Communications on Stochastic Analysis

Volume 9 | Number 2 Article 2

6-1-2015 Risk measures on Orlicz heart spaces Coenraad Labuschagne

Habib Ouerdiane

Imen Salhi

Follow this and additional works at: https://digitalcommons.lsu.edu/cosa Part of the Analysis Commons, and the Other Mathematics Commons

Recommended Citation Labuschagne, Coenraad; Ouerdiane, Habib; and Salhi, Imen (2015) "Risk measures on Orlicz heart spaces," Communications on Stochastic Analysis: Vol. 9 : No. 2 , Article 2. DOI: 10.31390/cosa.9.2.02 Available at: https://digitalcommons.lsu.edu/cosa/vol9/iss2/2 Communications on Stochastic Analysis Serials Publications Vol. 9, No. 2 (2015) 169-180 www.serialspublications.com

RISK MEASURES ON ORLICZ HEART SPACES

COENRAAD LABUSCHAGNE, HABIB OUERDIANE, AND IMEN SALHI

Abstract. In this paper, we are interested to find a robust representation of the risk measure which is defined via a convex Young function. We consider convex risk measures defined on Orlicz heart spaces with Banach lattice values and their dual representation.

1. Introduction The concept of risk measures has been studied by many authors. Artzner et al. [2] introduced the notion of coherent risk measure which is understood to be a measure of initial capital requirements that investors and managers should provide in order to overcome negative evolutions of the market. Delbaen [9, 10] extended this risk measure to more general settings. F¨olmerand Schied [13] generalized the notion of coherent risk measure on L(Ω) spaces. Fritelli and Rosazza Gianin [16] established the more general concepts of convex and monetary risk measures. Cheridito and Li [8] give a new result about convex risk measures on Orlicz heart spaces with real values. Jouini et al. [18] were the first to introduce set-valued coherent risk measures. Hamel extended the approach of Jouini et al. to define set-valued convex risk measures on the space Lp(Rd,P ) of Bochner p-integrable functions with valued in Rd. Offwood [20] considered the connection between utility functions and real valued Orlicz spaces as was noted by Fritelli and obtained a representation on set-valued convex risk measures on Orlicz heart spaces. This paper is organized as follows: in section 1, we introduce some notations required in the text and give preliminaries on Banach lattices E endowed with an ordering relation ≤. Then, we define the Lφ(Ω, E, µ), its dual Hφ(Ω, E, µ) (which is called the Orlicz heart space) and their associated norms. In the third part, we introduce a risk measure ρ from an Orlicz heart space to a Banach lattice E and we define its acceptance set A containing all acceptable financial positions. Then, we give a robust representation of our measure ρ. Fi- nally, we give some examples of this risk measure, namely value at risk V @R and average value at risk AV @R which are convex risk measures.

Received 2014-1-21; Communicated by the editors. 2010 Mathematics Subject Classification. Primary 06B75, 06F30; Secondary 91B30. Key words and phrases. Banach lattice, Orlicz space, Orlicz heart space, robust representation of risk measure. 169 170 COENRAAD LABUSCHAGNE, HABIB OUERDIANE, AND IMEN SALHI

2. Orlicz Spaces and Orlicz Heart Spaces Consider a real E which is ordered by some order relation ≤. This space E is called a vector lattice if any two element x, y ∈ E have a least upper bound denoted by x ∨ y = sup(x, y) and a greatest lower bound denoted by x ∧ y = inf(x, y) and satisfy the following properties: (L1) x ≤ y implies x + z ≤ y + z for all x, y, z ∈ E. (L2) 0 ≤ x implies 0 ≤ tx for all x ∈ E and t ∈ R+. We denote by E+ := {x ∈ E | 0 ≤ x} the positive cone of E. For x ∈ E, let: x+ := x ∨ 0, x− := (−x) ∨ 0 and |x| := x ∨ (−x) be the positive part, the negative part and the absolute value of x, respectively. A ∥.∥ on a vector lattice E is called a lattice norm if: |x| ≤ |y| ⇒ ∥x∥ ≤ ∥y∥ for x, y ∈ E. (2.1) In particular, a Banach lattice is a real E endowed with an ordering ≤ such that (E, ≤) is a vector lattice and the norm on E is a lattice norm. Let E be a Banach lattice and E+ be its non-negative cone. Its dual Banach ∗ ∗ ∗ lattice and its cone are denoted by E and E+ respectively and < x, x > will be the usual duality product. Definition 2.1. A map φ : [0, ∞] → R is called a Young function if φ is left φ(r) continuous, increasing, convex, φ(0) = 0 and lim = ∞. r→+∞ r Also, φ is continuous except possibly at a single point, where it jumps to ∞. So, the assumption of left-continuous is needed only at that one point. The conjugate φ∗(y) = sup{xy − φ(x)} ; y ≥ 0 x≥0 is a Young function. Definition 2.2. A map α : R → R is called a utility function if α is increasing, concave and continuous on R. We use concavity to minimize the risk and the growth to maximize the wealth. Let (Ω, F, µ) be a probability space, E a Banach lattice and consider the fol- lowing space for p ≥ 1 ∫ p { → | ∥ ∥p ∥ ∥p ∞} L (Ω, E, µ) = f :Ω E f(x) p := f(x) dµ(x) < . Ω Then the dual space of Lp(Ω, E, µ) is

1 1 (Lp(Ω, E, µ))∗ = Lq(Ω, E, µ) where + = 1. p q Let L∞(Ω, E, µ) be a Banach space with values in E of essentially bounded and measurable functions, with norm defined by:

∥y∥∞ = ess sup{∥y(ω)∥, ω ∈ Ω}. RISK MEASURES ON ORLICZ HEART SPACES 171

Definition 2.3. Let φ be a Young function and c > 0. Consider the space ∫ φ { → | ∥ ∥ ∞} Lc (Ω, E, µ) := X :Ω E Xis continuous and φ(c X(ω) )dµ(ω) < , Ω (2.2) The Orlicz space Lφ(Ω, E, µ) is given by: ∪ φ φ L (Ω, E, µ) = Lc (Ω, E, µ). (2.3) c>0 and the norm associated to Lφ(Ω, E, µ) is given by X ∥X∥ := inf{λ > 0 | E(φ(∥ ∥)) ≤ 1} φ λ The Orlicz heart space corresponding to φ is defined as: ∩ φ Hφ(Ω, E, µ) = Lc (Ω, E, µ). (2.4) c>0 φ∗ and the norm of an element Z ∈ L (Ω, E, µ) associated to Hφ(Ω, E, µ) is ∗ φ∗ ∥ ∥ {E | ∥ ∥ ∗ ≤ } ∈ Z φ∗ := sup < Z, X > X φ 1 ,X L (Ω, E, µ)

2.1. (∆2) condition. [23], [11]

Definition 2.4. A Young function φ satisfies the (∆2) condition if there exists a constant M > 0 such that φ(2u) ≤ Mφ(u) , ∀ u ≥ 0 (2.5) Using this relation, one can show that ∗ φ∗ ∗ (Hφ(Ω, E, µ)) = L (Ω,E , µ), (see [21] p 132 - 138).

3. Generalized Risk Measure With Banach Lattice Values In this section, we examine risk measures with values in a Banach lattice. Assuming that there exists x0 ∈ E strictly positive i.e, x0 ∧ x ∈ E+ \{0} for all x ∈ E+ \{0}.

Definition 3.1. A map ρ : Hφ(Ω, E, µ) → E is called a risk measure if ρ satisfies: (a) Monotonicity: for X,Y ∈ Hφ(Ω, E, µ), if Y ≥ X, then ρ(Y ) ≤ ρ(X). (b) Cash invariance: for any x0 ∈ E and all X ∈ Hφ(Ω, E, µ)

ρ(X + 1Ωx0) = ρ(X) − x0 If, in addition, ρ has the properties: (c) Positive homogeneity: for any β ≥ 0 and X ∈ Hφ(Ω, E, µ): ρ(βX) = βρ(X) (d) Sub-additive:

ρ(X + Y ) ≤ ρ(X) + ρ(Y ), ∀ X,Y ∈ Hφ(Ω, E, µ)

(e) Convexity: for any X,Y ∈ Hφ(Ω, E, µ) and λ ∈ [0, 1], ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) 172 COENRAAD LABUSCHAGNE, HABIB OUERDIANE, AND IMEN SALHI then ρ is called a coherent measure. Remark that the measure ρ(X) is understood as a capital requirement for X. In every state of the world the monotonicity means that the capital requirement for X will be smaller than Y µ-almost sure. The cash invariance means that if we add an amount of money x0 to a position X should reduce the capital requirement for X by x0. The positive homogeneity says that the capital requirement scale linearly when net worths are multiplied with non-negative constants. The sub-additivity means that the capital requirement for an aggregated discounted net worth X + Y should not exceed the sum of the capital requirement for X plus the capital requirement for Y . We define

A = {X ∈ Hφ(Ω, E, µ) | ρ(X) ≤ 0}. (3.1) as the set of acceptable positions. The following propositions summarize the re- lations between risk measures and their acceptance sets. We use the convention inf ∅ = ∞.

Proposition 3.2. Let Hφ(Ω, E, µ) be an Orlicz heart space and ρ : Hφ(Ω, E, µ) → E be a risk measure with an acceptance set A. Then:

(i) Let X ∈ A, Y ∈ Hφ(Ω, E, µ). If Y ≥ X then Y ∈ A. (ii)

ρ(X) = inf{x0 ∈ E | X + 1Ωx0 ∈ A}, ∀ X ∈ Hφ(Ω, E, µ) (3.2) (iii) inf{x0 ∈ E | 1Ωx0 ≥ Z for some Z ∈ A} ∈ E. (3.3)

Proof. ρ : Hφ(Ω, E, µ) → E , A is its acceptance set.

(i) ∀ X ∈ A, Y ∈ Hφ(Ω, E, µ). If Y ≥ X implies ρ(Y ) ≤ ρ(X) ≤ 0, then Y ∈ A. (ii)

inf{x0 ∈ E | X + 1Ωx0 ∈ A} = inf{x0 ∈ E | ρ(X + 1Ωx0) ≤ 0}

= inf{x0 ∈ E | ρ(X) − x0 ≤ 0}

= inf{x0 ∈ E | ρ(X) ≤ x0} = ρ(X). (iii) (3.3) is a consequence of monotonicity. It implies that for all X ∈ Hφ(Ω, E, µ) we obtain:

inf{x0 ∈ E | 1Ωx0 ≥ Z,Z ∈ A} = inf{x0 ∈ E | ρ(1Ωx0) ≤ ρ(Z) ≤ 0}

= inf{x0 ∈ E | ρ(0) − x0 ≤ 0}

= inf{x0 ∈ E | ρ(0) ≤ x0} = ρ(0) ∈ E. □

Corollary 3.3. ρ is a risk measure on Hφ(Ω, E, µ) with value in a Banach lattice. Then, (i) If ρ is convex, then A is convex. RISK MEASURES ON ORLICZ HEART SPACES 173

(ii) If ρ is coherent, then A is a . (iii) A has the following property:

∀X ∈ Hφ(Ω, E, µ), inf{x0 ∈ E | X + 1Ωx0 ≥ Z,Z ∈ A} ∈ E. (3.4)

Proof. (i) If ρ is convex, then for all X1,X2 ∈ Hφ(Ω, E, µ) and λ ∈ [0, 1] such that

ρ(λX1 + (1 − λ)X2) ≤ λρ(X1) + (1 − λ)ρ(X2)

Let X1 and X2 ∈ A, we have ρ(λX1+(1−λ)X2) ≤ λρ(X1)+(1−λ)ρ(X2) ≤ 0 because ρ(X1) ≤ 0 and ρ(X2) ≤ 0 for λ ∈ [0, 1], then λX1 + (1 − λ)X2 ∈ A. So A is convex. (ii) ρ is coherent implies ρ is positive homogenous. ∀X ∈ A, ρ(λX) = λρ(X) ≤ 0, ∀λ ≥ 0 then ρ(λX) ≤ 0 implies λX ∈ A. Therefore A is a cone and the convexity follows as in (i). (iii)

inf {x0 ∈ E | X + 1Ωx0 ≥ Z} = inf {x0 ∈ E | ρ(X + 1Ωx0) ≤ ρ(Z) ≤ 0} Z∈A Z∈A

= inf{x0 ∈ E | ρ(X) − x0 ≤ 0}

= inf{x0 ∈ E | ρ(X) ≤ x0} = ρ(X) ∈ E. Then the result holds. □

Proposition 3.4. Let Hφ(Ω, E, µ) be the Orlicz heart space and B a subset of Hφ(Ω, E, µ) with the property (iii) of Proposition 3.2. Then

ρ(X) = inf{x0 ∈ E | X + 1Ωx0 ≥ Z for some Z ∈ B}. (3.5) this application ρ defines a risk measure on Hφ(Ω, E, µ) whose acceptance set A is the smallest subset of Hφ(Ω, E, µ) that contains B and satisfies (i) of Proposition 3.2. If B is convex, then so is ρ. If B is a convex cone, then ρ is coherent. If B satisfies (3.4), then ρ is included in E. Proof. ρ is a risk measure and B is contained in A. Moreover, for each X ∈ A and ≥ n ∈ B 1 ≥ n n 1, there exists Z such that X + 1Ω n Z . This shows that A is contained in every subset of Hφ(Ω, E, µ) containing B and satisfying (iii) of Proposition 3.2. That ρ is convex when B is so, coherent when B is a convex cone, and with value in E when B satisfies (3.4). □

4. Robust Representation In this part, we give the robust representation for a coherent and a convex risk measure on Orlicz heart spaces that are Banach lattices valued. This also exploited for the representation of convex functionals in Cheridito et al. [6], Biagini and Fritelli [16] and Delbaen [7]. 174 COENRAAD LABUSCHAGNE, HABIB OUERDIANE, AND IMEN SALHI

The core of A, denoted by core(A), is the algebraic of a subset A of Hφ(Ω, E, µ) which consists of all x ∈ A that has an algebraic neighborhood contained in A; i.e.,

core(A) := {x0 ∈ A | ∀x ∈ E, ∃tx > 0, ∀t ∈ [0, tx] , x0 + t.x ∈ A} In general core(core(A)) ≠ core(A). If A is a , then core(core(A)) = core(A). int(A) is the interior of a subset A of a which is con- tained in its core(A). int(A) is the union of all open set contained in A. Lemma 4.1. If f : E → E is an increasing function, then core(domf) = int(domf). Proof. It is easy to check that int(domf) ⊂ core(domf). To prove the second inclusion, we assume that f : E → E where E is a Banach lattice on an algebraic neighborhood of x ∈ E but not on a neighborhood of x, ∈ ∥ ∥ ≤ 1 ∈ then there exists a sequence (zn) E such that zn 4n and f(x + zn) E for all x ∈ E. + − − ∥ +∥ ≤ 1 + ∈ (zn) can be written∑ as zn = zn zn , then zn 4n and f(x + zn ) E. n + Let define z := n≥1 2 zn . By assumption, there exists an ε > 0 such that f(x + εz) is finite. For all n with n2n ≥ 1, we have ∞ ≥ n + ≥ + ∈ > f(x + εz) f(x + ε2 zn ) f(x + zn ) E which is not true, then the result holds. □ Definition 4.2. A function f : E → R is called a proper function if f(x) < ∞ for at least one x and f(x) > −∞ for every x ∈ E. For a proper function f from a Banach lattice E to R , we have the following properties: (a) Every continuous map from a compact space to a Hausdorff space is both proper and closed. (b) A proper function satisfies the inequality of Young: xb ≤ f(x) + f ∗(b), ∀ x, b ∈ E The function f is called sub-differentiable at x ∈ E if f(x) ∈ R and there exists an element x∗ ∈ E∗, the dual topological of E, such that x∗(z) ≤ f(x + z) − f(x), ∀z ∈ E We define the conjugate of f: f ∗(x∗) := sup{x∗(x) − f(x)} x∈E f ∗ is σ(E∗,E) lower semi-continuous and convex on E∗. From the definition of f ∗, we have f(x) ≥ f ∗∗(x) := sup {x∗(x) − f ∗(x∗)}, ∀x ∈ E x∗∈E∗ Moreover, RISK MEASURES ON ORLICZ HEART SPACES 175

f(x) = max {x∗(x) − f ∗(x∗)}, ∀x ∈ E (4.1) x∗∈E∗ where f is sub-differentiable. let △(X) := {f : E → R | f is a proper } Theorem 4.3 (Theorem 2.4.9, [25]). Let f ∈ △(X). If f is continuous at ′ x ∈ domf, then Df(x) is non empty and ω∗-compact. Furthermore, f (x, .) is continuous and ∀ ∈ ′ { ∗ | ∗ ∈ } u E, fε(x, u) = max < u, x > x Df(x) (4.2) Theorem 4.4. Let f be an increasing and a convex function in E. Then for all x ∈ dom(f) the following holds: (i) f is Lipschitz and continuous in ϑ(x) with respect to the norm on E. (ii) f is sub-differentiable at x. (iii) f is given by f(x) = max {x∗(x) − f ∗(x∗)}. x∗∈E∗ Proof. (i) It follows from Corollary 2.2.12 [25]. (ii) We have core(domf) = int(dom(f), every x ∈ core(domf) has a neigh- borhood U such that U ⊂ dom(f). By Proposition 3.1 [24] it follows that f is continuous and sub-differentiable at x. (iii) (i) and (4.1) implies that f is continuous at x and there exists a neighbor- hood of x on which f is bounded. □

Definition 4.5. A probability measure Q in (Ω, F) is called absolutely continuous with respect to µ if Q(A) = 0 ⇒ µ(A) = 0. In the following, we identify a probability measure Q on (Ω, F) which is abso- dQ ∈ lutely continuous with respect to µ with its Radon-Nikodym derivative ξ = dµ L1(Ω, E, µ). Define

1 B := {ξ ∈ L (Ω, E, µ)|ξ ≥ 0, Eµ(ξ) = 1} which represents the set of all probability measure on (Ω, F, µ) that are absolutely continuous with respect to µ. ∗ ∗ ∗ We denote Bφ by Bφ = B ∩ Lφ (Ω, E, µ). ∗ Definition 4.6. A mapping α : Bφ → E is called a penalty function if it is bounded from below and not identically equal to ∞. We said that α satisfies the growth condition if there exist a and b ∈ E such that Q ≥ ∥Q∥∗ ∀ Q ∈ φ∗ α( ) a + b φ , B (4.3) φ∗ and for any penalty function α on B , we give the robust representation of ρα:

ρα(X) := sup EQ(−X) − α(Q) ,X ∈ Hφ(Ω, E, µ) ∗ Q∈Bφ 176 COENRAAD LABUSCHAGNE, HABIB OUERDIANE, AND IMEN SALHI

ρα is a risk measure on Hφ(Ω, E, µ) which is lower semi-continuous and convex. Now, let’s recall the Hahn-Banach theorem: Theorem 4.7. (Hahn Banach) Suppose that B and C are two non-empty, disjoint and convex subsets of a locally convex space E. Then, if B is compact and C is closed, there exists a continuous linear functional ℓ on E such that

sup ℓ(x) < inf ℓ(y) x∈C y∈B Theorem 4.8. Let lφ be an Orlicz with a norm given by ∞ ∑ t ∥X∥ = inf{λ > 0 | φ( i ) ≤ 1} λ i=1 φ for X = {ti} ∈ l . We say that a sequence of balls with centres {Xi} and redias r can be packed into the unit ball of a space E if {Xi} and r satisfy the following properties:

(a) ∥Xi∥ ≤ 1 − r , i = 1, 2, ...., (b) ∥Xi − Xj∥ ≤ 2r , i ≠ j , i, j = 1, 2.... The ball-packing constant of E is defined as Λ(E) = sup{r > 0 | there exists { }∞ ⊂ ∥ ∥ ≤ − ∥ − ∥ ≤ ̸ } Xi i=1 E such that Xi 1 r, Xi Xj 2r (i = j) . Then 1 Λ(E) ≤ (4.4) 2 1 { }∞ ⊂ Proof. In fact, if Λ(E) > 2 , then there exist r and Xi i=1 E satisfying Λ(E) > 1 r > 2 such that 1 ∥X ∥ ≤ 1 − r < i 2

∥Xi − Xj∥ ≥ 2r > 1 (i ≠ j)} so when ≠ j we have 1 1 1 > ∥X ∥ ≤ ∥X − X ∥ > 1 − = 2 i i j 2 2 with is a contradiction. □

we have the following result:

∗ Theorem 4.9. Let α be a penalty function on Bφ . Then there is an equivalence between the conditions: (i) α satisfies (4.3). (ii) core(domρα) is a non empty set. (iii) ρα is with value in a Banach lattice and every X ∈ Hφ(Ω, E, µ) has a neighborhood on which the risk measure is Lipschitz-continuous with re- spect to ∥.∥φ . Proof. (iii) ⇒ (ii) ⇒ (i) ⇒ (iii) • (iii) ⇒ (ii) is trivial. RISK MEASURES ON ORLICZ HEART SPACES 177

• (ii) ⇒ (i) Now, assume that ρα is with value in a Banach lattice space on an alge- braic neighborhood of X ∈ Hφ(Ω, E, µ). Since Y 7→ ρα(−Y ) is increasing and by core(domf) = int(domf) that there exists an ε > 0 such that ρα is a Banach lattice valued on the closed ball Bε(X) with redias ε around X. ∞ Therefore L (Ω, E, µ) is ∥.∥φ dense in Hφ(Ω, E, µ) implies there exists a n sequence (Y )n≥1 of bounded random variables such that n −n−2 ∥Y − X∥φ ≤ ε2 If α does not satisfy (4.3), then there exists a sequence of probability n φ∗ measure (Q )n≥1 in B such that Qn − − ∥ n∥ −n−2∥Qn∥∗ ≥ α( ) < n Y ∞ + ε2 φ for all n 1 φ∗ ∥ ∥∗ Since (L , . φ) is the of Hφ(Ω, E, µ), there exists for every n n n n n ≥ 1, Z ∈ Hφ(Ω, E, µ) such that Z ≤ 0, ∥Z ∥φ ≤ 1 and EQn (−Z ) ≥ 1 ∥Qn∥∗ 2 φ. ∑ −n n The Z := ε n≥1 2 Z is included in Hφ(Ω, E, µ) and ∥Z∥φ ≤ ε, then −n n ρα(X + Z) ≥ ρα(X + ε2 Z ) −n n n ≥ EQn (−X − ε2 Z ) − α(Q ) n n −n n n ≥ EQn (−Y ) + EQn (Y − X) + ε2 EQn (−Z ) − α(Q ) ≥ −∥ n − ∥ ∥Qn∥∗ −n−1∥Qn∥∗ − −n−2∥Qn∥∗ Y X φ φ + ε2 φ + n ε2 φ ≥ n ∀ n ≥ 1,

which is not true because of ρα(0) ∈ E on Bε(X). Therefore, α must fulfill (4.3) and (i) is proved. • (i) ⇒ (iii) We assume that there exist a and b ∈ E such that Q ≥ ∥Q∥∗ ∀Q ∈ φ∗ α( ) a + b φ, B

e φ We can choose X ∈ Hφ(Ω, E, µ). It exists X ∈ L (Ω, E, µ) such that e ∥X − X∥φ ≤ b and we obtain

e e EQ(−X) − α(Q) = EQ(−X) + EQ(X − X) − α(Q) ≤ ∥ e∥ ∥Q∥∗ ∥ e − ∥ ∥Q∥∗ − − ∥Q∥∗ X φ φ + X X φ φ a b φ ≤ ∥ e∥ ∥Q∥∗ − ∀ Q ∈ φ∗ X φ φ a, B , ≤ ∥ e∥ ∥Q∥∗ − ∈ which shows that ρα(X) X φ φ a E. Hence, ρα is a Banach lattice valued and the rest follows from Theorem 4.4 (i). □

5. Examples 5.1. Value at risk. A common approach to the problem of measuring the risk of a financial position X consists in specifying a quantile of X under the given 178 COENRAAD LABUSCHAGNE, HABIB OUERDIANE, AND IMEN SALHI probability measure µ. For λ ∈ (0, 1), a λ-quantile of a random variable X on Hφ(Ω, E, µ) is any q with the property µ[X ≤ q] ≥ λ and µ[X < q] ≤ λ − + and the set of all λ-quantile of X is an interval [qX (λ), qX (λ)], where − { | } { | ≤ ≥ } qX (t) = sup x µ(X < x) < t = inf x µ(X x) t is the lower and + { | ≤ } { | ≤ } qX (t) = inf x µ(X x) > t = sup x µ(X < x) t is the upper quantile function of X. Definition 5.1. Fix a level λ ∈ (0, 1). For a financial position X, we define its value at risk at level λ as − + − − { | ≤ } V @Rλ(X) := qX (λ) = q−X (1 λ) = inf x0 µ(X + 1Ωx0 < 0) λ (5.1)

In financial terms, V @Rλ(X) is the smallest amount of capital which, if added to X and invested in the risk-free asset, keeps the probability of a negative outcome below the level λ. However, value at risk only controls the probability of a loss, it does not capture the size of such a loss if it occurs. Clearly, V @Rλ is a risk measure on Hφ(Ω, E, µ), which is positively homogeneous. V @Rλ is not convex and so V @Rλ is not a convex risk measure (see Example 4.46 [14]).

Proposition 5.2. For each X ∈ Hφ(Ω, E, µ) and each λ ∈ (0, 1),

V @Rλ(X) = min{ρ(X)| ρ is convex, continuous from above and ≥ V @Rλ} − + ≤ ∈ Proof. Let q := V @Rλ(X) = qX (λ) so that µ(X < q) λ. Let A E. If A satisfies µ(A) > λ , then µ(A ∩ {X ≥ q}) > 0. Thus, we may define a measure QA by

QA := µ(. | A ∩ {X ≥ q}). E − ≤ − It follows that QA ( X) q = V @Rλ(X). Let Q := {QA | µ(A) > λ}, and we use this to define a coherent risk measure ρ via

E − ρ(X) := sup QA ( X) QA∈Q

Then ρ(X) ≤ V @Rλ(X). Hence, the assertion shows that ρ(X) ≥ V @Rλ(X) for each X ∈ Hφ(Ω, E, µ). Let ε > 0 and A := {X ≤ −V @Rλ(X) + 1Ωε}. Clearly µ(A) > λ, and so QA ∈ Q. Moreover, QA(A) = 1, and we obtain ≥ E − ≥ − ρ(X) QA ( X) V @Rλ(X) 1Ωε Since ε > 0 is arbitrary, the result follows. □

For the rest of this part, we concentrate on the following risk measure which is defined in terms of value at risk, but does not satisfy the axioms of a coherent risk measure. RISK MEASURES ON ORLICZ HEART SPACES 179

Definition 5.3. The average value at risk at level λ ∈ (0, 1) of a position X ∈ Hφ(Ω, E, µ) is given by ∫ 1 λ AV @Rλ(X) = V @Rγ (X)dγ. λ 0 where γ is a level on (0, 1). Sometimes, the average value at risk is also called the “conditional value at risk” or the “expected shortfall” and we write CV @Rλ(X) or ESλ(X). So, we prefer the term Average Value at Risk and note that ∫ −1 λ AV @Rλ(X) = qX (t)dt λ 0 For λ = 1 we have ∫ 1 − + E − AV @R1(X) = qX (t)dt = Q( X) 0

Remark 5.4. For X ∈ Hφ(Ω, E, µ), we have

limV @Rλ(X) = −ess inf X = inf{x0 | µ(X + 1Ωx0 < 0) ≤ 0}. λ↓0 Hence, it makes sense to define

AV @R0(X) := V @R0(X) := −ess inf X Recall that it is continuous from above but in general not from below. Lemma 5.5. For λ ∈ (0, 1) and any λ − quantile q of X, ( ) 1 + 1 + AV @Rλ(X) = EQ[(q − X) ] − q = inf EQ[(r − X) ] − λr (5.2) λ λ r∈Hφ(Ω,E,µ)

Proof. Let qX be a quantile function with qX (λ) = q. By Lemma A.19 in [14], ∫ ∫ 1 λ 1 + 1 + 1 EQ[(q − X) ] − q = (q − qX (t)) dt − q = − qX (t) = AV @Rλ(X) λ λ 0 λ 0 This proves the first identity. The second one follows from Lemma A.22 [14]. □

References 1. Arai, T.: Convex risk measure on Orlicz spaces: inf-convolution and shortfall, Mathematics and Financial Economics, vol. 3, no. 3, (2010) 73–88. 2. Artzner, P., Delbaen, F., Health, D. and Eber, J. M.: Coherent Measures of Risk, Mathe- matical Finance, vol. 9, no. 3, (1999) 203–228. 3. Balb´as,A., Balb´as,B. and Balb´as,R.: Minimizing Vector Risk Measures, Springer-Verlag, Berlin Heidelberg, 2010. 4. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M. and Montrucchio, L.: Complete Monotone Quasiconcave Duality, Mathematics of operations research. vol. 36, no. 2, (2011) 321–339. 5. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M. and Montrucchio L.: Risk Measures: Rationality and Diversification, Mathematical Finance, vol. 21, no. 4, (2011) 743–774. 6. Cheridito, P., Delbaen, F. and Kupper, M.: Coherent and Convex Monetary Risk measures for Bounded Processes, Stoch. Proc. Appl, vol. 112, no. 1, (2004) 1–22. 7. Cheridito, P., Delbaen, F. and Kupper, M. : Coherent and Convex Monetary Risk Measures for Unbounded C‘adl‘ag Processes, Finance and Stochastics, vol. 10, no. 3, (2006) 427–448. 180 COENRAAD LABUSCHAGNE, HABIB OUERDIANE, AND IMEN SALHI

8. Cheridito, P. and Li, T.: Risk Measures on Orlicz Hearts, Mathematical Finance, vol. 19, no. 2, (2009) 189–214. 9. Delbaen, F.: Coherent Risk measures, Bltter der DGVFM, vol.24, no. 4, (2000) 733–739. 10. Delbaen, F.: Coherent Risk measures on General Probability Spaces, Advances in finance and stochastics.Springer Berlin Heidelberg, (2002) 1–37. 11. Eisele, K-T. and Taieb, S.: Lattice Modules Over Rings of Bounded Random Variables, Communications on Stochastic Analysis, vol 6, no. 4, (2012) 525–545. 12. El Karoui, N. and Ravanelli, C.: Cash Subadditive Risk Measures and Intereset rate Ambi- guity, Mathematical Finance, vol. 19, no. 4, (2009) 561–590. 13. F¨olmer,H. and Schied, A.: Convex Measures of Risk and Trading Constraints, Finance and Stochastics, vol. 6, no. 4, (2002) 429–447. 14. F¨ollmer,H. and Schied, A.: Stochastic Finance: An Introduction in Discrete Time, second revised and extended edition, Walter de Gruyter, Berlin, 2004. 15. F¨ollmer,H. and Schied, A.: Stochastic Finance: An Introduction in Discrete Time, third revised and extended edition, Walter de Gruyter, Berlin, 2011. 16. Fritelli, M. and Gianin, E. R.: Law Invariant Convex Risk Measures, Advances in Mathe- matical Economics. Springer Tokyo, vol 7, (2005) 42–53. 17. Fritelli, M. and Rosazza Gianin, E.: On the penalty function and on continuity properties of risk measures, World Scientific Publishing Company, vol. 14, no. 1, (2011) 163–185. 18. Jouini, E. and Kallal, H.: Arbitrage in securities markets with short sales constraints, Math- ematical Finance, vol. 5, no. 3, (1995) 197–232. 19. Jouini, E., Schachermayer, W. and Touzi, N.: Optimal risk sharing for law invariant mon- etary utility functions, Mathematical Finance, vol. 18, no 2, (2008) 269292. 20. Labuschagne, C. C. A. and Offwood, T.M.: Pricing exotic options using the Wang transform, The North American Journal of Economics and Finance, vol. 25, (2013) 139–150. 21. Offwood, T.M.: No free lunch and risk measures on Orlicz spaces, Diss. University of the Witwatersrand, South Africa, 2012. 22. Orlicz, W.: Linear , World Scientific Publishing Co. Pte. Lid, Series In , 1963. 23. Rao, K.C and Subramanian, N.: The Orlicz space of entire sequences, International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 68, (2004) 3755–3764. 24. Ruszczy´nski,A. and Shapiro, A.: Optimization of Convex Risk functions, Mathematics of Operations Research, vol. 31, no. 3, (2006) 433–452. 25. Z˘alinescu,C.: Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, 2002.

Coenraad Labuschagne: Programme in Quantitative Finance, Department of Fi- nance and Investment Management, University of Johannesburg, PO Box 524, Auck- land Park 2006, South Africa E-mail address: [email protected] Habib Ouerdiane: Faculty of Sciences of Tunis, University Tunis El Manar, Tunisia E-mail address: [email protected] imen salhi: Faculty of Sciences of Tunis, University Tunis El Manar, Tunisia E-mail address: [email protected]