Mean- ρ portfolio selection and ρ -arbitrage for coherent risk measures
AA Dual Characterisation of Regulatory Arbitragefor Coherent Risk Measures ∗ Martin Herdegen † Nazem Khan ‡ September 14, 2020
Abstract
We revisit mean-risk portfolio selection in a one-period financial market where risk isquantified by a positively homogeneous risk measure ρ on L . We first show that undermild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact.However, unlike in classical mean-variance portfolio selection, it can happen that no efficientportfolios exist. We call this situation regulatory arbitrage, and prove that it cannot beexcluded – unless ρ is as conservative as the worst-case risk measure.After providing a primal characterisation, we focus our attention on coherent risk mea-sures, and give a necessary and sufficient characterisation for regulatory arbitrage. We showthat the presence or absence of regulatory arbitrage for ρ is intimately linked to the interplaybetween the set of equivalent martingale measures (EMMs) for the discounted risky assetsand the set of absolutely continuous measures in the dual representation of ρ . A specialcase of our result shows that the market does not admit regulatory arbitrage for ExpectedShortfall at level α if and only if there exists an EMM Q ≈ P such that (cid:107) d Q d P (cid:107) ∞ < α . Mathematics Subject Classification (2020):
JEL Classification:
G11, D81, C61
Keywords: portfolio selection, coherent risk measures, dual characterisation, regulatory arbi-trage, fundamental theorem of asset pricing
It has been widely argued that the financial crisis of 2007-2009 was a result of excessive risk-taking by banks; see e.g. [21, 44]. Consequently, the financial regulators have tried to imposebetter risk constraints on financial institutions, which for the banking sector are codified in theBasel accords. One of the key changes from Basel II to Basel III was updating the ‘official’ riskmeasure from Value at Risk (VaR) to Expected Shortfall (ES) in the hope of better financialregulation; cf. the discussion in [28].While Basel III is probably a step in the right direction, the following questions must beasked: Does an ES constraint really prevent banks from taking excessive risk? And if not,are there alternative coherent risk measures that are superior? In this paper we address thesequestions by revisiting the problem of portfolio optimisation in a one-period financial market –replacing the variance in the classical theory of Markowitz [32] by a positively homogeneous riskmeasure ρ , with ES a key example. We refer to this as mean- ρ portfolio selection in the sequel. ∗ The authors are grateful to John Armstrong for fruitful discussions. † University of Warwick, Department of Statistics, Coventry, CV4 7AL, UK, email [email protected] . ‡ University of Warwick, Department of Statistics, Coventry, CV4 7AL, UK, email [email protected] . For a thorough discussion of other “unexpected shortfalls” of ES we refer to [28]. a r X i v : . [ q -f i n . M F ] S e p ean-VaR portfolio selection was first studied by Alexander and Baptista [5] for multivariatenormal returns distributions, where all quantities can be calculated explicitly. Mean-ES portfolioselection was first studied by Rockafellar and Uryasev [36] for continuous returns distributions.They showed that the optimisation problem could be reduced to linear programming. Subse-quently, the results of [36] were extended to general returns distributions by the same authors in[40] and later generalised to spectral risk measures by Adam et al. [2]. We generalise the aboveresults by studying mean- ρ portfolio selection for positively homogeneous (but not necessarilyconvex) risk measures ρ . This allows us among others to also consider Value at Risk. In partic-ular, we prove existence of optimal portfolios (Theorem 3.11) without assuming convexity of ρ under minimal assumptions on the return distribution and the risk measure.Unlike in classical mean-variance portfolio selection (and even if optimal portfolios exist),it can happen that mean- ρ portfolio selection is ill-posed in the sense that there are no effi-cient portfolios or even worse that for each portfolio π , there is another portfolio π (cid:48) that hassimultaneously a higher expected return and a lower risk. Motivated by the idea that the riskmeasure is imposed by the regulator, we refer to these situations as regulatory arbitrage and strong regulatory arbitrage respectively. The occurrence of regulatory arbitrage was first recognised by Alexander and Baptista [5] forVaR, who considered multivariate normal returns distributions and gave necessary and sufficientconditions for the absence of regulatory arbitrage in this setup. Rockafellar et al. [38] recognisedthe occurrence of regulatory arbitrage for a wide class of coherent risk measures, noting thatminimising the risk subject to an inequality constraint on the expected return may fail to havea solution. They called this phenomenon an “acceptably free lunch”, but the main focus oftheir paper was the study of deviation risk measures (and the part on risk minimisation did notmake it into the published version [39]). Around the same time, the occurrence of regulatoryarbitrage for ES was noted by De Giorgi [15] in the case of elliptical returns distributions, andsubsequently been observed in a simulation study by Kondor et al. [29]. The latter paper ledto a more detailed study by Ciliberti et al. [14], who concluded that there is a phase transition ,i.e., for small values of α , mean-ES portfolio selection is well-posed, and from a certain criticalvalue α ∗ onwards, mean-ES portfolio selection becomes ill-posed. For example, if returns aremultivariate normal distributed and the maximal Sharpe ratio that can be attained in the marketis . , then α ∗ ≈ . – which is below the . of the Basel III accord; cf. Example 3.27 belowfor details. More recently, Armstrong and Brigo [7] argued that VaR and ES constraints may bevoid for behavioural investors with an S -shaped utility. In a recent preprint [6] they also studiedregulatory arbitrage (what they call ρ -arbitrage) for general coherent risk measures, focusing onmultivariate normal returns and looking at the issue from a numerical/statistical perspective.A different strand of literature closely related to strong regulatory arbitrage is the notion ofa good deal ; see e.g. [26]. For expectation bounded risk measures (cf. Remark 4.10), the twoconcepts are indeed equivalent if we define a good deal as Cherny in [12], who proves a dualcharacterisation of no good deals . However, Cherny’s result only applies to strong regulatoryarbitrage. There is a subtle, yet crucial difference between the two concepts: Strong regulatoryarbitrage is a generalisation of arbitrage of the second kind whereas regulatory arbitrage is a The term ‘regulatory arbitrage’ has been emphasised in the literature more intensively since 2004 as explainedin [43]. However, there is no universal definition for this concept. The general consensus is that it is a notionthat refers to actions performed by financial institutions to avoid unfavourable regulation. A deviation risk measure D is a generalisation of the standard deviation, and has been axiomatically studiedin [39]. Mean- D portfolio selection has first been studied by Tasche [42] and Bertsimas et al. [10] who looked atthe special case that D is Expected Shortfall Deviation. Their results were later extended to general deviationrisk measures by Rockafellar et al. [37], who showed existence (and non-uniqueness) of optimal portfolios, provedexistence of efficient portfolios and provided a full characterisation of the efficient frontier. In the conclusion section, [26] raise the question: “What parts of Markovitz’ theory carry over, what isdifferent in ( µ, ρ ) -optimisation?” (p. 199) Note that Value at Risk is not expectation bounded, so that this restrictions is practically relevant. ρ portfolio selection and provide necessary and sufficient conditions for theabsence of regulatory arbitrage for coherent risk measures ρ who admit a dual representation(Theorem 4.17). This is very delicate as it requires to find the right concept of “interior” for thedual set of absolutely continuous measures in the dual characterisation of ρ . While our prooftechniques are rather elementary, the ensuing results are not. Indeed, as a by-product of ourmain result, we get a refined version of the fundamental theorem of asset pricing in a one-periodmarket: for returns in L , there exists an EMM Q whose Radon Nikodým derivative is uniformlybounded away from (Theorem 6.2).The rest of the paper is organised as follows. In Section 2, we describe our financial marketmodel. Section 3 is devoted to a rigorous study of mean- ρ portfolio selection, which to the bestof our knowledge has not been carried out at this level of generality in the literature before.As an example we provide a full characterisation of the absence of (strong) regulatory arbitragefor elliptical returns in terms of the Sharpe ratio of portfolios (which has some non-obviouspitfalls; cf. Remark 3.26). In Section 4 we state and prove our main results on the dualcharacterisations of the absence of (strong) regulatory arbitrage. Section 5 is devoted to thecase that returns lie in some Orlicz space L Φ . Section 6 contains a large variety of examples andSection 7 concludes. Appendix A provides various counterexamples to complement our resultsand Appendix B contains some additional results. We consider a one-period (1 + d ) -dimensional market ( S t , . . . , S dt ) t ∈{ , } on some probabilityspace (Ω , F , P ) . We assume that S is riskless and satisfies S = 1 and S = 1 + r , where r > − denotes the riskless rate. We further assume that S , . . . , S d are risky assets, where S , . . . , S d > and S , . . . , S d are real-valued F -measurable random variables. We denote the(relative) return of asset i ∈ { , . . . , d } by R i := S i − S i S i , and set S := ( S , . . . , S d ) and R := ( R , . . . , R d ) for notational convenience.We may assume without loss of generality that the market is nonredundant in the sensethat (cid:80) di =0 ϑ i S i = 0 P -a.s. implies that ϑ i = 0 for all i ∈ { , . . . , d } . We also impose that therisky returns R , . . . , R d are P -integrable, which is a minimum requirement for mean- ρ portfolioselection. Thus, each asset i ∈ { , . . . , d } has a finite expected return µ i := E [ R i ] , and we set µ := ( µ , . . . , µ d ) ∈ R d . Finally, we may assume without loss of generality that the risky returnsare nondegenerate in the sense that for at least one i ∈ { , . . . , d } , µ i (cid:54) = r . Note that this impliesthat P itself is not an equivalent martingale measure for the discounted risky assets S/S . As S , . . . , S d > , we can parametrise trading in fractions of wealth , and we assume that tradingis frictionless. More precisely, we fix an initial wealth x > and describe any portfolio (forthis initial wealth) by a vector π = ( π , . . . , π d ) ∈ R d , where π i denotes the fraction of wealthinvested in asset i ∈ { , . . . , d } . The fraction of wealth invested in the riskless asset is in turn Indeed, if µ i = r for all i ∈ { , . . . , d } , then every portfolio π ∈ R d has zero expected excess return. Therewould be no incentive to invest and mean-risk portfolio optimisation becomes meaningless. π := 1 − (cid:80) di =1 π i = 1 − π · , where := (1 , . . . , ∈ R d . The return of a portfolio π ∈ R d can be computed by R π := (1 − π · ) r + π · R, and the excess return of a portfolio π ∈ R d over the riskless rate r is in turn given by X π := R π − r = (1 − π · ) r + π · R − r = π · ( R − r ) . (2.1)Note that X aπ + bπ (cid:48) = aX π + bX π (cid:48) for any portfolios π , π (cid:48) ∈ R d and any constants a, b ∈ R . The expected excess return of a portfolio π ∈ R d over the riskless rate r can be calculated as E [ X π ] = π · ( µ − r ) . For fixed ν ∈ R , we set Π ν := { π ∈ R d : E [ X π ] = ν } , (2.2)i.e., Π ν denotes the set of all portfolios with expected excess return ν . By nondegeneracy, Π ν (cid:54) = ∅ for all ν ∈ R . Moreover, it is easy to check that Π ν is closed and convex for each ν ∈ R . Finally,the definition of Π ν in (2.2) implies that Π k = (cid:40) k Π := { kπ : π ∈ Π } , if k > , ( − k )Π − := {− kπ : π ∈ Π − } , if k < . (2.3)In the following, we will only focus on nonnegative excess returns. ρ portfolio selection As Π ν (cid:54) = ∅ for all ν ∈ R , it is clear that in order to study portfolio selection, some kindof risk constraint has to be imposed. In the classical mean-variance approach pioneered byMarkowitz [32], risk is measured by variance. Here we follow the axiomatic approach of Artzneret al. [8] and focus on a positively homogeneous monetary measure of risk ρ : L −→ ( −∞ , ∞ ] ,which satisfies the axioms: • Monotonicity : For any X , X ∈ L such that X ≤ X P -a.s., ρ ( X ) ≥ ρ ( X ) . • Cash invariance: If X ∈ L and c ∈ R , then ρ ( X + c ) = ρ ( X ) − c . • Positive homogeneity:
For all X ∈ L and λ ≥ , ρ ( λX ) = λρ ( X ) . Remark 3.1.
Note that monotonicity of ρ is not used in any of the results of this section.We recall the definition of the two most prominent examples of risk measures satisfying theabove axioms, Value at Risk and Expected Shortfall. More examples are given in Section 6. Definition 3.2.
Let α ∈ (0 , be a confidence level and X ∈ L . • The
Value at Risk (VaR) of X at confidence level α is given byVaR α ( X ) := inf { m ∈ R : P [ m + X < ≤ α } . • The
Expected Shortfall (ES) of X at confidence level α is given byES α ( X ) := 1 α (cid:90) α VaR u ( X ) d u. Note that the (relative) return of a portfolio does not depend on the initial wealth x . Indeed, ν ( µ − r )( µ − r ) · ( µ − r ) ∈ Π ν for each ν ∈ R .
4e start our discussion on mean- ρ portfolio selection by introducing a partial preferenceorder on the set of portfolios. This preference order formalises the idea that return is “desirable”and risk is “undesirable”. Definition 3.3.
A portfolio π ∈ R d is strictly ρ -preferred over another portfolio π (cid:48) ∈ R d if E [ X π ] ≥ E [ X π (cid:48) ] and ρ ( X π ) ≤ ρ ( X π (cid:48) ) , with at least one inequality being strict. Remark 3.4.
Linearity of the expectation and cash-invariance of ρ imply that a portfolio π ∈ R d is strictly ρ -preferred over another portfolio π (cid:48) ∈ R d if and only if E [ R π ] ≥ E [ R π (cid:48) ] and ρ ( R π ) ≤ ρ ( R π (cid:48) ) , with at least one inequality being strict. This equivalent formulation mightseem more natural from an economic perspective. However, it turns out that working withexcess returns is mathematically more convenient. We approach the problem of mean- ρ portfolio selection by first looking at the slightly simplifiedproblem of finding the minimum risk portfolio given a fixed excess return. Since a negativeexcess return corresponds to an expected loss, we only focus on portfolios with nonnegativeexpected excess returns. Definition 3.5.
Let ν ≥ . A portfolio π ∈ R d is called ρ -optimal for ν if ρ ( X π ) < ∞ and ρ ( X π ) ≤ ρ ( X π (cid:48) ) for all π (cid:48) ∈ Π ν . We denote the set of all ρ -optimal portfolios for ν by Π ρν .Moreover, we set ρ ν := inf { ρ ( X π ) : π ∈ Π ν } ∈ [ −∞ , ∞ ] , (3.1)and define the ρ -optimal boundary by O ρ := { ( ρ ν , ν ) : ν ≥ } ⊂ [ −∞ , ∞ ] × [0 , ∞ ) . As the riskless portfolio has zero risk, ρ ≤ . Positive homogeneity implies that either ρ = −∞ (in which case Π ρ = ∅ ) or ρ = 0 (in which case ∈ Π ρ ). For ν > , positivehomogeneity gives Π ρν = ν Π ρ and ρ ν = νρ . Thus, the ρ -optimal boundary is given by O ρ = { ( ρ , } ∪ { ( kρ , k ) : k > } , (3.2)where ρ ∈ {−∞ , } and ρ ∈ [ −∞ , ∞ ] . Note that the ρ -optimal boundary is nonempty evenif ρ -optimal portfolios do not exist. Depending on the sign of ρ , Figure 1 gives a graphicalillustration of the three different shapes O ρ can take when ρ = 0 and ρ ∈ R . One might also wonder why we apply ρ to the (relative) return of a portfolio rather than the absolute return x R π or the final cash value x ( R π + 1) . As ρ is cash-invariant and positive homogeneous, this does not mattersince in each case we get exactly the same preference order as in Definition 3.3. We note in passing that if we dropthe assumption of positive homogeneity, this is no longer true and one has to be much more careful in thinkingabout which quantity the risk measure should be applied to. If ρ ν = −∞ , since ρ can only take values in ( −∞ , ∞ ] , for every portfolio in Π ν there is another portfolio in Π ν with strictly lower risk. Thus, Π ρν = ∅ . If ρ ν = ∞ , every portfolio in Π ν has infinite risk, and so Π ρν = ∅ . ρ -optimal boundary when ρ = 0 and ρ ∈ R We now seek to understand under which conditions ρ -optimal portfolios exist and whichproperties ρ -optimal sets have. First, we consider the case ν = 0 , which is also of key importancefor the case ν > . Proposition 3.6. Π ρ (cid:54) = ∅ if and only if ρ = 0 . Moreover in this case, either Π ρ = { } or Π ρ fails to be compact.Proof. If ρ = 0 , then ∈ Π ρ . If ρ (cid:54) = 0 , then ρ = −∞ and Π ρ = ∅ . Moreover, if ρ = 0 andthere is π (cid:54) = with ρ ( X π ) = 0 , it follows from positive homogeneity that λπ ∈ Π ρ for all λ ≥ and hence Π ρ fails to be compact.We proceed to provide sufficient conditions that guarantee ρ = 0 or even Π ρ = { } . Definition 3.7.
A risk measure ρ : L → ( −∞ , ∞ ] is called expectation bounded if ρ ( X ) ≥ E [ − X ] for all X ∈ L . It is called strictly expectation bounded if ρ ( X ) > E [ − X ] for all non-constant X ∈ L . Remark 3.8. (a) (Strict) expectation boundedness is a natural requirement on a risk measurethat is satisfied by Expected Shortfall and a large class of coherent risk measures; see [39] fordetails and cf. also Remark 4.1(d) and Proposition 4.8.(b) Value at Risk is not expectation bounded (apart from degenerate probability spaces). Forexample, if Z is a standard normal random variable, then VaR α ( Z ) < E [ − Z ] for α > / .This failure of expectation boundedness for Value at Risk has some undesirable consequenceslike the non-existence of optimal portfolios; cf. Remark 3.26.(c) By cash-invariance of ρ , it suffices to consider X ∈ L with E [ X ] = 0 in the definition of(strict) expectation boundedness.We proceed to show that under (strict) expectation boundedness of ρ , optimal portfolios for ν = 0 exist (and are unique). Corollary 3.9. If ρ is expectation bounded, then ρ = 0 . If ρ is even strictly expectationbounded, then Π ρ = { } .Proof. If ρ is expectation bounded, then for any π ∈ Π , ρ ( X π ) ≥ E [ − X π ] = 0 and we mayconclude that ρ = 0 . If ρ is strictly expectation bounded, fix π ∈ Π \ { } . Then X π is non-constant by nonredundancy of the financial market. Strict expectation boundedness of ρ gives ρ ( X π ) > E [ − X π ] = 0 . We may conclude that Π ρ = { } .We next consider ρ -optimal sets for ν > . To this end, we recall the Fatou property for ρ .6 efinition 3.10. Let
X ⊂ L . The risk measure ρ is said to satisfy the Fatou property on X if X n a.s. −−→ X for X n , X ∈ X and | X n | ≤ Y for some Y ∈ L implies ρ ( X ) ≤ lim inf n →∞ ρ ( X n ) .We now come to our main result of this section, which establishes existence of ρ -optimalportfolios under very weak assumptions on ρ , only requiring that ρ satisfies the Fatou propertyon X := { X π : π ∈ R d } and Π ρ = { } . In particular, we do not require ρ to be convex, whichis a key assumption in the extant literature; see e.g. [37, Proposition 4]. Theorem 3.11.
Assume that Π ρ = { } , ρ ∈ R and ρ satisfies the Fatou property on X := { X π : π ∈ R d } . Then for any ν ≥ , the set Π ρν of ρ -optimal portfolios for ν is nonempty andcompact.Proof. The key idea of the proof is to consider the function f ρ : R d → [0 , ∞ ] , defined by f ρ ( π ) = (cid:40) ρ ( X π ) + ( | ρ | + 1) E [ X π ] , if π ∈ ∪ k ≥ Π k , ∞ , if π ∈ ∪ k< Π k . Then f ρ is a nonnegative, positively homogeneous function and satisfies f − ρ ( { } ) = { } . More-over, if π n → π in R d , we have E [ X π ] = lim n →∞ E [ X π n ] as well as ρ ( X π ) ≤ lim inf n →∞ ρ ( X π n ) because ρ satisfies the Fatou property on X . This implies that f ρ is lower semi-continuous.We proceed to show that f ρ has compact sublevel sets. As ρ < ∞ , there is at least oneportfolio π ∗ ∈ Π with ρ ( X π ∗ ) < ∞ . Let S = { x ∈ R d : (cid:107) x (cid:107) = (cid:107) π ∗ (cid:107) } . As S is compactand f ρ lower semi-continuous, m := min { f ρ ( x ) : x ∈ S } is well defined. Note that m > since (cid:107) π ∗ (cid:107) > and f − π ( { } ) = { } . As f ρ is positively homogeneous, f ρ ( π ) ≥ m (cid:107) π ∗ (cid:107) (cid:107) π (cid:107) for anyportfolio π ∈ R d . Thus, f ρ has bounded sublevel sets, which are also closed since f ρ is lowersemi-continuous.We finish by a standard argument. For δ ≥ , set A δ := { π ∈ R d : f ρ ( π ) ≤ δ } ∩ Π and δ := inf { f ρ ( π ) : π ∈ Π } . Note that δ < ∞ since ρ ∈ R . Moreover, each A δ is compact andnonempty for δ > δ . As the A δ are nested (i.e., A δ ⊂ A δ (cid:48) for δ ≤ δ (cid:48) ), it follows that Π ρ = A δ = (cid:92) δ>δ A δ is nonempty and compact. Whence, so is Π ρν = ν Π ρ for any ν > . (For ν = 0 , the claim istrivial.) Remark 3.12. (a) The requirement that ρ satisfies the Fatou property on X is a mild assump-tion, which is satisfied by VaR and ES. Anticipating ourselves a bit, we note that it is satisfied byany risk measure ρ : L −→ ( −∞ , ∞ ] admitting a dual representation ρ ( X ) = sup Z ∈Q ( E [ − ZX ]) for some nonempty set Q of Radon-Nikodým derivatives satisfying ZR i ∈ L for all Z ∈ Q and i ∈ { , . . . , d } ; cf. Proposition 4.2.(b) By Corollary 3.9, the requirement that Π ρ = { } is automatically satisfied if ρ is strictlyexpectation bounded.(c) If ρ is in addition convex , i.e., ρ ( λX +(1 − λ ) X ) ≤ λρ ( X )+(1 − λ ) ρ ( X ) for X , X ∈ L and λ ∈ [0 , , then we also have convexity of ρ -optimal sets. Indeed, let ν ≥ , π, π (cid:48) ∈ Π ρν ,and λ ∈ [0 , . Then ρ ( X λπ +(1 − λ ) π (cid:48) ) = ρ ( λX π + (1 − λ ) X π (cid:48) ) ≤ λρ ( X π ) + (1 − λ ) ρ ( X π (cid:48) ) = ρ ν .Therefore, λπ + (1 − λ ) π (cid:48) ∈ Π ρν .(d) If | ρ | = ∞ , then Π ρν = ∅ for all ν > . If ρ ∈ R and { } (cid:40) Π ρ , then boundedness ofthe sublevel sets is lost (since f − ρ ( { } ) is unbounded) and Π ρν can be empty for all ν > ; seeExample A.1 for a concrete counterexample. 7 .2 Efficient portfolios We proceed to study the notion of ρ -efficient portfolios, which are defined in analogy to efficientportfolios in the classical mean-variance sense. Definition 3.13.
A portfolio π ∈ R d is called ρ -efficient if E [ X π ] ≥ and there is no otherportfolio π (cid:48) ∈ R d that is strictly ρ -preferred over π . We denote the ρ -efficient frontier by E ρ := { ( ρ ( X π ) , E [ X π ]) : π is ρ -efficient } ⊂ R . Remark 3.14. (a) If π ∈ R d is ρ -efficient, it follows that ρ ( X π ) < ∞ . Indeed, if E [ X π ] = 0 and ρ ( X π ) = ∞ , then is strictly ρ -preferred over π , and if E [ X π ] > and ρ ( X π ) = ∞ , then λπ is strictly ρ -preferred over π for λ > .(b) It follows from (a) that every ρ -efficient portfolio is ρ -optimal.(c) If ρ is expectation bounded, we may drop the assumption that E [ X π ] ≥ for π to beefficient since under expectation boundedness, for any portfolio π with E [ X π ] < , we have ρ ( X π ) ≥ E [ − X π ] > , and so the riskless portfolio is strictly ρ -preferred over π .Remark 3.14(b) implies that E ρ ⊆ O ρ . However, unlike in the case of mean-variance portfoliooptimisation, it can happen that there are no ρ -efficient portfolios – even if ρ -optimal portfoliosexist for all ν ≥ . The following result shows that when Π ρν (cid:54) = ∅ for all ν ≥ (which is satisfiedunder the conditions of Theorem 3.11), then the existence of the ρ -efficient frontier depends onlyon the sign of ρ . Proposition 3.15.
Assume Π ρν (cid:54) = ∅ for all ν ≥ . Then the following are equivalent: (a) ρ > . (b) E ρ (cid:54) = ∅ .Moreover, if ρ > , the ρ -efficient frontier is given by E ρ = { ( kρ , k ) : k ≥ } . Proof.
First assume that ρ > . We proceed to show that any ρ -optimal portfolio is ρ -efficient.It then follows from Remark 3.14(b) and Proposition 3.6 that E ρ = O ρ = { ( kρ , k ) : k ≥ } . Seeking a contradiction, let π ∈ Π ρν for some ν ≥ and assume that there is π (cid:48) ∈ R d suchthat E [ X π (cid:48) ] ≥ E [ X π ] = ν and ρ ( X π (cid:48) ) ≤ ρ ( X π ) = νρ , with one inequality being strict. Set ν (cid:48) := E [ X π (cid:48) ] . If ν (cid:48) = ν , then ρ ( X π (cid:48) ) < ρ ( X π ) and we arrive at a contradiction as π ∈ Π ρν .Otherwise, if ν (cid:48) > ν , let π ∗ ∈ Π ρν (cid:48) . Then ν (cid:48) ρ = ρ ( X π ∗ ) ≤ ρ ( X π (cid:48) ) ≤ ρ ( X π ) = νρ . Since ρ > ,we arrive at the contradiction that ν (cid:48) > ν and ν (cid:48) ≤ ν .Now assume that ρ ≤ . We proceed to show that there does not exist any ρ -efficientportfolio, even though Π ρν (cid:54) = ∅ for all ν ≥ . Seeking a contradiction, suppose that π ∈ R d is ρ -efficient. Then by Remark 3.14(b), π ∈ Π ρν for some ν ≥ . Pick ν (cid:48) > ν and let π (cid:48) ∈ Π ρν (cid:48) .Then E [ X π (cid:48) ] = ν (cid:48) > ν = E [ X π ] and ρ ( X π (cid:48) ) = ν (cid:48) ρ ≤ νρ = ρ ( X π ) by positive homogeneity of ρ and ρ ≤ . Hence, π (cid:48) is strictly ρ -preferred over π and we arrive at a contradiction. Remark 3.16.
A close inspection of the proof of Proposition 3.15 reveals that the equivalencebetween (a) and (b) remains true if we only require than Π ρν (cid:54) = ∅ for all ν > . However, if Π ρ = ∅ , the ρ -efficient frontier is given by E ρ = { ( kρ , k ) : k > } . It is an open question if there exists a risk measure satisfying Π ρν (cid:54) = ∅ for all ν > but Π ρ = ∅ . It is clearthat if it exists, ρ fails to be convex. ρ -optimal boundary (red) and ρ -efficient frontier (green) when Π ρν (cid:54) = ∅ for all ν ≥ We have seen above that mean- ρ portfolio selection is not always well defined as it can happenthat there are no ρ -efficient portfolios. We call this situation regulatory arbitrage . Definition 3.17.
The market ( S , S ) is said to satisfy regulatory arbitrage for ρ if there areno ρ -efficient portfolios. It is said to satisfy strong regulatory arbitrage for ρ if for any portfolio π ∈ R d , there exists another portfolio π (cid:48) such that E [ X π (cid:48) ] > E [ X π ] and ρ ( X π (cid:48) ) < ρ ( X π ) . It is clear that strong regulatory arbitrage implies regulatory arbitrage but not vice versa.
Remark 3.18. (a) In Section 6.1, we explain how (strong) regulatory arbitrage for the worst-case risk measure is equivalent to arbitrage of the first (second) kind. In this sense regulatoryarbitrage can be seen as an extension of the ordinary notion of arbitrage.(b) One might wonder if (strong) regulatory arbitrage is not just a pathology that disappearsfor reasonable (i.e., arbitrage-free) markets and risk measures. This is not the case. In fact, weshow in Theorem 6.4 that unless ρ is as conservative as the worst-case risk measure, one can always construct a financial market that is arbitrage-free but admits strong regulatory arbitragefor ρ .The following two results give primal characterisations for strong regulatory arbitrage andregulatory arbitrage. Whereas strong regulatory arbitrage is fully characterised by the sign of ρ ,defined in (3.1), the case of regulatory arbitrage is more subtle. Theorem 3.19.
The market ( S , S ) admits strong regulatory arbitrage for ρ iff ρ < .Proof. First, assume that the market satisfies strong regulatory arbitrage for ρ . As the risklessportfolio has zero risk and zero return, by definition of strong regulatory arbitrage, there is a π ∈ R d with E [ X π ] =: ν > and ρ ( X π ) < . Let π (cid:48) := ν π . Then π (cid:48) ∈ Π , and ρ ≤ ρ ( X π (cid:48) ) = ν ρ ( X π ) < . Conversely, assume that ρ < . Then there exists a portfolio π ∈ Π with ρ ( X π ) < .Thus, E [ X kπ ] → ∞ and ρ ( X kπ ) → −∞ as k → ∞ . Therefore, for any portfolio π (cid:48) (recallingthat E [ X π (cid:48) ] ∈ R and ρ ( X π (cid:48) ) ∈ ( −∞ , ∞ ] ), there exists k ∈ N such that E [ X kπ ] > E [ X π (cid:48) ] and ρ ( X kπ ) < ρ ( X π (cid:48) ) . Hence, the market satisfies strong regulatory arbitrage for ρ .9 heorem 3.20. We have the following three cases: (a) If Π ρ (cid:54) = ∅ , then the market ( S , S ) admits regulatory arbitrage for ρ iff ρ ≤ . (b) If Π ρ = ∅ and Π ρ (cid:54) = ∅ , then the market ( S , S ) admits regulatory arbitrage for ρ iff ρ < . (c) If Π ρ = ∅ and Π ρ = ∅ , then the market ( S , S ) admits regulatory arbitrage for ρ .Proof. (a) This follows from Proposition 3.15 and Remark 3.16.(b) If ρ < , by Theorem 3.19 the market admits strong regulatory arbitrage and a fortioriregulatory arbitrage for ρ . Conversely, if ρ ≥ , any portfolio π ∈ R d with E [ X π ] =: ν > has ρ ( X π ) > νρ = 0 because Π ρν = ν Π ρ = ∅ . Thus, any portfolio in Π ρ is ρ -efficient because Π ρ (cid:54) = ∅ (and therefore ρ = 0 ). Thus, the market does not admit regulatory arbitrage for ρ .(c) This follows from Remark 3.14(b). The primal characterisations of (strong) regulatory arbitrage in Theorems 3.19 and 3.20 areparticularly useful when returns are elliptically distributed with finite second moments and therisk measure is law invariant. We briefly recall both concepts.
Definition 3.21. An R d -valued random vector X = ( X , . . . , X d ) has an elliptical distribution if there exists a location vector (cid:101) µ ∈ R d , a d × d nonnegative definite dispersion matrix (cid:101) Σ ∈ R d × d ,and a characteristic generator ψ : [0 , ∞ ) → R such that the characteristic function of X , φ X can be expressed as φ X ( t ) = e it (cid:62) (cid:101) µ ψ ( t T (cid:101) Σ t ) for all t ∈ R d . In this case we write X ∼ ˜ E d ( (cid:101) µ, (cid:101) Σ , ψ ) .Elliptical distributions are generalisations of the multivariate normal distribution, whichallow for heavy tail models while possessing many useful properties. Indeed, the fat tails of mostof their members make them natural candidates in modelling the distribution of speculativereturns. Examples of elliptical distributions include the multivariate normal distribution, themultivariate t-distribution and the multivariate symmetric Laplace distribution. For a thoroughdescription of elliptical distributions refer to [22, 31]. Remark 3.22. If X has an elliptical distribution with finite second moments, X is also char-acterised by its mean vector µ ∈ R d , covariance matrix Σ ∈ R d × d and characteristic generator ψ . Therefore, we may write X ∼ E d ( µ, Σ , ψ ) ; see [33, Remark 3.27] for details. Definition 3.23.
A risk measure ρ : L → ( −∞ , ∞ ] is called law-invariant if ρ ( X ) = ρ ( X ) whenever X , X ∈ L have the same law.The following result shows why elliptical distributions and law-invariant risk measures workparticularly nicely together. Lemma 3.24.
Suppose ρ is law-invariant and the return vector R has an elliptical distributionwith mean vector µ ∈ R d , covariance matrix Σ ∈ R d × d and characteristic generator ψ . Let Z ∼ E (0 , , ψ ) . Then for any portfolio π ∈ R d , ρ ( X π ) = − E [ X π ] + ρ ( Z ) (cid:112) Var( X π ) = − π (cid:62) ( µ − r ) + ρ ( Z ) √ π (cid:62) Σ π. (3.3) Moreover, ρ ( Z ) is nonnegative (positive) if ρ is (strictly) expectation bounded. Note that ρ ( Z ) ∈ ( −∞ , ∞ ] . We employ the convention that ∞ × , so that ρ ( X π ) = − E [ X π ] if Var( X π ) = 0 . roof. Standard properties of elliptical distributions imply that π · ( R − r ) ∼ E (cid:0) π · ( µ − r ) , π T Σ π, ψ (cid:1) for any portfolio π ∈ R d . This means that X π d = π (cid:62) ( µ − r ) + Z √ π (cid:62) Σ π , where Z ∼ E (0 , , ψ ) .As ρ is a law-invariant, ρ ( X π ) = − π (cid:62) ( µ − r ) + ρ ( Z ) √ π (cid:62) Σ π . The final claim follows from thefact that E [ Z ] = 0 because Z ∼ E (0 , , ψ ) has a symmetric distribution.With the help of Lemma 3.24, we can give a very simple characterisation for the absence of(strong) regulatory arbitrage in terms of the maximal Sharpe ratio . Corollary 3.25.
Suppose ρ is law-invariant and the return vector R has an elliptical distributionwith mean vector µ ∈ R d satisfying µ (cid:54) = r , positive definite covariance matrix Σ ∈ R d × d andcharacteristic generator ψ . Let Z ∼ E (0 , , ψ ) . Define the maximal Sharpe ratio as SR max := max π ∈ R d \{ } E [ X π ] (cid:112) Var( X π ) = (cid:113) ( µ − r ) (cid:62) Σ − ( µ − r ) . (3.4) Then we have the following trichotomy: (a) If SR max < ρ ( Z ) , the market ( S , S ) does not admit regulatory arbitrage for ρ . (b) If SR max = ρ ( Z ) , the market ( S , S ) admits regulatory arbitrage for ρ but does not admitstrong regulatory arbitrage for ρ . (c) If SR max > ρ ( Z ) , the market ( S , S ) admits strong regulatory arbitrage for ρ .In particular, if ρ ( Z ) ≤ , the market ( S , S ) admits strong regulatory arbitrage for ρ , indepen-dent of µ or Σ . Moreover, if ρ ( Z ) < and d ≥ , ρ -optimal portfolios fail to exist for any ν ≥ ,independent of µ or Σ .Proof. For π ∈ R d \ { } , set SR π := E [ X π ] / (cid:112) Var( X π ) and note that this is well definedbecause µ (cid:54) = r and Σ is positive definite. It follows from linearity of the expectation andpositive homogeneity of the standard deviation that SR max := max π ∈ Π SR π . It is not difficultto check that the portfolio π ∗ := µ − r ) T Σ − ( µ − r ) Σ − ( µ − r ) ∈ Π has maximal Sharpe ratiogiven by the right-hand side of (3.4).If ρ ( Z ) ∈ (0 , ∞ ) , then by Lemma 3.24 for any π ∈ Π , ρ ( X π ) = − ρ ( Z ) (cid:112) Var( X π ) = − ρ ( Z ) SR π . Thus, minimising ρ ( X π ) over π ∈ Π is equivalent to maximising SR π over Π . Whence ρ := − ρ ( Z ) SR max = − ρ ( Z ) SR π ∗ = ρ ( X π ∗ ) . Parts (a), (b) and (c) now follow from Theorems 3.19 and Theorem 3.20(a).If ρ ( Z ) = ∞ , every portfolio has infinite risk except the riskless portfolio which has zero risk.Whence Π ρ = { } , Π ρ = ∅ and ρ = ∞ . Now part (a) follow from Theorem 3.20(b).If ρ ( Z ) = 0 , ρ ( X π ) = − E [ X π ] for every portfolio π ∈ R d . Thus, ρ ν = − ν for any ν ≥ andthe market admits strong regulatory arbitrage for ρ by Theorem 3.19.Finally, if ρ ( Z ) < , Lemma 3.24 gives for ν ≥ , ρ ν = inf π ∈ Π ν ρ ( X π ) = inf π ∈ Π ν ( − ν + ρ ( Z ) (cid:112) Var( X π )) = − ν + ρ ( Z ) sup π ∈ Π ν (cid:112) Var( X π ) < , whence, the market admits strong regulatory arbitrage for ρ by Theorem 3.19. If d ≥ , it isnot difficult to check that sup π ∈ Π ν (cid:112) Var( X π ) = ∞ , and hence ρ ν = −∞ , which implies that Π ρν = ∅ . 11 emark 3.26. Corollary 3.25 shows that in general it is not true that for elliptically distributedreturns and a law-invariant risk measure ρ , the ρ -optimal portfolios coincide with the Markowitzoptimal portfolios. Indeed, Corollary 3.25 shows that in every elliptical market,
VaR α -optimalportfolios fail to exist if α > P [ Z ≤
0] = 1 / / P [ Z = 0] , where Z ∼ E (0 , , ψ ) . Inparticular,
VaR α -optimal portfolios fail to exist for α > / in every multivariate Gaussianmarket. The underlying reason is that Value at Risk fails to be expectation bounded.We illustrate the above result by considering the case that R has multivariate Gaussianreturns and the risk measure is either Value at Risk or Expected Shortfall. Example 3.27.
Assume the return vector R has a multivariate normal distribution with meanvector µ ∈ R d satisfying µ (cid:54) = r and a positive definite covariance matrix Σ ∈ R d × d . Let Z ∼ N (0 , . Then for α ∈ (0 , , we haveVaR α ( Z ) = Φ − (1 − α ) and ES α ( Z ) = φ (Φ − ( α )) α , where φ and Φ denote the pdf and cdf of a standard normal distribution, respectively. ByCorollary 3.25, we can fully characterise (strong) regulatory arbitrage in this market for bothrisk measures by looking at the maximal Sharpe ratio. Figure 3 gives a graphical illustration. Figure 3: Regulatory arbitrage for ES (blue) and VaR (orange), for multivariate normal returnsIf SR max lies above the blue (orange) curve, then this Gaussian market admits strong regula-tory arbitrage for ES α (VaR α ). If it lies below the blue (orange) curve then the market does notsatisfy regulatory arbitrage for ES α (VaR α ). And in the intermediate case, the market admitsregulatory arbitrage (but not strong regulatory arbitrage) for ES α (VaR α ).Also note that for Value at Risk, if α > / , then Φ − (1 − α ) < . Hence, in this case wealways have strong regulatory arbitrage for VaR α and VaR α -optimal portfolios fail to exist for d ≥ , independent of µ or Σ . Theorems 3.19 and 3.20 provide a full characterisation of strong regulatory arbitrage and reg-ulatory arbitrage for ρ , respectively. However, the criterion is rather indirect as it requires tocalculate ρ , which relies on a nontrivial optimisation problem. In this section, we consider the This is for instance claimed in [20, Theorem 1]. Note that Z ∼ E (0 , , ψ ) has a symmetric distribution. ρ is in addition convex (and hence coherent) and admits a dual representation. Wethen derive a dual characterisation of (strong) regulatory arbitrage.Let D := { Z ∈ L : Z ≥ P -a.s. and E [ Z ] = 1 } be the set of all Radon-Nikodým derivativesof probability measures that are absolutely continuous with respect to P . Throughout thissection, we assume that ρ : L −→ ( −∞ , ∞ ] is an expectation bounded, coherent risk measureand admits a dual representation ρ ( X ) = sup Z ∈Q ( E [ − ZX ]) , (4.1)for some Q ⊂ D . Since ρ is expectation bounded, we may assume without loss of generality that ∈ Q . Moreover, taking the supremum over Q is equivalent to taking the supremum over itsconvex hull, and therefore, we may assume without loss of generality that Q is convex. Remark 4.1. (a) Since − ZX may not be integrable, we define E [ − ZX ] := E [ ZX − ] − E [ ZX + ] ,with the conservative convention that if E [ ZX − ] = ∞ , then E [ − ZX ] = ∞ .(b) Apart from the (natural) assumption that ρ is expectation bounded, this is the mostgeneral class of coherent risk measures on L that admit a dual representation. For instance, wedo not impose L -closedness or uniformly integrability of Q (which is for instance assumed in[12]). A wide range of examples of risk measures satisfying (4.1) are given in Sections 5 and 6.(c) The representation in (4.1) is not unique. However, it is not difficult to check that the maximal dual set for which (4.1) is satisfied is given by Q ρ := { Z ∈ D : E [ ZX ] ≥ and E (cid:2) ZX − (cid:3) < ∞ for all X ∈ A ρ } , where A ρ := { X ∈ L : ρ ( X ) ≤ } is the acceptance set of ρ . However, it turns out that it issometimes useful not to consider the maximal dual set; cf. some of the examples in Section 6.(d) If we define ρ by (4.1) for some convex set Q containing , it follows that ρ is ( −∞ , ∞ ] -valued, expectation bounded and a coherent risk measure (i.e., it is monotone, cash-invariant,positively homogeneous and convex). Moreover, if R ⊂ L ⊂ L is a subspace and ρ : L → ( −∞ , ∞ ] is defined by (4.1) on L , we can naturally extend ρ to an expectation bounded coherentrisk measure on L via the right hand side of (4.1). In this section, we introduce and discuss some additional conditions that are needed (and nec-essary) for our main results, Theorems 4.12 and 4.17.We start by introducing two conditions concerning the (uniform) integrability of the returnsunder the probability measures “contained” in the dual set Q . Condition I.
For all i ∈ { , . . . , d } and any Z ∈ Q , ZR i ∈ L . Condition UI. Q is uniformly integrable, and for all i ∈ { , . . . , d } , R i Q is uniformly integrable,where R i Q := { R i Z : Z ∈ Q} . While Condition I is quite weak, it has some important consequences.
Proposition 4.2.
Suppose that Condition I is satisfied. Then the set C Q := { E [ − Z ( R − r )] : Z ∈ Q} (4.2) is a convex subset of R d and for any portfolio π ∈ R d , ρ ( X π ) = sup c ∈ C Q ( π · c ) . (4.3) Moreover, the risk measure ρ satisfies the Fatou property on X = { X π : π ∈ R d } . Note that this extension is not necessarily unique, i.e., if ρ : L → ( −∞ , ∞ ] has the representations (4.1) fortwo dual sets Q (cid:48) and Q (cid:48)(cid:48) on L , the corresponding extensions ρ (cid:48) and ρ (cid:48)(cid:48) to L may not coincide on L . roof. The set C Q is real valued by Condition I and convex by convexity of Q . This togetherwith linearity of the expectation implies that ρ ( X π ) = sup Z ∈Q ( E [ − ZX π ]) = sup Z ∈Q ( E [ − Z ( π · ( R − r ))]) = sup c ∈ C Q ( π · c ) . Finally, to establish the Fatou property on X = { X π : π ∈ R d } , assume that X π n → X π P -a.s.Nondegeneracy of the market implies that π n → π ∈ R d . Then for any Z ∈ Q , Condition I,linearity of the expectation and the definition of ρ in (4.1) gives E [ − ZX π ] = π · E [ − Z ( R − r )] = lim n →∞ π n · E [ − Z ( R − r )] = lim n →∞ E [ − ZX π n ] ≤ lim inf n →∞ ρ ( X π n ) . Taking the supremum over Z ∈ Q gives us ρ ( X π ) ≤ lim inf n →∞ ρ ( X π n ) . Remark 4.3.
Example A.2 shows that without Condition I, C Q may fail to be convex and (4.3)may break down.Condition UI is a uniform version of Condition I. The following result shows that underCondition UI, the supremum in (4.3) can be replaced by a maximum, if we replace Q in (4.2)by its L -closure. Proposition 4.4.
Suppose that Condition UI is satisfied. Denote by ¯ Q the L -closure of Q .Then the set C ¯ Q := { E [ − Z ( R − r )] : Z ∈ ¯ Q} (4.4) is a convex and compact subset of R d . Moreover, for any portfolio π ∈ R d , ρ ( X π ) = max c ∈ C ¯ Q ( π · c ) . (4.5) Proof.
Since Condition UI implies Condition I, (4.3) gives ρ ( X π ) = sup c ∈ C Q ( π · c ) ≤ sup c ∈ C ¯ Q ( π · c ) . (4.6)Since Q is UI and convex, ¯ Q is convex and weakly compact by the Dunford-Pettis theorem.To show that the supremum on the right side of (4.6) is attained, let ( Z n ) n ∈ N be a maximisingsequence in ¯ Q . Since ¯ Q is weak sequentially compact by the Eberlein-Šmulian theorem, afterpassing to a subsequence, we may assume that Z n converges weakly to some Z ∈ ¯ Q . Since themap ˜ Z (cid:55)→ E [ − ˜ Z ( R − r )] is weakly continuous on ¯ Q by Proposition B.2, Z is a maximiser. Thesame argument, but now for a maximising sequence in Q ⊂ ¯ Q , shows that we have have equalityin (4.6). Finally, using again that the map ˜ Z (cid:55)→ E [ − ˜ Z ( R − r )] is weakly continuous on ¯ Q and ¯ Q is weakly compact, it follows that C ¯ Q is compact. Remark 4.5.
In [12], it is assumed that Q is uniformly integrable and that R i ∈ L ( Q ) , where L ( Q ) := { X ∈ L : lim a →∞ sup Z ∈Q E [ Z | X | {| X | >a } ] = 0 } . (4.7)By Proposition B.1, this is equivalent to Condition UI. However, we believe that Condition UIis slightly more natural (and easier to understand) than (4.7). Note that Z ∈ L for all Z ∈ Q even though this does not appear explicitly in Condition I.
14e next aim to introduce a notion of “interior” for Q , which is crucial for the dual charac-terisation of regulatory arbitrage. This turns out to be rather subtle since neither algebraic nortopological notions of interior work in general; cf. Remark 4.6. Instead, we define our notion of“interior” in an abstract way. More precisely, we look for (nonempty) subsets ˜ Q ⊂ Q satisfying
Condition POS. ˜ Z > P -a.s. for all ˜ Z ∈ ˜ Q . Condition MIX. λZ + (1 − λ ) ˜ Z ∈ ˜ Q for all Z ∈ Q , ˜ Z ∈ ˜ Q and λ ∈ (0 , . Condition INT.
For all ˜ Z ∈ ˜ Q , there is an L ∞ -dense subset E of D ∩ L ∞ such that for all Z ∈ E , there is λ ∈ (0 , such that λZ + (1 − λ ) ˜ Z ∈ Q .A few comments are in order. Remark 4.6. (a) Condition MIX implies in particular that ˜ Q is convex.(b) Condition INT of ˜ Q is inspired by the definition of the core/algebraic interior. Indeed,recall that for a vector space V , the algebraic interior of a set M ⊂ V with respect to a vectorsubspace X ⊂ V is defined by aint X M := { m ∈ M : for all x ∈ X, there is λ > such that m + δx ∈ M for all δ ∈ [0 , λ ] } . When M is convex, one can show that aint X M = { m ∈ M : for all x ∈ X, there is λ > such that m + λx ∈ M } , and any strict convex combination of a point in M and aint X M belongs to aint X M . To see thelink to our setup, assume that Q ⊂ L ∞ . Set M := Q , V := L ∞ and X := { Z ∈ L ∞ : E [ Z ] = 0 } .Then aint X M satisfies conditions POS, MIX and INT. Moreover, for certain examples (e.g.Expected Shortfall), aint X M (cid:54) = ∅ . Note, however, that if Q (cid:54)⊂ L ∞ , it is not possible to define anonempty set ˜ Q satisfying Conditions POS, MIX and INT via the algebraic interior.(c) One might wonder if one could define ˜ Q as the topological interior of Q in a suitablesubspace topology of D ∩ V , where L ∞ ⊂ V ⊂ L is a vector subspace. Again if Q ⊂ L ∞ , forcertain examples (e.g. Expected Shortfall), the topological interior of Q in the subspace topologyof D ∩ L ∞ is nonempty and satisfies Conditions POS, MIX and INT. However, if Q (cid:54)⊂ L ∞ , thisapproach does not work since the topological interior may fail to satisfy Condition MIX (because D ∩ V is not a vector space).(d) In light of Propositions 4.8 and B.6 one could slightly relax Condition INT, by requiringthat the sets E are only σ ( L ∞ , L ) -dense in D ∩ L ∞ . However, this additional level of generalitydoes not seem to be useful in concrete examples. However, considering L ∞ -dense subsets of D ∩ L ∞ is useful; cf. Section 6.3.We proceed to characterise the maximal subset of Q satisfying Conditions POS, MIX andINT. This is surprisingly simple and shows that we can expect ˜ Q max to be nonempty for mostrisk measures ρ . Proposition 4.7.
Define the set ˜ Q max by ˜ Q max := { ˜ Z > ∈ Q : there is an L ∞ -dense subset E of D ∩ L ∞ such that for all Z ∈ E , there is λ ∈ (0 , such that λZ + (1 − λ ) ˜ Z ∈ Q} . Then ˜ Q max satisfies Conditions POS, MIX and INT. Moreover, if ˜ Q ⊂ Q satisfies ConditionsPOS, MIX and INT, then ˜ Q ⊂ ˜ Q max . We refer the reader to [46] for details. The case that X = V is more standard and called the core/algebraicinterior of M . roof. ˜ Q max satisfies Conditions POS and INT by definition. To establish Condition MIX, let Z ∈ Q , ˜ Z ∈ ˜ Q max and µ ∈ (0 , . Clearly µZ + (1 − µ ) ˜ Z > P -a.s. It remains to show thatthere exists an L ∞ -dense subset E (cid:48) of D ∩ L ∞ such that for all Z (cid:48) ∈ E (cid:48) , there is λ (cid:48) > such that λ (cid:48) Z (cid:48) +(1 − λ (cid:48) )( µZ +(1 − µ ) ˜ Z ) ∈ Q . Let E be the L ∞ -dense subset of D∩ L ∞ for ˜ Z in the definitionof ˜ Q max . Set E (cid:48) := E . Let Z (cid:48) ∈ E (cid:48) . Then there is λ > such that λZ (cid:48) + (1 − λ ) ˜ Z ∈ ˜ Q max ⊂ Q .Set µ (cid:48) := − µ − µλ ∈ (0 , and λ (cid:48) := λµ (cid:48) ∈ (0 , . Then by convexity of Q , µ (cid:48) (cid:0) λZ (cid:48) + (1 − λ ) ˜ Z (cid:1) + (1 − µ (cid:48) ) Z = λ (cid:48) Z (cid:48) + (1 − λ (cid:48) )( µZ + (1 − µ ) ˜ Z ) ∈ Q . The additional claim follows immediately from the definition of ˜ Q max .We finish this section by explaining the role of Conditions POS and INT for establishingexistence of ρ -optimal portfolios. Proposition 4.8.
Suppose Condition I is satisfied. Let ˜ Q ⊂ Q satisfy Conditions POS andINT. If ∈ ˜ Q , then ρ is strictly expectation bounded and Π ρ = { } . If in addition ρ < ∞ ,then for all ν ≥ , Π ρν is nonempty, compact and convex.Proof. Strict expectation boundedness of ρ follows from Lemma B.3 (with ˜ Z = 1 ) and Re-mark 3.8(c). Corollary 3.9 then gives Π ρ = { } . Finally, if ρ < ∞ , it follows that ρ ∈ R since ρ ≥ − by expectation boundedness of ρ . Now the remaining claim follows from Proposition4.2, Theorem 3.11 and Remark 3.12(c). In this section, we provide a dual characterisation of strong regulatory arbitrage for ρ in termsof absolutely continuous martingale measures (ACMMs) for the discounted risky assets S/S .To this end, set M = { Z ∈ D : E [ Z ( R i − r )] = 0 for all i = 1 , . . . , d } , (4.8)and note that each Z ∈ M is the Radon-Nikodým derivative of an ACMM for S/S .A first step towards a dual characterisation is the following equivalent characterisation ofstrong regulatory arbitrage. Proposition 4.9.
The market ( S , S ) satisfies strong regulatory arbitrage for ρ if and only if ρ ( X π ) < for some portfolio π ∈ R d .Proof. If the market admits strong regulatory arbitrage for ρ , then ρ < by Theorem 3.19.Hence, ρ ( X π ) < for some portfolio π ∈ R d .Conversely, if ρ ( X π ) < for some portfolio π , E [ X π ] ≥ − ρ ( X π ) > because ∈ Q . Thus, ρ < , and the market satisfies strong regulatory arbitrage for ρ . Remark 4.10.
The condition ρ ( X π ) < for some π ∈ R d is referred to as a Good Deal in theliterature, see e.g. [12]. Note, however, that the equivalence of Proposition 4.9 crucially relieson ρ being expectation bounded (via ∈ Q ) since otherwise a portfolio with negative risk mayhave a negative expected excess return. Our next result shows that if Q contains an ACMM, the market does not satisfy strongregulatory arbitrage for ρ . Proposition 4.11. If Q ∩ M (cid:54) = ∅ , then the market ( S , S ) does not admit strong regulatoryarbitrage for ρ . Note that ˜ Q does not need to satisfy Condition MIX. Note that Proposition 4.9 holds more generally for positively homogeneous (not necesarily convex) riskmeasures that are expectation bounded. Also note that assuming that ρ is expectation bounded is a real restrictionas it is not satisfied by Value at Risk. roof. Let Z ∈ Q ∩ M . Then for any portfolio π ∈ R d , ρ ( X π ) ≥ E [ − ZX π ] = 0 . Therefore, by Proposition 4.9 the market does not admit strong regulatory arbitrage for ρ .The converse of Proposition 4.11 is false. Example A.3 shows that even under Condition I, Q ∩ M = ∅ is not enough to imply strong regulatory arbitrage for ρ . However, under conditionUI, the converse of Proposition 4.11 is essentially true. Theorem 4.12.
Assume Q satisfies UI. Denote by ¯ Q the L -closure of Q . The following areequivalent: (a) The market ( S , S ) does not admit strong regulatory arbitrage for ρ . (b) ¯ Q ∩ M (cid:54) = ∅ .Proof. First, assume ¯ Q ∩ M (cid:54) = ∅ . Let Z ∈ ¯ Q ∩ M . Then Proposition 4.4 gives ρ ( X π ) ≥ E [ − ZX π ] = 0 for any π ∈ R d . Therefore, the market does not admit strong regulatory arbitragefor ρ by Proposition 4.9.Conversely, assume ¯ Q ∩ M = ∅ . By Proposition 4.4, { } and C ¯ Q are two nonempty disjointconvex and compact subsets of R d . By the strict separation theorem (cf. [9, Proposition B.14]),there exists π ∈ R d \ { } with π · c < b < for all c ∈ C ¯ Q . Thus, Proposition 4.4 gives ρ ( X π ) = max c ∈ C ¯ Q ( π · c ) < , and so the market admits strong regulatory arbitrage for ρ by Proposition 4.9. Remark 4.13. (a) By virtue of Proposition 4.9, Theorem 4.12 is identical to Cherny’s equivalentcharacterisation of No Good Deals in [12, Theorem 3.1]. However, our proof is simpler sincewe are working with a finite number of assets. We have included it for the convenience of thereader.(b) Example A.4 shows that when Q is uniformly integrable but R Q is not, then Theorem4.12 is false. Example A.5 shows that Theorem 4.12 is also false if R Q is uniformly integrablebut Q is not. Thus, we need both parts of Condition UI simultaneously.Characterising the absence of strong regulatory arbitrage is important. However, it is notenough as the risk constraint also remains void if there is no portfolio with negative risk but aportfolio π ∈ Π with zero risk. This is illustrated by the following example. Example 4.14.
Consider a binomial model with one riskless asset and one risky asset withreturns r = 0 and R , respectively, where P [ R = 1] = P [ R = 0] = 1 / . Set π = π := 2 . Thenfor α ∈ (0 , , Π ES α = { π } and ES α = ES α ( X π ) = (cid:40) , if α ≤ − αα < , if α > . Hence, by either Theorem 3.19 or Theorem 4.12, the market does not admit strong regulatoryarbitrage for ES α if α ≤ . However, by Theorem 3.20(a), the market admits regulatory arbitragefor ES α if α ≤ .Therefore, to see the whole picture, it is important to also have a dual characterisation ofregulatory arbitrage for ρ . 17 .3 Dual characterisation of regulatory arbitrage In this section, we provide a dual characterisation of regulatory arbitrage for ρ in terms ofequivalent martingale measures (EMMs) for the discounted risky assets S/S . To this end, set P = { Z ∈ M : Z > P -a.s. } . and note that each Z ∈ P is the Radon-Nikodým derivative of an EMM for S/S .As we did for strong regulatory arbitrage, we start by providing an equivalent characterisationof regulatory arbitrage. However, for regulatory arbitrage, we need to assume that is the unique ρ -optimal portfolio. Proposition 4.15.
Assume Π ρ = { } . Then the market ( S , S ) satisfies regulatory arbitragefor ρ if and only if ρ ( X π ) ≤ for some portfolio π ∈ R d \ { } .Proof. First assume the market satisfies regulatory arbitrage for ρ . As the riskless portfolio has zero risk, by definition of regulatory arbitrage, there must be another portfolio π ∈ R d \ { } with ρ ( X π ) ≤ .Conversely, if ρ ( X π ) ≤ for some portfolio π ∈ R d \ { } , then E [ X π ] ≥ by expectationboundedness of ρ , which in turn gives E [ X π ] > by Π ρ = { } . Now for any portfolio π (cid:48) ∈ R d , π + π (cid:48) is strictly ρ -preferred over π (cid:48) because E [ X π + π (cid:48) ] = E [ X π ] + E [ X π (cid:48) ] > E [ X π (cid:48) ] and ρ ( X π + π (cid:48) ) ≤ ρ ( X π ) + ρ ( X π (cid:48) ) ≤ ρ ( X π (cid:48) ) . It follows that the market admits regulatory arbitrage for ρ .We proceed to give a preliminary dual characterisations of regulatory arbitrage for ρ . Notethat this characterisation does not rely on the set ˜ Q max to be nonempty. Proposition 4.16.
Assume Π ρ = { } and Q satisfies Condition I. If Q ∩ M = ∅ , then themarket ( S , S ) admits regulatory arbitrage for ρ .Proof. Condition I implies that the set C Q in (4.2) is convex. If Q ∩ M = ∅ then / ∈ C Q . Bythe supporting hyperplane theorem (cf. [9, Proposition B.12]), there exists π ∈ R d \ { } with π · c ≤ for all c ∈ C Q . By (4.3), ρ ( X π ) = sup c ∈ C Q ( π · c ) ≤ , and the claim follows from Proposition 4.15.We are now in a position to state and prove the main result of this paper, the dual charac-terisation of regulatory arbitrage. Theorem 4.17.
Suppose Π ρ = { } , Q satisfies Condition I and ˜ Q max (cid:54) = ∅ . Then the followingare equivalent: (a) The market ( S , S ) does not admit regulatory arbitrage for ρ . (b) ˜ Q ∩ P (cid:54) = ∅ for some ∅ (cid:54) = ˜ Q ⊂ Q satisfying Conditions POS, MIX and INT. (c) ˜ Q ∩ P (cid:54) = ∅ for all ∅ (cid:54) = ˜ Q ⊂ Q satisfying Conditions POS, MIX and INT.Proof. ( b ) = ⇒ ( a ) . Let ∅ (cid:54) = ˜ Q ⊂ Q satisfying Conditions POS, MIX and INT and π ∈ R d \ { } .By Proposition 4.15, we have to show that ρ ( X π ) > . Let ˜ Z ∈ ˜ Q ∩ P . Then E [ − ˜ ZX π ] = 0 .Since X π (cid:54) = 0 by nonredundancy of the market, this implies that X π is a non-constant randomvariable. Now the claim follows from Lemma B.3.18 a ) = ⇒ ( c ) . We argue by contraposition. So assume that there exists ∅ (cid:54) = ˜ Q ⊂ Q satisfyingConditions POS, MIX and INT such that ˜ Q ∩ P = ∅ . This implies that ˜ Q ∩ M = ∅ by ConditionPOS. Refining the argument of Proposition 4.16, it suffices to show that is not in the interiorof C Q . Seeking a contradiction, assume that ∈ C o Q . Then there is ε > such that B ( , ε ) ⊂ Q ,where B ( , ε ) denotes the open ball of of radius ε > around with respect to some norm (cid:107) · (cid:107) .Set C ˜ Q := { E [ − Z ( R − r )] : Z ∈ ˜ Q} ⊂ C Q ⊂ R d . Then C ˜ Q is convex by Remark 4.6(a) and does not contain the origin because ˜ Q ∩ M = ∅ .Hence, B ( , ε ) (cid:54)⊂ C ˜ Q . As ˜ Q (cid:54) = ∅ , there is x ∈ C ˜ Q . Set y := − ε/ (2 (cid:107) x (cid:107) ) x ∈ B ( , ε ) . Then λ x + (1 − λ ) y = for λ := ε/ (2 (cid:107) x (cid:107) + ε ) . Letting Z x ∈ ˜ Q and Z y ∈ Q denote Radon-Nikodýmderivatives corresponding to x and y , respectively, it follows from definition of M in (4.8) andCondition MIX that λZ x + (1 − λ ) Z y ∈ ˜ Q ∩ M , in contradiction to ˜ Q ∩ M = ∅ . ( c ) = ⇒ ( b ) . This is trivial. Remark 4.18. (a) While the proof of Theorem 4.17 is elementary, it is by no means trivial asit relies on finding the “correct” conditions for the sets ˜ Q .(b) If we choose for ρ the worst-case risk measure, we recover a refined version of the fun-damental theorem of asset pricing in a one-period model; see Theorem 6.2 below for details. Inthis case, the proof is particularly simple. For the nontrivial direction, the argument is – to thebest of our knowledge – new, even simpler than any of the existing proofs (cf. e.g. [23, Theorem1.7]) and yields a much sharper result.(c) If ˜ Q max = ∅ , Theorem 4.17 fails. A concrete counterexample is given in Example A.6. In this section, we discuss how the previous theory applies to the case where the returns lie insome Orlicz space L Φ and ρ is finite valued on L Φ . We proceed to recall some key definitions and results relating to Orlicz spaces and Orlicz hearts;see [45, Chapter 10] and [19, Chapter 2] for details. • A function
Φ : [0 , ∞ ) → [0 , ∞ ] is called a Young function if it is convex and satisfies lim x →∞ Φ( x ) = ∞ and lim x → Φ( x ) = Φ(0) = 0 . A Young function Φ is called superlinear if Φ( x ) /x → ∞ as x → ∞ . • Given a Young function Φ , the Orlicz space corresponding to Φ is given by L Φ := { X ∈ L : E [Φ( c | X | )] < ∞ for some c > } , and the Orlicz heart is the linear subspace H Φ := { X ∈ L Φ : E [Φ( c | X | )] < ∞ for all c > } . • L Φ and H Φ are Banach spaces under the Luxemburg norm given by (cid:107) X (cid:107) Φ := inf (cid:8) λ > E (cid:2) Φ (cid:0)(cid:12)(cid:12) Xλ (cid:12)(cid:12)(cid:1)(cid:3) ≤ (cid:9) . Note that a Young function is continuous except possibly at a single point, where it jumps to ∞ . Thus afinite Young function is continuous. For any Young function Φ , its convex conjugate Ψ : [0 , ∞ ) → [0 , ∞ ] defined by Ψ( y ) := sup x ≥ { xy − Φ( x ) } is also a Young function and its conjugate is Φ . • If X ∈ L Φ and Y ∈ L Ψ , we have the generalised Hölder inequality: E [ | XY | ] ≤ (cid:107) X (cid:107) Φ (cid:107) Y (cid:107) Ψ . (5.1) • Using the conjugate Ψ and (5.1), we may define the Orlicz norm on L Φ by (cid:107) X (cid:107) ∗ Ψ := sup { E [ XY ] : Y ∈ L Ψ , (cid:107) Y (cid:107) Ψ ≤ } . This norm is equivalent to the Luxemburg norm on L Φ . • When Φ jumps to infinity, then L Φ = L ∞ (and (cid:107) · (cid:107) Φ is equivalent to (cid:107) · (cid:107) ∞ ) and H Φ = { } . • When Φ is finite, the norm dual of the Orlicz heart ( H Φ , (cid:107)·(cid:107) Φ ) (with the Luxemburg norm)is the Orlicz space ( L Ψ , (cid:107)·(cid:107) ∗ Φ ) (with the Orlicz norm). • Φ is said to satisfy the ∆ -property if there exists a finite constant K > such that Φ(2 x ) ≤ K Φ( x ) for all x ∈ [0 , ∞ ) . Φ satisfies the ∆ property if and only if L Φ = H Φ . After these preparations, we consider the following setup: Let
Φ : [0 , ∞ ) → [0 , ∞ ] be a Youngfunction and ρ : L Φ → R a coherent risk measure that is expectation bounded. We firstsummarise criteria under which ρ admits a dual representation. To this end, we first need torevisit the Fatou property from Definition 3.10. Definition 5.1.
Let
Φ : [0 , ∞ ) → [0 , ∞ ] be a Young function and ρ : L Φ → ( −∞ , ∞ ] a map.Then ρ is said to satisfy the • Fatou property on L Φ , if X n → X P -a.s. for X n , X ∈ L Φ and | X n | ≤ Y P -a.s. for some Y ∈ L Φ implies that ρ ( X ) ≤ lim inf n →∞ ρ ( X n ) . • strong Fatou property on L Φ , if X n → X P -a.s. for X n , X ∈ L Φ and sup n (cid:107) X n (cid:107) Φ < ∞ implies that ρ ( X ) ≤ lim inf n →∞ ρ ( X n ) .The strong Fatou property implies the Fatou property but the converse is not true. Note,however, that the two are equivalent if L Φ = L ∞ . Remark 5.2.
The notion of strong Fatou property has been introduced by Gao et al [24] whonoted in [25] that for a general normed vector space L , the Fatou property for risk measures(which was originally only formulated on L ∞ ) could either be understood in terms of orderbounded sequences (giving the Fatou property) or norm bounded sequences (giving the strongFatou property).We proceed to summarize the existing dual representation results for (finite) coherent riskmeasures on Orlicz spaces from the literature. Theorem 5.3.
Let
Φ : [0 , ∞ ) → [0 , ∞ ] be a Young function with conjugate Ψ and ρ : L Φ → R a coherent risk measure. Then ρ admits a dual representation under the following conditions: If ρ admits a dual representation on L Φ , then it can be naturally extended to L ; see Remark 4.1(d). For Orlicz hearts, the representation result for (finite) coherent risk measures is given in [11, Corollary 4.2]. Φ satisfies the ∆ -condition. (b) Ψ satisfies the ∆ -condition and ρ satisfies the Fatou property. (c) Φ is a superlinear Young function and ρ satisfies the strong Fatou property.Proof. (a) In this case L Φ = H Φ and the result follows from [11, Corollary 4.2].(b) This follows from [24, Theorem 2.5] or [17, Proposition 2.5] and Fenchel–Moreau duality. (c) This follows from [16, Theorem 3.2] in the case that L Φ = L ∞ and from [25, Theorem 2.4]in the general case. Remark 5.4. (a) If a coherent risk measure ρ : L Φ → R admits a dual representation, it isstraightforward to check that it satisfies the Fatou property. The converse is false if both Φ and Ψ fail to satisfy the ∆ -condition; see [24, Theorem 4.2] for a generic counterexample. (b) A coherent risk measure that admits a dual characterisation does not need to satisfy thestrong Fatou property; in fact if Φ is a superlinear Young function and ρ : L Φ → R admits adual characterisation such that Q ρ (cid:54)⊂ H Ψ , then ρ fails to satisfy the strong Fatou property by[25, Theorem 2.4].Next, we show that all coherent risk measures on Orlicz spaces that satisfy a dual represen-tation (independent of whether one of the conditions of Theorem 5.3 is satisfied) have a nicemaximal dual set. Proposition 5.5.
Let
Φ : [0 , ∞ ) → [0 , ∞ ] be a Young function with conjugate Ψ and ρ : L Φ → R a coherent risk measure. If ρ admits a dual representation, then the maximal dual set Q ρ is L Ψ -closed and L Ψ -bounded if Φ is finite. If Φ satisfies the ∆ -condition, Q ρ is also L -closed.Moreover, if Q ⊂ Q ρ has L Ψ -closure Q ρ , then Q represents ρ , and if Q ⊂ Q ρ represents ρ , then Q ρ ⊂ ¯ Q , where ¯ Q denotes the L -closure.Proof. If Φ jumps to ∞ , i.e., L Φ = L ∞ , then the result follows from [16, Theorem 3.2]. Soassume for the rest of the proof that Φ is finite. Denote by ρ H the restriction of ρ to H Φ . Then A ρ H ⊂ A ρ and hence Q ρ H ⊃ Q ρ . It follows from [11, Corollary 4.2] and Proposition B.5(a) that Q ρ H is L Ψ -bounded and L -closed. Hence, Q ρ is L Ψ -bounded. This together with the definitionof Q ρ and the generalised Hölder inequality (5.1) implies that Q ρ is L Ψ -closed. If Φ satisfiesthe ∆ -condition then A ρ H = A ρ and so Q ρ = Q ρ H is L -closed. Moreover, if Q ⊂ Q ρ has L Ψ -closure Q ρ , then Q represents ρ by the generalised Hölder inequality (5.1) and if Q ⊂ Q ρ represents ρ , then ¯ Q ρ = ¯ Q because otherwise by the Hahn-Banach separation theorem (for thepairing ( L ∞ , L ) ), there exists X ∈ L ∞ such that sup Z ∈ ¯ Q ρ E [ − ZX ] (cid:54) = sup Z ∈ ¯ Q E [ − ZX ] . We proceed to provide the dual characterisation of (strong) regulatory arbitrage for (finite)expectation bounded coherent risk measures on Orlicz spaces.We first consider the case L Φ = L ∞ , i.e., when Φ jumps to infinity, which is different fromall other Orlicz spaces in that the corresponding Orlicz heart is the null space. Corollary 5.6.
Let ρ : L ∞ → R be an expectation bounded coherent risk measure on L ∞ thatsatisfies the Fatou property. Let Q ⊂ Q ρ be a convex subset with ∈ Q and ¯ Q = Q ρ . Supposethat R i ∈ L ∞ . If ρ is strictly expectation bounded, then Π ρν is nonempty, compact and convexfor all ν ≥ . Moreover: (a) If ρ is continuous from below, the market ( S , S ) does not admit strong regulatory arbitragefor ρ if and only if Q ρ ∩ M (cid:54) = ∅ . Note that for “(4) ⇒ (1)” in [24, Theorem 3.7], the assumption of an atomless probability space is not needed. Note, however, that ρ in [24, Theorem 4.2] is ( −∞ , ∞ ] -valued. If there exists ˜ Q ⊂ Q satisfying Conditions POS, MIX and INT with ∈ ˜ Q , then themarket ( S , S ) does not admit regulatory arbitrage for ρ if and only if ˜ Q ∩ P (cid:54) = ∅ .Proof. The first assertion follows from Theorem 3.11 and Corollary 3.9. Next, since R i ∈ L ∞ ,Condition UI is satisfied if and only if the dual set Q is uniformly integrable, which by [23,Corollary 4.35] is equivalent to ρ being continuous from below. Since ¯ Q = Q ρ by Proposition5.5, part (a) follows from Theorem 4.12. Finally, Condition I is trivially satisfied and so part(b) follows from Theorem 4.17.We now consider the case of Orlicz spaces for a finite Young function. Corollary 5.7.
Let
Φ : [0 , ∞ ) → [0 , ∞ ) be a finite Young function with conjugate Ψ and ρ : L Φ → R an expectation bounded coherent risk measure that admits a dual representation. Let Q ⊂ Q ρ be a convex subset with ∈ Q and whose closure in L Ψ is Q ρ . Suppose that R i ∈ L Φ .If ρ is strictly expectation bounded, then Π ρν is nonempty, compact and convex for all ν ≥ .Moreover: (a) If R i ∈ H Φ , the market ( S , S ) does not admit strong regulatory arbitrage for ρ if and onlyif ¯ Q ρ ∩ M (cid:54) = ∅ . (b) If there exists ˜ Q ⊂ Q satisfying Conditions POS, MIX and INT with ∈ ˜ Q , then themarket ( S , S ) does not admit regulatory arbitrage for ρ if and only if ˜ Q ∩ P (cid:54) = ∅ .Proof. The first assertion follows from Theorem 3.11 and Corollary 3.9. Next, since R i ∈ H Φ , Condition UI is satisfied by Proposition B.5(b), and (a) follows from Proposition 5.5 andTheorem 4.12. Finally, Condition I follows from R i ∈ L Φ and the generalised Hölder inequality(5.1). Now part (b) follows from Theorem 4.17. In this section, we apply our main results to various examples of risk measures. Recall that wehave already investigated the case of elliptically distributed returns in Section 3.4. Here, we donot make any assumptions on the returns, other than our standing assumptions that returns arein L and that the market ( S , S ) is nonredundant and nondegenerate. The worst-case risk measure
WC : L → ( −∞ , ∞ ] is given by WC ( X ) := ess sup( − X ) . It is acoherent risk measure and is – among others – represented by Q := D ∩ L ∞ ; see Proposition B.6.This implies that Condition I is satisfied. By contrast, Condition UI is not satisfied unless Ω isfinite. It is not difficult to check that ˜ Q = { Z ∈ D ∩ L ∞ : Z > P -a.s. } satisfies conditions POS,MIX and INT. However, it turns out that we get a stronger dual characterisation of regulatoryarbitrage for the worst-case risk measure if we consider the set ˆ Q := { Z ∈ D ∩ L ∞ : Z ≥ ε P -a.s. for some ε > } . It is easy to check that ˆ Q satisfies Conditions POS, MIX and INT. Since ∈ ˆ Q , it follows fromProposition 4.8 that Π WC0 = { } . Theorems 4.17 and 4.12 now give the following result. Corollary 6.1.
The market ( S , S ) does not admit regulatory arbitrage for WC if and only ifthere is Z ∈ P ∩ L ∞ with Z ≥ ε P -a.s. for some ε > . Moreover, if Ω is finite, the market ( S , S ) does not admit strong regulatory arbitrage for WC if and only if M (cid:54) = ∅ . Here ¯ Q ρ denotes the closure of Q ρ in L .
22t is not difficult to check that (strong) regulatory arbitrage for WC is equivalent to ordinaryarbitrage of the first (second) kind for markets ( S , S ) that satisfy our standing assumptions.Indeed, recall that the market ( S , S ) is said to admit an • arbitrage of the first kind if there exists a trading strategy ( ϑ , ϑ ) ∈ R d such that ϑ S + ϑ · S ≤ , ϑ S + ϑ · S ≥ P -a.s. and P [ ϑ S + ϑ · S > > . • arbitrage of the second kind if there exists a trading strategy ( ϑ , ϑ ) ∈ R d such that ϑ S + ϑ · S < , and ϑ S + ϑ · S ≥ P -a.s.Now if ( ϑ , ϑ ) ∈ R d is an arbitrage of the first (second) kind, then π := ( ϑ S , . . . , ϑ d S d ) (cid:54) = satisfies X π = π · ( R − r ) = ( ϑ S + ϑ · S ) − (1 + r )( ϑ S + ϑ · S ) , which implies that WC( X π ) is nonpositive (negative). Hence, the market satisfies (strong) regulatory arbitrage for WC byProposition 4.15 (4.9). The converse direction can be argued similarly.Using this equivalence, Corollary 6.1 gives the following refined version of the one-periodfundamental theorem of asset pricing for L -markets (with trivial initial information). Therefinement is that we show the existence of an EMM with a positive lower bound . Theorem 6.2.
Suppose that the market ( S , S ) has finite first moments. (a) The market does not admit arbitrage of the first kind if and only if there exists Z ∈ P ∩ L ∞ with Z ≥ ε P -a.s. for some ε > . (b) If Ω is finite, the market does not admit arbitrage of the second kind if and only if M (cid:54) = ∅ . Remark 6.3.
To the best of our knowledge, a simple proof for the existence of an EMM withpositive lower bound for arbitrage-free L -markets (with trivial initial information) has not beengiven before. In fact, the only extant result that we are aware of that gives this lower boundfor L -markets is [41, Corollary 2], which uses very heavy machinery from functional analysis. By contrast our proof is elementary and short, and might even be given in a classroom setting.It follows from the above discussion that if the market ( S , S ) satisfies no-arbitrage, thenit does not satisfy (strong) regulatory arbitrage for the worst-case risk measure. We now ad-dress the question if there exists a positive homogeneous (not necessarily convex or expectationbounded) risk measure ρ such that regulatory arbitrage (or at least strong regulatory arbitrage)for ρ can be ruled out if the market satisfies no-arbitrage. The following result shows that theanswer is negative. Theorem 6.4.
Let ρ be a positively homogeneous risk measure that is not as conservative asthe worst-case risk measure. Then there exists a market ( S , S ) that satisfies no-arbitrage butadmits strong regulatory arbitrage for ρ .Proof. We are going to construct a random variable R with E [ R ] > and P [ R < > and ρ ( R ) < . Define the market ( S , S ) by S ≡ and S := S , where S = 1 and S = 1 + R .Then ( S , S ) is nonredundant, nondegenerate, and arbitrage-free but admits strong regulatoryarbitrage for ρ by Proposition 4.9 (with π = 1 ).First, if ρ is not expectation bounded, there exists X ∈ L such that E [ − X ] − ρ ( X ) := ε > .By cash-invariance of ρ , this implies that X cannot be constant so ess sup( − X + E [ X ]) > . Set δ ∈ (0 , ess sup( − X + E [ X ])) and let R := X − E [ X ] + δ . Then E [ R ] = δ > , P [ R < > and ρ ( R ) = − ε − δ < .Next, if ρ is expectation bounded, there exists X ∈ L such that ρ ( X ) < ess sup( − X ) ≤ ∞ .Let m ∈ ( ρ ( X ) , ess sup( − X )) and R := X + m . Then P [ R < > , ρ ( R ) < and E [ R ] ≥− ρ ( R ) > by expectation boundedness of ρ . Note that if ( S , S ) has finite first moments, we may assume without loss of generality that it is nonredundant,nondegenerate and satisfies S i > for all i ∈ { , . . . , d } . Under stronger integrability conditions on the market, the result has also been established by [35, Remark 7.5]. .2 Value at Risk and Expected Shortfall We have already introduced VaR and ES in Definition 3.2. Since VaR has no dual representation,we cannot apply the results from Section 4. However, using VaR α ( X ) ≤ ES α ( X ) for α ∈ (0 , and X ∈ L , it follows that if there is (strong) regulatory arbitrage for ES α , then there is (strong)regulatory arbitrage for VaR α .Unlike VaR, ES is coherent and admits for α ∈ (0 , the following dual representation: ES α ( X ) = sup Z ∈Q α E [ − ZX ] = max Z ∈Q α E [ − ZX ] , where Q α := { Z ∈ D : (cid:107) Z (cid:107) ∞ ≤ α } . (6.1)This can be extended to include α ∈ { , } , where Q := D ∩ L ∞ and Q := { } only “contains”the real-word measure P . Note that ES ( X ) = E [ − X ] ; ES corresponds to the worst-case riskmeasure considered in Section 6.1, where the supremum in (6.1) is no longer attained.For α ∈ (0 , , Conditions I and UI are satisfied for ES α and Q α is closed in L . Moreover,Proposition B.7 shows that ˜ Q α := { Z ∈ D : Z > P -a.s. and (cid:107) Z (cid:107) ∞ < α } (6.2)satisfies Conditions POS, MIX and INT. Note that ∈ ˜ Q α . Using Proposition 4.8 together withTheorems 4.12 and 4.17, we arrive at the following complete description of mean-ES portfolioselection: Theorem 6.5.
Fix α ∈ (0 , . Π ES α ν is nonempty, compact and convex for ν ≥ . Moreover: (a) The market ( S , S ) does not admit strong regulatory arbitrage for ES α if and only if thereexists Z ∈ M such that (cid:107) Z (cid:107) ∞ ≤ α . (b) The market ( S , S ) does not admit regulatory arbitrage for ES α if and only if there exists Z ∈ P such that (cid:107) Z (cid:107) ∞ < α . Remark 6.6.
It straightforward to check that ˆ Q α := { Z ∈ D : there exists ε > such that Z ≥ ε P -a.s. and (cid:107) Z (cid:107) ∞ < α } (6.3)is nonempty and satisfies Conditions POS, MIX and INT. Thus, Theorem 6.5(b) can be strength-ened: The market ( S , S ) does not admit regulatory arbitrage for ES α if and only if there exists Z ∈ P with Z ≥ ε P -a.s. for some ε > and (cid:107) Z (cid:107) ∞ < α . Spectral risk measures are mixtures of Expected Shortfall risk measures that were introducedby Acerbi in [1]. Here, we follow the definition of Cherny [13], who has studied their finerproperties in great detail. For a probability measure µ on ([0 , , B [0 , ) , the spectral risk measure ρ µ : L → ( −∞ , ∞ ] with respect to µ is given by ρ µ ( X ) := (cid:90) [0 , ES α ( X ) µ ( d α ) . Remark 6.7. (a) If µ does not have an atom at , we can define the non-increasing function φ µ : [0 , → R + by φ µ ( u ) := (cid:82) [ u,
1] 1 α µ (d α ) and write ρ µ ( X ) := (cid:82) φ µ ( u )VaR u ( X ) d u . This is See e.g. [23, Theorem 4.47] (which extends to the case X ∈ L ). For this risk measure, it is clear that the set of optimal portfolios for any ν ≥ is given by Π ν , and ES ν = − ν .Hence by Theorem 3.19, the market admits strong regulatory arbitrage for ES . Also note that ES α < ∞ because ES α is real-valued. µ (or more precisely φ µ ) are given in [18].(b) It was shown in [30, Theorem 7] for the domain L ∞ that on a standard probability spacewhere P is non-atomic, spectral risk measures coincide with law-invariant, comonotone, coherentrisk measure that satisfy the Fatou property. It was then shown in [27] that the Fatou propertyis automatically satisfied by law-invariant coherent risk measures. The result has then beengeneralised to L by [34, Theorem 2.45].Spectral risk measures admit a dual representation. It follows from [13, Theorem 4.4] thatthe maximal dual set Q ρ µ is L -closed and given by Q ρ µ = (cid:26) (cid:90) [0 , ζ α µ ( d α ) : ζ α ( ω ) is jointly measurable and ζ α ∈ Q α for all α ∈ [0 , (cid:27) , where Q α is as in (6.1). Here, we consider an L -dense subset of Q ρ µ given by Q µ = (cid:26) (cid:90) [0 , ζ α µ ( d α ) : ζ α ( ω ) is jointly measurable and there is > ε > such that ζ α ∈ Q α for α ∈ [0 , − ε ] and ζ α ≡ for α ∈ (1 − ε, (cid:27) . It is shown in Proposition B.8(a) that Q µ also represents ρ µ . If µ does not have an atom at and (cid:82) (0 ,
1] 1 α µ (d α ) < ∞ , it follows that Q µ (and Q ρ µ ) is bounded in L ∞ . Hence, ρ µ is real-valuedand Condition I and UI are satisfied.If µ does not have an atom at , it follows from Proposition B.8(b) that the set ˜ Q µ = (cid:26) (cid:90) [0 , ζ α µ ( d α ) : ζ α ( ω ) is jointly measurable and there is < ε < and < δ < ε − ε such that ζ α ∈ ˜ Q α (1+ δ ) for α ∈ [0 , − ε ] and ζ α ≡ for α ∈ (1 − ε, (cid:27) , where ˜ Q α (1+ δ ) is as in (6.2), satisfies Conditions POS, MIX and INT. Note that ∈ ˜ Q µ . UsingProposition 4.8 together with Theorems 4.12 and 4.17 we arrive at the following result: Corollary 6.8.
Let µ be a probability measure on ([0 , , B [0 , ) such that µ ( { } ) = 0 and (cid:82) (0 ,
1] 1 α µ (d α ) < ∞ . Then Π ρ µ ν is nonempty, compact and convex for ν ≥ . Moreover: (a) The market ( S , S ) does not admit strong regulatory arbitrage for ρ µ if and only if thereexists Z ∈ M such that Z = (cid:82) [0 , ζ α µ ( d α ) , where ζ α ( ω ) is jointly measurable and satisfies ζ α ∈ D and (cid:107) ζ α (cid:107) ∞ ≤ α . (b) If µ does not have an atom at , the market ( S , S ) does not admit regulatory arbitragefor ρ µ if and only if there exists Z ∈ P , < ε < and < δ < ε − ε such that Z = µ ((1 − ε, (cid:82) [0 , − ε ] ζ α µ ( d α ) , where ζ α ( ω ) is jointly measurable and satisfies ζ α ∈ D and (cid:107) ζ α (cid:107) ∞ ≤ α (1+ δ ) for α ∈ [0 , − ε ] . g -entropic risk measures The class of g -entropic risk measures was introduced by Ahmadi-Javid [3, Definition 5.1]. It isbest understood when presented in the context of Orlicz spaces. Let Φ : [0 , ∞ ) → R be a finitesuperlinear Young function and Ψ its conjugate. Let g : [0 , ∞ ) → [0 , ∞ ) be a convex functionthat dominates Ψ . For β > g (1) , define the risk measure ρ g,β : L Φ → R by ρ g,β ( X ) = sup Z ∈Q g,β E [ − ZX ] , where Q g,β := { Z ∈ D : E [ g ( Z )] ≤ β } , Note that our definition slightly differs from the definition in [3], who considers the domain L ∞ and assumesthat g is convex, ( −∞ , ∞ ] -valued and satisfies g (1) = 0 . g -entropic risk measure with divergence level β . By convexity and nonnegativity of g and the fact that g dominates Ψ , it follows that Q g,β is convex, L Ψ -bounded and L -closed. By Proposition 5.5, we may deduce that Q g,β = Q ρ g,β . Moreover, Proposition B.9 shows that ˜ Q g,β := { Z ∈ D : Z > P -a.s. and E [ g ( Z )] < β } satisfies Conditions POS, MIX and INT. Note that ∈ ˜ Q g,β ⊂ Q g,β . Applying Corollary 5.7,we get the following result: Corollary 6.9.
Let
Φ : [0 , ∞ ) → R be a superlinear finite Young function with conjugate Ψ , g : [0 , ∞ ) → [0 , ∞ ) a convex function that dominates Ψ and β > g (1) . Suppose that R i ∈ L Φ .Then Π ρ g,β ν is nonempty, compact and convex for all ν ≥ . Moreover: (a) If R i ∈ H Φ , the market ( S , S ) does not admit strong regulatory arbitrage for ρ g,β if andonly if there is Z ∈ M with E [ g ( Z )] ≤ β . (b) The market ( S , S ) does not admit regulatory arbitrage for ρ g,β if and only if there is Z ∈ P with E [ g ( Z )] < β . We finish this section, by providing two specific examples of g -entropic risk measures. Let p ∈ (1 , ∞ ) and α ∈ (0 , . Define the transformed L p -norm risk measure with sensitivityparameter α as ρ ( X ) := min s ∈ R { α (cid:107) ( s − X ) + (cid:107) p − s } , X ∈ L p . It is shown in [11, Section 5.3] that this is a real-valued coherent risk measure on L p and admitsthe dual representation with Q ρ = { Z ∈ D : (cid:107) Z (cid:107) q ≤ α } , where q := p/ ( p − . Hence ρ = ρ g,β , where Φ( x ) = x p /p , Ψ( y ) = y q /q , g = Ψ and β := ( α ) q /q . The entropic value at risk (EVaR) was introduced in Ahmadi-Javid [3] and further studied in [4].Consider the Young function Φ( x ) = exp( x ) − and fix α ∈ (0 , . Then the entropic value atrisk at level α is a risk measure on L Φ given by EVaR α ( X ) := inf z> (cid:26) z log (cid:18) E (cid:20) exp( − zX ) α (cid:21)(cid:19)(cid:27) . It is shown in [4, Section 4.4] that it admits a dual representation with dual set Q := { Z ∈ D : E [ Z log( Z )] ≤ − log( α ) } . Hence, EVaR α = ρ g,β , where Ψ( y ) = ( y log( y ) − y + 1) { y ≥ } , g ( y ) = y log( y ) − y + 1 and β := − log( α ) . More precisely, (cid:107) Z (cid:107) Ψ ≤ max(1 , β ) for all Z ∈ Q g,β and L -closedness follow from Fatou’s lemma. The case p = 1 corresponds to Expected Shortfall, see Section 6.2. Note that the parametrisation in [4] is different: α is replaced by − α and X by − X . Conclusion and outlook
It has been said (cf. [12]) that there have been three major revolutions in finance: the first onewas Markowitz’ mean-variance analysis [32], which led to the CAPM of Treynor, Sharpe, Lintnerand Mossin; the second revolution was the Black-Scholes-Merton formula; and the third one wasthe theory of coherent risk measures developed by Artzner, Delbaen, Eber and Heath [8]. In thispaper, we have endeavoured to link the first and third revolution by substituting the variancein classical portfolio selection with a positively homogeneous risk measure ρ .We have shown that under mild assumptions, ρ -optimal portfolios for a fixed return exist.However, contrary to what many people seem to take for granted, ρ -efficient portfolios may fail to exist. We referred to this situation as regulatory arbitrage. The first aim of this paper hasbeen to make regulators aware of this pitfall, which in a sense is a generalisation of arbitrage ofthe first kind.The second aim of this paper has been to explain why this complication arises and howto avoid it. The fundamental theorem of asset pricing states that the market does not satisfyarbitrage of the first kind (i.e., does not admit regulatory arbitrage for the the worst-case riskmeasure) if and only if P ∩ L ∞ (cid:54) = ∅ . Our main result, Theorem 4.17, extends this. We have shownthat for coherent risk measures under mild assumptions on the dual set Q , the market does notadmit regulatory arbitrage for ρ if and only if P ∩ ˜ Q (cid:54) = ∅ for some nonempty ˜ Q ⊂ Q satisfyingproperties POS, MIX and INT. We have also demonstrated that ˜ Q can be computed explicitlyfor a large variety of risk measures. Furthermore, we have shown that amongst markets that donot admit arbitrage (of the first kind), regulatory arbitrage for ρ cannot be excluded unless ρ is the worst-case risk measure. Since a worst-case approach to risk is infeasible in practise, thisshows that regulators cannot avoid the existence of (strong) regulatory arbitrage when imposinga positively homogeneous risk measure.Going back to the two question posed in the introduction, we see that it is certainly possiblefor ES constraints to be ineffective. The root of this issue stems specifically from positive homo-geneity. Therefore, if there is an alternative superior risk measure, it cannot be coherent. Nat-urally, this leads to the following questions regarding convex, but not positively homogeneous,measures of risk: Do optimal portfolios exist? Does (strong) regulatory arbitrage occur? If so,can we give a dual characterisation? We intend to return to these question in a subsequentpublication. A Counterexamples
In this appendix, we give several counterexamples to complement the results in Sections 3 and 4.
Example A.1.
In this example we show that if all assumptions of Theorem 3.11 hold, but { } (cid:40) Π ρ , the result fails.Let Ω = [ − , × [1 , ⊂ R with the Borel σ -algebra and the uniform probability measure P . Let r = 0 and assume there are two risky assets with returns R i ( ω ) := ω i for ω = ( ω , ω ) ∈ Ω and i ∈ { , } . Let C be the closed ball of radius 2 centred at (2 , , and for each ( x, y ) ∈ C , let C ( x,y ) be the closed ball of radius centred at ( x, y ) , and Z ( x,y ) the Radon-Nikodým derivativeof the uniform probability measure on C ( x,y ) with respect to P . Define the risk measure ρ viaits dual set Q = { Z ( x,y ) : ( x, y ) ∈ C } , and note that E [ Z ( x,y ) R ] = x and E [ Z ( x,y ) R ] = y . For this financial market (that is nonredun-dant and nondegenerate), E [ R ] = 0 , E [ R ] = 4 , and Π = { ( π , π ) : π ∈ R , π = 1 / } . Thus,for every π ∈ Π and ( x, y ) ∈ C , E [ − Z ( x,y ) X π ] = − π · ( x, y ) = − ( π x + y ) .
27t follows that for any π ∈ Π , ρ ( X π ) = sup ( x,y ) ∈ C − ( π x + y ) = (cid:112) π ) + 1 − π − g ( π ) . Therefore, ρ = inf { ρ ( X π ) : π ∈ Π } = inf { g ( π ) : π ∈ R } = − is not attained, since g isstrictly decreasing. Thus, Π ρ is empty, even though ρ satisfies the Fatou property and ρ ∈ R .The reason Theorem 3.11 fails is because Π ρ = { ( π , π ) : π ≥ , π = 0 } (cid:41) { } .For the rest of the counterexamples, we take Ω = [0 , , with the Borel σ -algebra and theLebesgue measure P . In each example, the financial market is nonredundant and the riskyreturns are integrable and nondegenerate. Moreover, we always have ∈ Q . Example A.2.
In this example we show that when Condition I is not satisfied, the set C Q from(4.2) may fail to be convex and (4.3) may break down.Suppose r = 0 and there are two risky assets with returns R ( ω ) := (cid:40) √ ω , if ω < , − , if ω ≥ , and R ( ω ) := (cid:40) − √ ω , if ω < , , if ω ≥ . Let Q := { λY + (1 − λ ) : λ ∈ [0 , } , where Y ( ω ) := (cid:40) √ ω , if ω < , , if ω ≥ . Note that E [ − R ] = E [ − R ] = − , E [ − Y R ] = −∞ and E [ − Y R ] = ∞ . Thus, Π = { ( π , π ) : π + π = 1 } , and for any π = ( π , π ) ∈ Π , X π ( ω ) = (cid:40) π − √ ω , if ω < , (3 − π ) , if ω ≥ , and so ρ ( X π ) = E [ − Y X π ] = ∞ , if π < , E [ − Y X π ] = 0 , if π = , E [ − ZX π ] = − , if π > . It follows that sup c ∈ C Q ( π · c ) = max {− , −∞ π + ∞ π } . Thus, C Q is not a convex subset of R and (4.3) does not hold. Example A.3.
In this example we show the converse of Proposition 4.16 fails.Let r = 0 and assume there is one risky asset whose return R is uniformly distributedon [0 , . Let ρ be the worst-case risk measure, cf. Section 6.1. Then Condition I is satisfied, Q ∩ M = ∅ (because M = ∅ ), but ρ ( X π ) ≥ for any portfolio π . Therefore, by Theorem 3.19,this market does not admit strong regulatory for ρ , even though Q ∩ M = ∅ . Example A.4.
In this example we show that when Q is uniformly integrable but R Q is not,Theorem 4.12 may fail.Let the risk-free rate be given by r = 1 + 12 c , where c := (cid:82) / / log(1 /x ) d x . Suppose there isone risky asset whose return is given by R ( ω ) = ln (cid:0) ω (cid:1) , if ω < , , if ω ∈ [ , ] , − , if ω > . n ≥ , set Z n ( ω ) = n ln(1 /ω ) , if ω < n , , if ω ∈ [ n , ] ,k n , if ω ∈ ( , ] , , if ω > , where k n is chosen so that E [ Z n ] = 1 . Note that k n ↑ , and that Z n converges in L to Z = 12 (1 / , / . Therefore, ( ∪ n ≥ { Z n } ) ∪ { Z } is uniformly integrable, and whence, if we let Q be the L -closed convex hull of ( Z n ) n ≥ , Z and , it will also be uniformly integrable. Moreover, E [ Z n R ] = 1 + k n c ↑ c but E [ ZR ] = 12 c. It follows that the set C Q is given by C Q = { E [ − Y ( R − r )] : Y ∈ Q} = (0 , d ] , where d := E [ − ( R − r )] > . Thus Condition I is satisfied, Q is uniformly integrable and ¯ Q ∩ M = Q ∩ M = ∅ , but the market does not admit strong regulatory arbitrage for ρ : ρ ( X π ) = sup c ∈ C Q ( π · c ) ≥ , for any portfolio π ∈ R . Example A.5.
In this example we show that when R Q is uniformly integrable but Q is not,Theorem 4.12 may fail.Let r = 0 and suppose there is one risky asset whose return is given by R ( ω ) = , if ω ≤ , , if ω ∈ ( , ) , − , if ω ≥ . For n ≥ , define the intervals A n := ( , + n ) and set Z n ( ω ) = n − n , if ω ∈ A n , , if ω ∈ ( , ) \ A n ,k n , if ω ∈ [0 , ] ∪ [ , , where k n is chosen so that E [ Z n ] = 1 . Note that k n ↓ . Let Q be the closed convex hull of ( Z n ) n ≥ and . Then Q is not uniformly integrable but R Q is. Moreover, E [ R ] = , E [ Z n R ] = k n ↓ and k < . It follows that C Q = { E [ − Z ( R − r )] : Z ∈ Q} = [ − , . Thus ¯ Q ∩ M = Q ∩ M = ∅ , but the market does not admit strong regulatory arbitrage for ρ : ρ ( X π ) = sup c ∈ C Q ( π · c ) ≥ , for any portfolio π ∈ R . Example A.6.
In this example we show that when ˜ Q max = ∅ , Theorem 4.17 fails.Consider the financial market described in Example A.5. Let Q be the convex hull of thetwo densities and Y ( ω ) = 2 (1 / , ( ω ) , Q = { µY + (1 − µ ) : µ ∈ [0 , } . Then E [ Y R ] = − and E [ R ] = , so C Q = [ − , ] and there is no regulatory arbitrage.However, ˜ Q ∩ P = ∅ because ˜ Q max = ∅ . Indeed, any Z ∈ Q is of the form Z ( ω ) = (cid:40) − µ, if ω ≤ , µ, if ω > , for some µ ∈ [0 , . Therefore if Z ∈ Q and λ > , then λZ + (1 − λ ) ˜ Z ∈ Q for some ˜ Z ∈ D ∩ L ∞ implies that ˜ Z ∈ Q and since Q is not dense in D ∩ L ∞ , the result follows.29 Additional results
Proposition B.1.
For
Q ⊂ D , set L ( Q ) := { X ∈ L : lim a →∞ sup Z ∈Q E [ Z | X | {| X | >a } ] = 0 } .If Q is UI and X ∈ L , the following are equivalent: (a) X ∈ L ( Q ) (b) X Q is UI.Proof. First assume that X Q is uniformly integrable. Then, for any a > and Z ∈ Q , E [ Z | X | {| X | >a } ] = E [ Z | X | {| X | >a } { Z ≤ } ] + E [ Z | X | {| X | >a } { Z> } ] ≤ E [ | X | {| X | >a } ] + E [ Z | X | { Z | X | >a } ] . Taking the supremum over Q on both sides, letting a → ∞ and using that X ∈ L and X Q isUI yields lim a →∞ sup Z ∈Q E [ Z | X | {| X | >a } ] ≤ lim a →∞ E [ | X | {| X | >a } ] + lim a →∞ sup Z ∈Q E [ Z | X | { Z | X | >a } ] = 0 . Conversely, assume that X ∈ L ( Q ) . For any a, b > and Z ∈ Q , E [ Z | X | { Z | X | >a } ] ≤ E [ Z | X | { Z>a } ] + E [ Z | X | {| X | >a } ] ≤ E [ Z | X | { Z>a } {| X |≤ b } ] + E [ Z | X | { Z>a } {| X | >b } ] + E [ Z | X | {| X | >a } ] ≤ b E [ Z {| Z | >a } ] + E [ Z | X | {| X | >b } ] + E [ Z | X | {| X | >a } ] . Taking the supremum over Q on both sides, letting a → ∞ and using that Q is UI and X ∈ L ( Q ) yields lim a →∞ sup Z ∈Q E [ Z | X | { Z | X | >a } ] ≤ sup Z ∈Q E [ Z | X | {| X | >b } ] . Now, the result follows when letting b → ∞ and using again that X ∈ L ( Q ) . Proposition B.2.
Suppose Condition UI is satisfied. (a)
The set ¯ Q and R i ¯ Q for i ∈ { , . . . , d } are uniformly integrable. (b) The R d -valued map F : ¯ Q → R d given by F ( Z ) = E [ − Z ( R − r )] is weakly continuous.Proof. (a) Fix i ∈ { , . . . , d } . The Dunford-Pettis theorem implies that ¯ Q and R i Q are UI. Itsuffices to show that R i ¯ Q ⊂ R i Q . So let Z ∈ ¯ Q . Then there exists a sequence ( Z n ) n ∈ N ⊂ Q such that Z n converges to Z in L and hence in probability. It follows that R i Z n converges to R i Z in probability and hence also in L as ( R i Z n ) n ∈ N ⊂ R i Q is UI. It follows that R i Z ∈ R i Q .(b) Since F ( λZ + (1 − λ ) Z ) = λF ( Z ) + (1 − λ ) F ( Z ) for Z , Z ∈ ¯ Q and λ ∈ [0 , ,preimages under F of convex sets are convex. Since ¯ Q is convex as Q is convex, it thereforesuffices to show that F is strongly continuous. So let ( Z n ) n ∈ N ⊂ ¯ Q be a sequence that convergesto Z in L and hence in probability. Then − Z n ( R i − r ) converges to − Z ( R i − r ) in probabilityand hence also in L by part (a) for each i ∈ { , . . . , d } . Lemma B.3.
Assume ˜ Q ⊂ D satisfies Conditions POS and INT. Let ˜ Z ∈ ˜ Q and X ∈ L be anon-constant random variable. If E [ − ˜ ZX ] = 0 , then there exists Z ∈ Q such that E [ − ZX ] > .Proof. Note that ˜ Z > P -a.s. by Condition POS. Define ˜ Q ≈ P by d˜ Q d P := ˜ Z and A := { X < } .Since X is non-constant and E ˜ Q [ X ] = 0 , it follows that ˜ Q [ A ] ∈ (0 , . Seeking a contradiction,suppose that E [ − ZX ] ≤ for all Z ∈ Q . Let E be an L ∞ -dense subset of D ∩ L ∞ corresponding30o ˜ Z in Condition INT. Let Z (cid:48) ∈ E . Then there exists λ > such that λZ (cid:48) + (1 − λ ) ˜ Z ∈ Q .Thus, Since Z (cid:48) was chosen arbitrarily, we may deduce that sup Z ∈E ( E [ − ZX ]) ≤ , which together with Proposition B.6 below implies that X ≥ P -a.s. Since ˜ Q ≈ P , it followsthat ˜ Q [ A ] = 0 and we arrive at a contradiction. Proposition B.4.
Let
Φ : [0 , ∞ ) → [0 , ∞ ] be a Young function. Let ( Y n ) n ∈ N be a sequence in L Φ that converges in probability to some random variable Y . Then (cid:107) Y (cid:107) Φ ≤ lim inf n →∞ (cid:107) Y n (cid:107) Φ . Proof.
Set K := lim inf n →∞ (cid:107) Y n (cid:107) Φ . We may assume without loss of generality that K < ∞ .After passing to a subsequence, we may assume without loss of generality that ( Y n ) n ∈ N convergesto Y P -a.s. If Φ jumps to infinity, then (cid:107) · (cid:107) Φ is equivalent to (cid:107) · (cid:107) ∞ and the result follows. Soassume that Φ is finite and hence continuous. For any ε > , we can pass to a further subsequenceand assume without loss of generality that (cid:107) Y n (cid:107) Φ ≤ K + ε for all n . Then by the definition ofthe Luxemburg norm, E [Φ( | Y n / ( K + ε ) | )] ≤ for all n . Fatou’s lemma gives E (cid:104) Φ (cid:16)(cid:12)(cid:12)(cid:12) YK + ε (cid:12)(cid:12)(cid:12)(cid:17)(cid:105) ≤ lim inf n →∞ E (cid:104) Φ (cid:16)(cid:12)(cid:12)(cid:12) Y n K + ε (cid:12)(cid:12)(cid:12)(cid:17)(cid:105) ≤ . This implies that (cid:107) Y (cid:107) Φ ≤ K + ε . By letting ε → , we conclude that (cid:107) Y (cid:107) Φ ≤ K . Proposition B.5.
Let
Φ : [0 , ∞ ) → [0 , ∞ ) be a finite Young function with conjugate Ψ . Let ρ : H Φ → R be a coherent risk measure. Denote by Q ρ the maximal dual set. Then (a) Q ρ is L -closed and L Ψ -bounded. (b) If R ∈ H Φ , then R Q ρ is uniformly integrable.Proof. (a) It follows from [11, Corollary 4.2] that Q ρ ∩ L Ψ is L Ψ bounded and represents ρ . Itsuffices to show that Q ρ ∩ L Ψ is L -closed. Indeed, this implies that Q ρ ⊂ L Ψ because otherwise,by the Hahn-Banach separation theorem (for the pairing ( L , L ∞ ) ), there exists X ∈ L ∞ suchthat sup Z ∈Q ρ ∩ L Ψ E [ − ZX ] < sup Z ∈Q ρ E [ − ZX ] , in contradiction to the fact that both Q ρ ∩ L Ψ and Q ρ represent ρ on L ∞ .Set K := sup Z ∈Q ρ (cid:107) Z (cid:107) Ψ < ∞ . Let ( Z n ) n ≥ be a sequence in Q ρ ∩ L Ψ that converges to Z ∈ L . Then Z ∈ D and (cid:107) Z (cid:107) Ψ ≤ K by Proposition B.4. Let X ∈ A ρ ⊂ H Φ . We have to showthat E [ ZX ] ≥ . Since E [ Z n X ] ≥ by the fact that Z n ∈ Q ρ ∩ L Ψ , it suffices to show that E [ ZX ] = lim n →∞ E [ Z n X ] . For any n ∈ N and a n > , the generalised Hölder inequality (5.1)yields | E [ Z n X ] − E [ ZX ] | ≤ E [ | X || Z n − Z | ] = E [ | X || Z n − Z | {| X | >a n } ] + E [ | X || Z n − Z | {| X |≤ a n } ] ≤ E [ | X | Z n {| X | >a n } ] + E [ | X | Z {| X | >a n } ] + a n (cid:107) Z n − Z (cid:107) ≤ (2 K + 2 K ) (cid:107) X {| X | >a n } (cid:107) Φ + a n (cid:107) Z n − Z (cid:107) . (B.1)Now if we choose a n := min( n, √ (cid:107) Z n − Z (cid:107) ) and let n → ∞ , the right hand side of (B.1) convergesto by order continuity of of H Φ (see e.g. [19, Theorem 2.1.14]).(b) First, consider the case that R = 1 . If Φ is not superlinear, then Ψ jumps to infinity,and hence Q ρ is L ∞ -bounded by part (a) and therefore UI. If Φ is superlinear (and finite), then Ψ is superlinear and finite. Set K := sup Y ∈Q (cid:107) Y (cid:107) Ψ < ∞ and define the superlinear function ˜Ψ by ˜Ψ( y ) := Ψ( y/K ) . By the definition of the Luxenbourg norm, E [ ˜Ψ( Y )] = E [Ψ( Y /K )] ≤ , for all Y ∈ Q ρ . sup Y ∈Q ρ E [ ˜Ψ( Y )] ≤ < ∞ . Since ˜Ψ is superlinear, the de la Vallée-Poussin theoremimplies that Q ρ is UI.Next, assume that R ∈ H Φ . By Proposition B.1, it is enough to show that R ∈ L ( Q ρ ) where L ( Q ρ ) := { X ∈ L : lim a →∞ sup Z ∈Q ρ E [ Z | X | {| X | >a } ] = 0 } . Since R ∈ H Φ , the generalised Hölder inequality and order continuity of H Φ give lim a →∞ sup Z ∈Q E [ Z | X | {| X | >a } ] ≤ lim a →∞ Z ∈Q (cid:107) Z (cid:107) Ψ (cid:107) X {| X | >a } (cid:107) Φ = 0 . Proposition B.6.
Let E be an σ ( L ∞ , L ) -dense subset of D ∩ L ∞ . Then for all X ∈ L . sup Z ∈E E [ − ZX ] = WC ( X ) . Proof.
Define the coherent risk measure ρ : L → ( −∞ , ∞ ] by ρ ( X ) := sup Z ∈E ( E [ − ZX ]) . Toshow that ρ = WC, let X ∈ L and set c := ess sup( − X ) = WC ( X ) .First, assume that c < ∞ . Then monotonicity of the expectation gives ρ ( X ) ≤ WC ( X ) .For the reverse inequality, let ε > and set Z := {− X ≥ c − ε } / P [ − X ≥ c − ε ] ∈ D ∩ L ∞ . Then E [ − ZX ] ≥ c − ε . Since E is σ ( L ∞ , L ) -dense in D ∩ L ∞ , there exists a net ( Z i ) i ∈ I in E whichconverges to Z in σ ( L ∞ , L ) . Thus, ρ ( X ) ≥ lim i ∈ I E [ − Z i X ] = E [ − ZX ] ≥ c − ε = WC ( X ) − ε. Letting ε → yields ρ ( X ) ≥ WC ( X ) .Finally, assume that c = ∞ . Let N > be given and set X N := max( X, − N ) . Then X N ≥ X and WC ( X N ) = N . By monotonicity of ρ and the first part, ρ ( X ) ≥ ρ ( X N ) = WC ( X N ) = N. Letting N → ∞ yields ρ ( X ) = ∞ = WC ( X ) . Proposition B.7.
Fix α ∈ (0 , . Then ˜ Q α := { Z ∈ D : Z > P -a.s. and (cid:107) Z (cid:107) ∞ < α } is anonempty subset of Q α satisfying Conditions POS, MIX and INT.Proof. It is clear that ∈ ˜ Q α ⊂ Q α , and by definition ˜ Q α satisfies POS. If Z ∈ Q α , ˜ Z ∈ ˜ Q α and λ ∈ (0 , , then λZ + (1 − λ ) ˜ Z > P -a.s., and by the triangle inequality (cid:107) λZ + (1 − λ ) ˜ Z (cid:107) ∞ ≤ λ (cid:107) Z (cid:107) ∞ + (1 − λ ) (cid:107) ˜ Z (cid:107) ∞ < α , so ˜ Q α satisfies Condition MIX. To show Condition INT, let ˜ Z ∈ ˜ Q α . Set E := D ∩ L ∞ and let Z ∈ E . Since (cid:107) Z (cid:107) ∞ < ∞ and (cid:107) ˜ Z (cid:107) ∞ < α there is λ ∈ (0 , such that λ (cid:107) ˜ Z (cid:107) ∞ +(1 − λ ) (cid:107) Z (cid:107) ∞ ≤ α .By the triangle inequality it follows that λ ˜ Z + (1 − λ ) Z ∈ Q α . Proposition B.8.
Assume µ is a probability measure on ([0 , , B [0 , ) and ρ µ the correspondingspectral risk measure. (a) ρ µ is represented by Q µ = (cid:26) (cid:90) [0 , ζ α µ ( d α ) : ζ α ( ω ) is jointly measurable and there is > ε > such that ζ α ∈ Q α for α ∈ [0 , − ε ] and ζ α ≡ for α ∈ (1 − ε, (cid:27) . Note that
D ∩ L ∞ is σ ( L ∞ , L ) -closed. If µ does not have an atom at , the set ˜ Q µ = (cid:26) (cid:90) [0 , ˜ ζ α µ ( d α ) : ˜ ζ α ( ω ) is jointly measurable and there is ε ∈ (0 , , δ ∈ (0 , ε − ε ) such that ˜ ζ α ∈ ˜ Q α (1+ δ ) for α ∈ [0 , − ε ] and ˜ ζ α ≡ for α ∈ (1 − ε, (cid:27) , is nonempty and satisfies Conditions POS, MIX and INT.Proof. (a) It follows from [13] that ρ µ is represented by Q ρ µ = (cid:26) (cid:90) [0 , ζ α µ ( d α ) : ζ α ( ω ) is jointly measurable and ζ α ∈ Q α for all α ∈ [0 , (cid:27) . Let Z = (cid:82) [0 , ζ α µ ( d α ) ∈ Q ρ µ . Set Z n := (cid:82) [0 , − /n ] ζ α µ ( d α ) + µ ((1 − /n, ∈ Q µ . Then lim n →∞ (cid:107) Z n − Z (cid:107) ∞ ≤ lim n →∞ nn − µ ((1 − /n, . This implies that E [ − ZX ] = lim n →∞ E [ − Z n X ] for all X ∈ L .(b) Since ∈ ˜ Q β for all β ∈ [0 , and ˜ Q β only contains positive random variables, it followsthat ∈ ˜ Q µ and Condition POS is satisfied.To show Condition MIX, let Z ∈ Q µ , ˜ Z ∈ ˜ Q µ and λ ∈ (0 , . Then there is ε ∈ (0 , and δ ∈ (0 , ε − ε ) such that Z = (cid:82) [0 , − ε ] ζ α µ ( d α )+ µ ((1 − ε, and ˜ Z = (cid:82) [0 , − ε ] ˜ ζ α µ ( d α )+ µ ((1 − ε, ,where ζ α ∈ Q α and ˜ ζ α ∈ ˜ Q α (1+ δ ) for α ∈ [0 , − ε ] . Set δ (cid:48) := δ (1 − λ )1+ δλ ∈ (0 , δ ) . A simple calculationshows that λζ α + (1 − λ )˜ ζ α ∈ ˜ Q α (1+ δ (cid:48) ) for all α ∈ [0 , − ε ] . Thus, λZ + (1 − λ ) ˜ Z = (cid:90) [0 , − ε ] λζ α + (1 − λ )˜ ζ α µ ( d α ) + µ ((1 − ε, ∈ ˜ Q µ . Finally, to show Condition INT, let ˜ Z ∈ ˜ Q µ and set E := (cid:110) (cid:90) [0 , ζ α µ ( d α ) : ζ α ( ω ) is jointly measurable and there is > γ, ε > such that ζ α ∈ Q γ for α ∈ [0 , − ε ] and ζ α ≡ for α ∈ (1 − ε, (cid:111) . It is straightforward to check that E is a dense subset of D ∩ L ∞ . Let Z ∈ E . Then thereexists ε, γ ∈ (0 , and δ ∈ (0 , ε − ε ) such that ˜ Z = (cid:82) [0 , − ε ] ˜ ζ α µ ( d α ) + µ ((1 − ε, and Z = (cid:82) [0 , − ε ] ζ α µ ( d α ) + µ ((1 − ε, , where ˜ ζ α ∈ ˜ Q α (1+ δ ) and ζ α ∈ Q γ for α ∈ [0 , − ε ] . Set λ (cid:48) := δγ (2+ δ )(1+ δ − γ ) ∈ (0 , . A simple calculation shows that λ (cid:48) ζ α + (1 − λ (cid:48) )˜ ζ α ∈ ˜ Q α (1+ δ/ for all α ∈ [0 , − ε ] . Thus, λ (cid:48) Z + (1 − λ (cid:48) ) ˜ Z = (cid:90) [0 , − ε ] λ (cid:48) ζ α + (1 − λ (cid:48) )˜ ζ α µ ( d α ) + µ ((1 − ε, ∈ ˜ Q µ ⊂ Q µ . Proposition B.9.
Let g : [0 , ∞ ) → R be a convex function and β > g (1) . Let Q g,β := { Z ∈ D : E [ g ( Z )] ≤ β } . Then ˜ Q g,β := { Z ∈ D : Z > P -a.s. and E [ g ( Z )] < β } is nonempty and satisfiesConditions POS, MIX and INT.Proof. It is clear that ˜ Q g,β satisfies Condition POS and ∈ ˜ Q g,β since β > g (1) . To showcondition MIX, let Z ∈ Q g,β , ˜ Z ∈ ˜ Q g,β and λ ∈ (0 , . By the convexity of g , E [ g ( λ ˜ Z + (1 − λ ) Z )] ≤ E [ λg ( ˜ Z ) + (1 − λ ) g ( Z )] = λ E [ g ( ˜ Z )] + (1 − λ ) E [ g ( Z )] < β. To show Condition INT, let ˜ Z ∈ ˜ Q g,β . Set E := D ∩ L ∞ and let Z ∈ E . Since E [ g ( Z )] < ∞ and E [ g ( ˜ Z )] < β there is λ ∈ (0 , such that λ E [ g ( ˜ Z )] + (1 − λ ) E [ g ( Z )] ≤ β . Now convexity of g implies that λ ˜ Z + (1 − λ ) Z ∈ Q g,β . 33 eferences [1] C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion , J. Bank.Finance (2002), no. 7, 1505–1518.[2] A. Adam, M. Houkari, and J. P. Laurent, Spectral risk measures and portfolio selection , J. Bank.Finance (2008), no. 9, 1870–1882.[3] A. Ahmadi-Javid, Entropic value-at-risk: A new coherent risk measure , J. Optim. Theory Appl. (2012), no. 3, 1105–1123.[4] A. Ahmadi-Javid and A. Pichler,
An analytical study of norms and banach spaces induced by theentropic value-at-risk , Math. Financial Econ. (2017), no. 4, 527–550.[5] G. J. Alexander and A. M. Baptista, Economic implications of using a mean-var model for portfolioselection: A comparison with mean-variance analysis , J. Econ. Dyn. Control (2002), no. 7-8,1159–1193.[6] J. Armstrong and D. Brigo, Statistical arbitrage of coherent risk measures , preprint arXiv:1902.10015(2019).[7] J. Armstrong and D. Brigo,
Risk managing tail-risk seekers: Var and expected shortfall vs s-shapedutility , J. Bank. Finance (2019), 122–135.[8] P. Artzner, F. Delbaen, J.-M. Ebner, and D. Heath,
Coherent measures of risk , Math. Finance (1999), no. 3, 203–228.[9] D. P. Bertsekas, Nonlinear programming , Athena Scientific, Belmont, MA, 1995.[10] D. Bertsimas, G. J. Lauprete, and A. Samarov,
Shortfall as a risk measure: properties, optimizationand applications , J. Econ. Dyn. Control (2004), no. 7, 1353–1381.[11] P. Cheridito and T. Li, Risk measures on Orlicz hearts , Math. Finance (2009), no. 2, 189–214.[12] A. S. Cherny, Pricing with coherent risk , Theory Probab. its Appl. (2008), no. 3, 389–415.[13] A. S. Cherny, Weighted V@R and its properties , Finance Stoch. (2006), no. 3, 367–393.[14] S. Ciliberti, I. Kondor, and M. Mézard, On the feasibility of portfolio optimization under expectedshortfall , Quant. Finance (2007), no. 4, 389–396.[15] E. De Giorgi, A note on portfolio selections under various risk measures , Working paper/Institutefor Empirical Research in Economics (2002).[16] F. Delbaen,
Coherent risk measures on general probability spaces , Advances in finance and stochas-tics, Springer, 2002, pp. 1–37.[17] F. Delbaen and K. Owari,
Convex functions on dual Orlicz spaces , Positivity (2019), no. 5,1051–1064.[18] K. Dowd, J. Cotter, and G. Sorwar, Spectral risk measures: properties and limitations , J. FinancialServ. Res. (2008), no. 1, 61–75.[19] G. A. Edgar and L. Sucheston, Stopping Times and Directed Processes , Encyclopedia of Mathematicsand its Applications, vol. 47, Cambridge University Press, 1992.[20] P. Embrechts, A. McNeil, and D. Straumann,
Correlation and dependence in risk management:properties and pitfalls , Risk management: value at risk and beyond (2002), 176–223.[21] D. H. Erkens, M. Hung, and P. Matos, Corporate governance in the 2007–2008 financial crisis:Evidence from financial institutions worldwide , J. Corp. Finance (2012), no. 2, 389–411.[22] K. T. Fang, S. Kotz, and K. W. Ng, Symmetric Multivariate and Related Distributions , Chapman& Hall, London, 1987.[23] H. Föllmer and A. Schied,
Stochastic finance , fourth ed., de Gruyter Studies in Mathematics, vol. 27,Walter de Gruyter & co., Berlin, 2016.[24] N. Gao, D. H. Leung, and F. Xanthos,
Closedness of convex sets in Orlicz spaces with applicationsto dual representation of risk measures , Studia Math. (2019), no. 3, 329–347.
25] N. Gao and F. Xanthos,
On the C-property and w ∗ -representations of risk measures , Math. Finance (2018), no. 2, 748–754.[26] S. Jaschke and U. Küchler, Coherent risk measures and good-deal bounds , Finance Stoch. (2001),no. 2, 181–200.[27] E. Jouini, W. Schachermayer, and N. Touzi, Law invariant risk measures have the Fatou property ,Advances in mathematical economics, vol. 9, Springer, Tokyo, 2006, pp. 49–71.[28] P. Koch-Medina and C. Munari,
Unexpected shortfalls of expected shortfall: Extreme default profilesand regulatory arbitrage , J. Bank. Finance (2016), 141–151.[29] I. Kondor, S. Pafka, and G. Nagy, Noise sensitivity of portfolio selection under various risk measures ,J. Bank. Finance (2007), no. 5, 1545–1573.[30] S. Kusuoka, On law invariant coherent risk measures , Advances in mathematical economics,Springer, 2001, pp. 83–95.[31] Z. M. Landsman and E. A. Valdez,
Tail conditional expectations for elliptical distributions , N. Am.Actuar. J. (2013), 55–71.[32] H. Markowitz, Portfolio selection , J. Finance (1952), no. 1, 77–91.[33] A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management , Princeton UniversityPress, Princeton, New Jersey, 2005.[34] G. Ch. Pflug and W. Römisch,
Modeling, measuring and managing risk , World Scientific PublishingCo. Pte. Ltd., Hackensack, NJ, 2007.[35] M. Rásonyi and L. Stettner,
On utility maximization in discrete-time financial market models , Ann.Appl. Probab. (2005), no. 2, 1367–1395.[36] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk , J. Risk Finance (2000),21–42.[37] R. T. Rockafellar, S. Uryasev, and M. Zabarankin, Master funds in portfolio analysis with generaldeviation measures , J. Bank. Finance (2006), no. 2, 743–778.[38] R. T. Rockafellar, S. Uryasev, and M. Zabarankin, Deviation measures in risk analysis and opti-mization , University of Florida, Department of Industrial & Systems Engineering Working Paper(2002).[39] R. T. Rockafellar, S. Uryasev, and M. Zabarankin,
Generalized deviations in risk analysis , FinanceStoch. (2006), no. 1, 51–74.[40] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions , J. Bank.Finance (2002), no. 7, 1443–1471.[41] D. Rokhlin and W. Schachermayer, A note on lower bounds of martingale measure densities , IllinoisJ. Math. (2006), no. 1-4, 815–824.[42] D. Tasche, Risk contributions and performance measurement , Report of the Lehrstuhl für mathe-matische Statistik, TU München (1999).[43] M. Willesson,
What is and what is not regulatory arbitrage? A review and syntheses , FinancialMarkets, SME Financing and Emerging Economies, Springer, 2017, pp. 71–94.[44] M. T. Williams,
Uncontrolled risk: the lessons of lehman brothers and how systemic risk can stillbring down the world financial system , McGraw-Hill, 2010.[45] M. Wilson,
Weighted Littlewood-Paley Theory and Exponential-Square Integrability , Springer Science& Business Media, 2008.[46] C. Zălinescu,
Convex analysis in general vector spaces , World Scientific Publishing Co., Inc., RiverEdge, NJ, 2002., World Scientific Publishing Co., Inc., RiverEdge, NJ, 2002.