial positions that carry infinite risk. It has a nice interpretation in terms of liquidity risk in the sense of Lacker [24]. In that paper the author considers liquidity risk profiles, i.e. curves of the form ρ R(tX) t 0 capturing how risk scales when increasing the leverage. C R consists of financial positions X such that the liquidity risk profiles of X + or X breach the infinite risk regimes. Whereas an agent could at least hypothetically hedge any position in H R at finite cost, no matter what the leverage, she has to be very careful in the case of elements in C R that have finite risk themselves but which produce potentially completely non-hedgeable losses under incautious scaling. Recalling that for any X L R there is λ > 0 such that ρ( X /λ) <, we obtain that C R =, if and only if H R = L R, and ρ R is continuous. Moreover, if H R L R, both H R and M R are nowhere dense (as true subspaces of L R ) and by Baire s Theorem C R {ρ R = } is a dense open set. Note that the inclusions M R H R L R can all be strict, as is illustrated by Example 5.2. Remark Having introduced M R we can now discuss the extension given by the norm closure operation (4.6). Seen as a subset of L R, A is unfortunately not an acceptance set in the sense of Definition 2.1, since X Y and Y A does not necessarily imply X A, so the monotonicity property is violated. However, one can show that R := (A, S, p) is a risk measurement regime on the Banach lattice M R. By Proposition 4.9 it follows that ρ A (X) = ρ R(X) = η(x) for all X M R, and ρ A is continuous on M R The dual of L R In this short interlude we discuss a few properties of the norm dual (L R, R ) of (L R, R ), the space of continuous linear functionals on the Minkowski domain, which will be essential when we study subgradients in Section 4.4. Theorem L R is the direct sum of two subspaces CA and P A, i.e. L R = CA P A. 21