3 214 5 ( ) Journal of East China Normal University (Natural Science) No. 3 May 214 Article ID: 1-5641(214)3-23-7 Representation theorem for AVaR under a submodular capacity TIAN De-jian, JIANG Long, JI Rong-lin (School of Sciences, China University of Mining and Technology, Xuzhou 221116, China) Abstract: From the viewpoints of quantile functions, we gave the definition of AVaR of financial positions under a capacity. Then, using the classical results of AVaR under the probability measure, we established the representation theorem for AVaR under the submodular capacity. As a byproduct of this representation theorem, we proved that AVaR under a submodular capacity is a coherent risk measure, which generalized the classical results. Key words: AVaR; quantile function; representation theorem; submodular capacity CLC number: O211 Document code: A DOI: 1.3969/j.issn.1-5641.214.3.4 AVaR,, (, 221116) :, VaR AVaR. Choquet AVaR, AVaR. AVaR,. : AVaR; ; ; Introduction From a regulatory perspective, the axiomatic-based monetary risk measures have been largely investigated because most axioms embed desirable economic properties. Coherent risk measures have been introduced by Artzner et al. [1]. Later, Follmer and Schied [2,3] and, independently, Frittelli and Rasazza Gianin [4] introduced the broader class of convex risk measures. Besides, coherent risk measures and convex risk measures can be induced by g-expectations [5-8]. : 213-5 : (11371362, 1111422); (2919); (21LKSX4). :,,,. E-mail:tiandejian1985@163.com. :,,,. E-mail: jianglong365@hotmail.com.
24 ( ) 214 One of the most widely known risk measures in the area of financial risk management is the Value at Risk (VaR) measure. (see, for instance, [3]). Due to its intuitive definition and wide utilization by many banking institutions, the VaR measure has been adopted as a standard for measuring risk exposure of financial positions. However, one of the major deficiencies of VaR, from the methodological point of view, is that it does not reflect the extreme losses beyond the λ-quantile level. In order to remedy the shortcomings of VaR (most importantly, its nonconvexity), the Average Value at Risk (AVaR) is introduced. Furthermore, AVaR is a coherent risk measure, and [3] established the representation theorem of AVaR. This paper starts from the observation that VaR of a financial position is a quantile function with a confidence level λ (, 1) under a given probability measure. Once a priori probability measure is replaced with a capacity, VaR and AVaR of financial positions under a capacity should be defined from the viewpoints of quantile functions in order to maintain the original interpretation. Under some additional assumptions on the capacity, we establish the representation theorem for AVaR under a submodular capacity, and we also obtain that AVaR under a submodular capacity is a coherent risk measure. The paper is organized as follows. We introduce some preliminaries in Section 1. The definitions of VaR and AVaR of financial positions under a capacity and the representation results for AVaR under a submodular capacity are presented in Section 2. 1 Preliminaries Let (Ω, F) be a measurable space, and let L := L (Ω, F) be the space of bounded F-measurable functions, endowed with the supremum norm. We denote by P the set of all probability measures on (Ω, F). We emphasize that no a priori probability measure is given on the measurable space (Ω, F). Firstly, we introduce some properties of set functions µ : F [, 1] by [9]. Monotonicity: if A, B F and A B, then µ(a) µ(b). Normalization: if µ( ) = and µ(ω) = 1. Continuous from below: if A n, A F, and A n A, then µ(a n ) µ(a). Continuous from above: if A n, A F, and A n A, then µ(a n ) µ(a). Submodular (or 2-alternating): if A, B F, then µ(a B) + µ(a B) µ(a) + µ(b). Now we introduce the definitions of capacity and Choquet integral (see, for instance, [9]). Definition 1 A set function µ : F [, 1] is called a capacity if it is monotonic, normalized and continuous from below. Definition 2 Let µ be a capacity, X L, and denote µ[x] by µ[x] := (µ(x > t) 1)dt + + µ(x > t)dt. We call µ[x] the Choquet integral of X with respect to the capacity µ. Let µ be a capacity and X L. Put G µ,x (x) := µ(x > x).
3, AVaR ( ) 25 We call G µ,x the decreasing distribution function of X with respect to µ. Taking into account the continuity property form below of the capacity µ, we derive that G µ,x is right continuous. We introduce the definition of quantile functions of X with respect to µ by [9]. Definition 3 Let µ be a capacity and X L. Then we say that е,X is a quantile function of X under the capacity µ if sup{x G µ,x (x) > λ} е,X(λ) sup{x G µ,x (x) λ}, λ (, 1). Any two quantile functions coincide for all levels λ, except on at most a countable set. We also have the following properties about the quantile functions of X under the capacity µ (see Chapter 1 and Chapter 4 of [9]). Lemma 1 Let µ be a capacity, X L, we have (i) е,X( ) is a decreasing function; (ii) If ν is an another capacity such that G ν,x G µ,x, then Ğν,X е,X except on at most a countable set; (iii) µ[x] = 1 е,X(t)dt; (iv) If u is an increasing continuous function, then е,u(X) = u е,X. We end this section by introducing the relationship between a submodular capacity and the coherent risk measures. The proof of the result can be found in Theorem 4.88 and Remark 4.9 of [3]. Lemma 2 Let µ be a submodular capacity, then ρ(x) := µ[ X] is a coherent risk measure on L. If the capacity µ has the additional continuity property from above, then we have where µ[x] = max Q Q µ E Q [X], X L, Q µ = {Q P Q(A) µ(a) for all A F}. (1.1) 2 Main results Let s recall the definition of quantile functions of X under the capacity µ (see Definition 3). It is easy to see that for any λ (, 1), all of the quantile functions of X under the capacity µ at λ forms a closed interval with Ğ µ,x (λ) and Ğ+ µ,x (λ). These endpoints can be defined as We also have the following proposition. Ğ µ,x (λ) := sup{x µ(x > x) > λ}, Ğ + µ,x (λ) := sup{x µ(x > x) λ}. Proposition 1 Let µ be a capacity, X L. For any λ (, 1), we have Ğ µ, X (λ) = inf{y µ( X > y) λ}.
26 ( ) 214 In particular, if µ = P P, then VaR P,λ (X) = Ğ P, X (λ). Proof We need to show that sup{x µ( X > x) > λ} = inf{y µ( X > y) λ}. (2.1) Indeed, for any x and y such that µ( X > x) > λ and µ( X > y) λ, we have µ( X > x) > µ( X > y). Due to the fact that the distribution function G µ, X ( ) of X under µ is decreasing, we derive that x < y. Thus we have sup{x µ( X > x) > λ} inf{y µ( X > y) λ}. On the other hand, for any y > sup{x µ( X > x) > λ}, we have µ( X > y) λ, and Let y sup{x µ( X > x) > λ}. Then we have y inf{y µ( X > y) λ}. sup{x µ( X > x) > λ} inf{y µ( X > y) λ}. Then (2.1) follows. If µ = P P, from (2.1) and the definition of VaR under P, we have that The proof is complete. Ğ P, X (λ) = inf{y P( X > y) λ} = inf{y P(X + y < ) λ} = VaR P,λ (X). Motivated by the classical definitions of VaR and AVaR and Proposition 1, we give the following definitions, which generalize the definitions of VaR and AVaR under a priori probability measure. Definition 4 Let µ be a capacity, X L. For λ (, 1), we define the Value at Risk with a confidence level λ (, 1) of X under the capacity µ as VaR µ,λ (X) := Ğ µ, X (λ) = inf{x µ(x + x < ) λ}. Definition 5 The average value at risk under a capacity µ at λ (, 1] of a position X L is given by AVaR µ,λ (X) = 1 λ VaR µ,t (X)dt. Remark 1 Since for any two quantiles coincide for all levels λ, except on at most a countable set, then the AVaR under a capacity µ at λ (, 1] of a financial position X L can be defined by any quantile function of X under µ, i.e., AVaR µ,λ (X) = 1 λ Ğ µ, X (t)dt.
3, AVaR ( ) 27 In the case of AVaR under a probability measure, we have known that AVaR P,λ is a coherent risk measure (see Theorem 4.47 of [3]). A natural question is: can we establish the representation theorem for AVaR µ,λ? The following theorem gives the positive answer under some additional assumptions on µ. Theorem 1 Let µ be a capacity, then we have (i) For any X L, AVaR µ,1 (X) = µ[ X]; (ii) For any X L, λ (, 1) and for any quantile function е, X of X under µ, letting q := е, X(λ), we have AVaR µ,λ (X) = 1 λ µ[(q X)+ ] q; (iii) If µ is also a submodular capacity with the continuity property from above, then for λ (, 1], AVaR µ,λ is a coherent risk measure. And it has the representation where AVaR µ,λ (X) = max Q Q µ,λ E Q [ X], X L, Q µ,λ = P Q µ and where Q µ is defined by (1.1) in Lemma 2. 1 (iii). have { Q P Q P and dq dp 1 λ, P-a.s. }, Proof (i) It is obvious from the definitions of AVaR µ,1 (X) and VaR µ,λ (X) and Lemma (ii) For any X L and λ (, 1), form the Lemma 1 (i), (iii), (iv) and Remark 1, we 1 λ µ[(q X)+ ] q = 1 λ = 1 λ = 1 λ = 1 λ 1 1 Ğ µ,(q X) +(t)dt q [q + е, X(t)] + dt q [е, X(t) е, X(λ)]dt + е, X(λ) Ğ µ, X (t)dt = AVaR µ,λ (X). (iii) For λ = 1, we derive that Q µ,1 = Q µ. By virtue of the first part of Theorem 1, we have AVaR µ,1 (X) = µ[ X]. Then from Lemma 2, the claim holds. For < λ < 1, we first show that AVaR µ,λ (X) sup Q Q µ,λ E Q [ X], X L. (2.2) For any Q Q µ,λ, there exists a probability measure P Q Q µ such that Q P Q and dq/dp Q 1 λ, P Q a.s.. From Theorem 4.47 of [3], we have AVaR PQ,λ(X) E Q [ X], X L. (2.3)
28 ( ) 214 On the other hand, by the definition of AVaR µ,λ and AVaR PQ,λ, for all X L, we have AVaR µ,λ (X) AVaR PQ,λ(X) = 1 λ (е, X(t) ĞP Q, X(t))dt. Due to the fact that P Q Q µ, we have G µ, X G PQ, X. By Lemma 1 (ii), we have е, X Ğ PQ, X except on an at most a countable set. Hence, we have AVaR µ,λ (X) AVaR PQ,λ(X), X L. (2.4) Then (2.2) follows from (2.3) and (2.4). Then we only need to show that for < λ < 1 and for any X L, there exists a probability measure Q Q µ,λ such that AVaR µ,λ (X) = E Q [ X]. (2.5) For any given X L, by the translation invariance of AVaR µ,λ and the linear expectation, we may assume without loss of generality that X >. By Lemma 2, we obtain that there exists a probability measure P µ,x Q µ such that µ[ X] = E Pµ,X [ X]. From the above equality, we have µ[ X] E Pµ,X [ X] = + [µ( X > x) P µ,x ( X > x)] dx =. Due to the fact P µ,x Q µ and the right continuity of the decreasing distribution functions G µ, X ( ) and G Pµ,X, X( ), for any x, we have µ( X > x) = P µ,x ( X > x). (2.6) For any λ (, 1), by the equality (2.6), the definitions of quantile functions of X under µ and quantile functions of X under P µ,x, we have sup{x µ( X > x) > λ} = sup{x P µ,x ( X > x) > λ}, sup{x µ( X > x) λ} = sup{x P µ,x ( X > x) λ}. Hence all of the quantile functions of X under µ and the quantile funtions of X under P µ,x are coincide except on at most a countable set. Thus by the definitions of AVaR µ,λ and AVaR Pµ,X,λ, we have AVaR µ,λ (X) AVaR Pµ,X,λ(X) = 1 λ So we have [е, X(t) ĞP µ,x, X(t)] dt =. AVaR µ,λ (X) = AVaR Pµ,X,λ(X). (2.7) By Theorem 4.47 of [3], we have there exists a probability measure Q P with Q P µ,x and dq /dp µ,x 1/λ, P µ,x -a.s. such that AVaR Pµ,X,λ(X) = E Q [ X]. (2.8)
3, AVaR ( ) 29 Combining (2.7) and (2.8), we obtain that there exists a probability measure Q Q µ,λ such that AVaR µ,λ (X) = E Q [ X]. Hence (2.5) holds. It follows from (2.7) and (2.8), we have that AVaR µ,λ (X) = max Q Q µ,λ E Q [ X], X L. We obtain that AVaR µ,λ is a coherent risk measure from the above representation formula. The proof is complete. Here we give a simple example of the submodular capacity. Example 1 Let ψ : [, 1] [, 1] be an increasing and concave function such that ψ() = and ψ(1) = 1. Given a probability space (Ω, F, P), we define the set function µ(a) := ψ(p(a)), for all A F. Then µ is a submodular capacity by Proposition 4.69 of [3]. In fact, µ is called distorted probability, and it has important applications in economics. Acknowledgements The authors would like to thank the anonymous referees for their careful reading and helpful suggestions, and the authors also thank SHI Xue-jun for his help. [ References ] [ 1 ] ARTZNER P, DELBAEN F, EBER J M, et al. Coherent measures of risk [J]. Math Finance, 1999, 9: 23-228. [ 2 ] FÖLLMER H, SCHIED A. Convex measures of risk and trading constraints [J]. Finance and Stochastics, 22, 6(4): 429-447. [ 3 ] FÖLLMER H, SCHIED A. Stochastic Finance, An Introduction in Discrete Time [M]. 2nd ed. Berlin: Welter de Gruyter, 24. [ 4 ] FRITTELLI M, ROSAZZA GIANIN E. Putting order in risk measures[j]. Journal of Banking and Finance, 22, 26(7): 1473-1486. [ 5 ] ROSAZZA GIANIN E. Risk measures via g-expectations [J]. Insurance: Mathematics and Economics, 26, 39: 19-34. [ 6 ] JIANG L. Convexity, translation invariance and subadditivity for g-expectations and related risk measures [J]. Annals of Applied Probability, 28, 18(1): 245-258. [ 7 ] HE K, CHEN Z, HU M. The relationship between risk measures and choquet expectations in the framework of g-expectations [J]. Statist Probab Lett, 29, 79: 58-512. [ 8 ] ZHOU Q. Cooperative hedging in the complete market under g-expectation constraint [J]. Acta Mathematicae Applicatae Sinica, 211, 27: 373-38. [ 9 ] DENNEBERG, D. Non-additive Measure and Integral [M]. Boston: Kluwer Academic Publishers, 1994. ( )