Monetary Risk Measures and Generalized Prices Relevant to Set-Valued Risk Measures
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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 109, HIKARI Ltd, Monetary Risk Measures and Generalized Prices Relevant to Set-Valued Risk Measures Christos E. Kountzakis Department of Mathematics University of the Aegean, Karlovassi GR , Samos, Greece Copyright c 2014 Christos E. Kountzakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this article is to provide some results under which a coherent set-valued risk measure may imply the existence and the coherence of a monetary coherent risk measure related to it. Also, set-valued risk measures may imply the existence of a generalized price functional, in the sense of [2], which is also extended in the case of correspondences. The space of financial positions is a partially ordered normed vector space. Mathematics Subject Classification: 46B40; 47H04; 91B30 Keywords: set-valued risk measure; well-based cone; derivative monetary risk measure; generalized price 1 Preliminaries We consider two periods of time (0 and 1) and a non-empty set of states of the world Ω which is supposed to be an infinite set. The true state ω Ω that the investors face is contained in some A F, where F is some σ-algebra of subsets of Ω which gives the information about the states that may occur at time-period 1. A financial position is a F-measurable random variable x : Ω R. This random variable is the profile of this position at time-period 1. We suppose that the probability of any state of the world to occur is given by a probability measure µ : F [0, 1]. The financial positions are supposed
2 5440 Christos E. Kountzakis to lie in some subspace E of L 0 (Ω, F, µ) of the measurable random variables with respect to the probability space (Ω, F, µ), being a partially ordered vector space. The elements of the topologiical dual space E of E, which are strictly positive, denote the spot- price functionals. Namely, if the investors face the price f E in the market, this indicates that the amount of money that has to be paid at time-period 0 by some investor in order to receive the payoff of the position x E is equal to f(x). First, we remind the definitions of monetary risk measures, in which C is a wedge (or a cone) of E. Definition 1.1 A real-valued function ρ : E R which satisfies the properties (i) ρ(x + ae) = ρ(x) a (e-translation Invariance) (ii) ρ(λx + (1 λ)x) λρ(x) + (1 λ)ρ(y) for any λ [0, 1] (Convexity) and (iv) y C x implies ρ(y) ρ(x) (C-Monotonicity) where x, y E is called (C, e)-convex monetary risk measure. Definition 1.2 A real-valued function ρ : E R is a (C, e)-coherent monetary risk measure if it is an (C, e)-convex monetary risk measure and it satisfies the following property: ρ(λx) = λρ(x) for any x E and any λ R + (Positive Homogeneity). Second, we remind the definition of the coherent and the convex risk correspondence ρ : E 2 E. Definition 1.3 A correspondence ρ : E 2 E which satisfies the properties (i) ρ(x + ae) = ρ(x) {ae} (e-translation Invariance) (ii) ρ(λx + (1 λ)x) λρ(x) + (1 λ)ρ(y) for any λ [0, 1] (Convexity) and (iv) y C x implies ρ(y) ρ(x) (C-Monotonicity) where x, y E is called (C, e)-convex risk correspondence. Definition 1.4 A correspondence ρ : E 2 E is a (C, e)-coherent risk correspondence if it is an (C, e)-convex risk correspondence and it satisfies the following property: ρ(λx) = λρ(x) for any x E and any λ R + (Positive Homogeneity). We assume that e is a quasi-interior point of E with respect to the partial ordering induced by C. Also, A + B, where A, B 2 E is the set {t E t = a + b, a A, b B}. λ A = {x E x = λ a, a A}, λ R.
3 Monetary risk measures and generalized prices Connection to previous existing results Set -Valued Risk Measures are introduced in [10] and were studied by several researchers, as an attempt to face the problem of a complete ordering for the values of a risk measure in the case where it is a vector -valued one. This approach fits the case of the credit risk, where the solvancy capital for a position varies according to the position x and the posibility of different events related to liquidity risk to occur. The spaces in which the values of a set-valued risk measure belong, are usually finite -dimensional (see [6], [7]). 3 Coherent Risk Measures and Generalized Prices relevant to Coherent Risk Correspondences Definition 3.1 A financial position x is safe with respect to rho : E 2 E, if for this x E, ρ(x) E +. Definition 3.2 A financial position x is consistent with respect to ρ : E 2 E, if for this x E, ρ(x) E + = {0}. Lemma 3.3 The subset of the safe financial positions of E with respect to a coherent ρ : E 2 E is a wedge. Proof: Let x, y be safe with respect to ρ. This indicates ρ(x), ρ(y) E +, namely ρ(x) + ρ(y) ( E + ) + ( E + ) E +. Since ρ(x + y) ρ(x) + ρ(y) E +, x + y is safe with respect to ρ. Also, if x is safe with respect to ρ, then ρ(x) E + and let λ R +. Then hence λ x is safe with respect to ρ. ρ(λ x) λρ(x) λ( E + ) E +, Definition 3.4 The wedge of the safe financial positions is called acceptance set of ρ. A ρ = {x E ρ(x) E + },
4 5442 Christos E. Kountzakis Corollary 3.5 The derivative monetary coherent risk measure ˆρ : E R, which arises from ρ : E 2 E is defined as follows: ˆρ(x) = sup π( x), x E, π A 0 ρ B e where e is a quasi-interior point of E, which defines a closed, bounded base B e on E +. Proposition 3.6 The derivative monetary coherent risk measure ˆρ : E R, which arises from ρ : E 2 E, ˆρ(x) = sup π( x), x E, π A 0 ρ B e is well-defined in the case where e is a quasi- interior point of E, which defines a closed, bounded base B e on E +, which has to be a cone with the 0-Schur Property. Proof: The existence of a bounded base which is defined by a continuous linear functional is equivalent to the 0-Schur Property for E + and the existence of quasi-interior points of E, by [11, Thm.10]. Corollary 3.7 The derivative monetary coherent risk measure ˆρ : E R, which arises from ρ : E 2 E is well-defined in the case where E is a reflexive space and e is an interior point of E +, which defines a closed, bounded base B e on E 0 +. Proof: It arises from the previous Proposition if we notice that since E is reflexive, E + = E Example 3.8 We consider some coherent risk correspondence ρ : E 2 E, where E = L 1 (Ω, F, µ), E + = L 1 + e = 1 being an interior point in L +. An example of a coherent risk correspondence ρ : L 1 2 L1 + is given by ρ(x) = x + L 1 +. Example 3.9 We may also pick E = L 2 (Ω, F, µ) being ordered by E + = {x E E(x) 1 2 x 2}, where E(x) denotes the mean-value of the random variable x. This is actually a Bishop-Phelps cone, which actually has interior points, see [8, p ]. Lemma 3.10 The subset of the consistent financial positions of E with respect to a coherent ρ : E 2 E is a wedge.
5 Monetary risk measures and generalized prices 5443 Proof: Let x, y be consistent with respect to ρ. This indicates ρ(x) E + = {0}, ρ(y) E + = {0}, namely (ρ(x) + ρ(y)) E + = {0}. This is true, because if r 1 ρ(x), r 2 ρ(y), t 1 E +, t 2 E + and r 1 t 1 = 0, r 2 t 2 = 0, then r 1 +r 2 (t 1 +t 2 ) = 0, where t 1 = 0, t 2 = 0. Hence r 1, r 2 = 0 and consequently ρ(x + y) E + = (ρ(x) + ρ(y)) E + = {0}. Finally, x + y is consistent with respect to ρ. Also, if x is consistent with respect to ρ, then ρ(x) E + = {0}. If r ρ(x) and t E +, where r t = 0, then λ (r t) = 0, hence λ r λ t = 0, with λ R +. Let us set λ t = t 1 E +, which implies λ r t 1 = 0. Definition 3.11 The set of financial positions is called consistence range of ρ. C ρ = {x E ρ(x) E + = {0}}, Lemma 3.12 Cρ 0 linear functional. if E + has a base which is defined by a continuous Proof: See [9, Thm.3.22]. The consistence range of ρ is used for the definition of a pricing functional ψ : E R, which satisfies the properties of a generalized price indicated in [2, Lem.3.3]. These properties are listed in the following Definition 3.13 A functional ψ : E R is a generalized price if and only if satisfies the following properties: (i) If x y, then ψ(x) ψ(y) (E + -Monotonicity) (ii) ψ(x + y) ψ(x) + ψ(y) (Super-Additivity) (iii) ψ(a x) = aψ(x), x E, a R + (Positive Homogeneity) We obtain the following duality between monetary coherent risk measures and generalized prices.
6 5444 Christos E. Kountzakis Lemma 3.14 If ψ : E R is a generalized price which is normalized at the numeraire financial position e E + (ψ(e) = 1) and satsifies the e- Translation Invariance ψ(x + a e) = ψ(x) + a, a R, then ψ = ρ ψ is a (E +, e)- monetary coherent risk measure. On the other hand if ρ : E R is a monetary coherent risk measure, then ρ = ψ ρ : E R is a generalized price functional. Proof: We have to verify that ψ satifies the properties of a (E +, e) - coherent risk measure. Since ψ is Super-additive, ψ is Sub-additive. Since ψ satisfies Positive Homogeneity, then ψ also satisfies it. Since ψ satisfies the E + -Monotonicity in the sense that if x y, then ψ(x) ψ(y), then ψ satisfies the E + - Monotonicity of a (E +, e)-convex risk measure, since if x y, then ψ(y) ψ(x). Since ψ satisfies the above version of e-translation Invariance, then ψ = ρ ψ satisfies the e-traslation Invariance in the sense which is stated in the properties of a (E +, e)-convex risk measure risk measure, namely ρ ψ (x + a e) = ρ ψ (x) a, x E, a R. For the ψ ρ the proof is equivalent. Theorem 3.15 The functional ψ Cρ : E R +, where ψ Cρ,g(x) = inf f(x) f Cρ 0 Bg is a generalized price, where E is such that E + has a base defined by a continuous linear functional g. Proof: It suffices to prove that the functional ψ Cρ,g : E R + satisfies the following properties: (i) If x y, then ψ Cρ,g(x) ψ Cρ,g(y) (E + -Monotonicity) (ii) ψ Cρ,g(x + y) ψ Cρ,g(x) + ψ Cρ,g(y) (Super-Additivity) (iii) ψ Cρ,g(a x) = aψ Cρ,g(x), x E, a R + (Positive Homogeneity) About (i), if x y, then f(x) f(y), hence inf f C 0 ρ f(x) inf f C 0 ρ f(y). About (ii) hence f(x + y) = f(x) + f(y) inf f(x) + inf f(y), f Cρ 0 f Cρ 0 About (iii), f(a x) = af(x), hence inf f(x + y) inf f(x) + inf f(y). f Cρ 0 f Cρ 0 f Cρ 0 inf f(a x) = a inf f(y). f Cρ 0 f Cρ 0
7 Monetary risk measures and generalized prices Appendix In this Section, we give some essential notions and results from the theory of partially ordered linear spaces which are used in this paper. For these notions and definitions, see [8, Ch.1, Ch.2, Ch.3]. Let E be a (normed) linear space. A set C E satisfying C + C C and λc C for any λ R + is called wedge. A wedge for which C ( C) = {0} is called cone. A pair (E, ) where E is a linear space and is a binary relation on E satisfying the following properties: (i) x x for any x E (reflexive) (ii) If x y and y z then x z, where x, y, z E (transitive) (iii) If x y then λx λy for any λ R + and x + z y + z for any z E, where x, y E (compatible with the linear structure of E), is called partially ordered linear space. The binary relation in this case is a partial ordering on E. The set P = {x E x 0} is called (positive) wedge of the partial ordering of E. Given a wedge C in E, the binary relation C defined as follows: x C y x y C, is a partial ordering on E, called partial ordering induced by C on E. If the partial ordering of the space E is antisymmetric, namely if x y and y x implies x = y, where x, y E, then P is a cone. E denotes the linear space of all linear functionals of E, called algebraic dual while E is the norm dual of E, in case where E is a normed linear space. Suppose that C is a wedge of E. A functional f E is called positive functional of C if f(x) 0 for any x C. f E is a strictly positive functional of C if f(x) > 0 for any x C\C ( C). A linear functional f E where E is a normed linear space, is called uniformly monotonic functional of C if there is some real number a > 0 such that f(x) a x for any x C. In case where a uniformly monotonic functional of C exists, C is a cone. C 0 = {f E f(x) 0 for any x C} is the dual wedge of C in E. Also, by C 00 we denote the subset (C 0 ) 0 of E. It can be easily proved that if C is a closed wedge of a reflexive space, then C 00 = C. If C is a wedge of E, then the set C 0 = {x E ˆx(f) 0 for any f C} is the dual wedge of C in E, whereˆ: E E denotes the natural embedding map from E to the second dual space E of E. Note that if for two wedges K, C of E, K C holds, then C 0 K 0. If C is a cone, then a set B C is called base of C if for any x C\{0} there exists a unique λ x > 0 such that λ x x B. The set B f = {x C f(x) = 1} where f is a strictly positive functional of C is the base of C defined by f. B f
8 5446 Christos E. Kountzakis is bounded if and only if f is uniformly monotonic. If B is a bounded base of C such that 0 / B then C is called well-based. If C is well-based, then a bounded base of C defined by a g E exists. If E = C C then the wedge C is called generating, while if E = C C it is called almost generating. If C is generating, then C 0 is a cone of E in case where E is a normed linear space. Also, f E is a uniformly monotonic functional of C if and only if f intc 0, where intc 0 denotes the norm-interior of C 0. If E is partially ordered by C, then any set of the form [x, y] = {r E y C r C x} where x, y C is called order-interval of E. If E is partially ordered by C and for some e E, E = n=1[ ne, ne] holds, then e is called order-unit of E. I e = n=1[ ne, ne] is the solid subspace of E generated by e, then e is a quasi-interior point of E if E = I e. If E is a normed linear space, then if every interior point of C is an order-unit of E. If E is moreover a Banach space and C is closed, then every order-unit of E is an interior point of C. The partially ordered vector space E is a vector lattice if for any x, y E, the supremum and the infimum of {x, y} with respect to the partial ordering defined by P exist in E. In this case sup{x, y} and inf{x, y} are denoted by x y, x y respectively. If so, x = sup{x, x} is the absolute value of x and if E is also a normed space such that x = x for any x E, then E is called normed lattice. If a normed lattice is a Banach space, then it is called Banach lattice. A Banach lattice E whose norm has the property x+y = x + y, x, y E + is called AL-space. A set S in a vector lattice E is called solid if y x and x S implies y S. A solid vector subspace of a vector lattice is called ideal. An ideal I is a sublattice of E, i.e. a subspace of E such that x y I, x y I if x, y I respectively. A subset F of a convex set C in E is called extreme set or else face of C, if whenever x = az + (1 a)y F, where 0 < a < 1 and y, z C implies y, z F. If F is a singleton, F is called extreme point of C (see [1, Def.5.111]). References [1] Aliprantis, C.D., Border, K.C. (1999) Infinite Dimensional Analysis, A Hitchhiker s Guide (second edition), Springer [2] Aliprantis, C.D., Tourky, R., Yannelis, N.C. (2001) A Theory of Value with Non-Linear Prices: Equilibrium Analysis beyond Vector Latttices. Journal of Economic Theory, [3] Artzner, P., Delbaen, F., Eber, J.M., Heath, D. (1999) Coherent measures of risk. Mathematical Finance,
9 Monetary risk measures and generalized prices 5447 [4] Delbaen, F. (2002) Coherent risk measures on general probability spaces. In Advances in finance and stochastics: essays in honour of Dieter Sondermann, Springer-Verlag: Berlin, New York, [5] Föllmer, H., Schied, A. (2002) Convex measures of risk and trading constraints. Finance and Stochastics, [6] Hamel, A.H., Heyde, F. (2010) Duality for Set-Valued Measures of Risk. SIAM Journal of Financial Mathematics, [7] Hamel, A.H., Heyde, F., Rudloff, B. (2011) Set-Valued Risk Measures for Conical Markets. Mathematics and Financial Economics, [8] Jameson, G. (1970) Ordered Linear Spaces, Lecture Notes in Mathematics vol. 141, Springer-Verlag. [9] Jahn, J. (2004) Vector Optimization, Springer-Verlag. [10] Jouini, E., Meddeb, M., Touzi, N. (2004) Vector -valued coherent risk measures. Finance and Stochastics, [11] Kountzakis, C., Polyrakis, I.A. (2006) Geometry of Cones and an Application in the theory of Pareto Effiecient Points. Journal of Mathematical Analysis and Applications, Sent: March 13, 2014
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