Ordering risk bounds in factor models

Abstract Conditionally comonotonic risk vectors have been proved in [4] to yield worst case dependence structures maximizing the risk of the portfolio sum in partially specified risk factor models. In this paper we investigate the question how risk bounds depend on the specification of the pairwise copulas of the risk components Xiwith the systemic risk factor. As basic toolwe introduce a new ordering based on sign changes of the derivatives of copulas. This together with discretization by n-grids and the theory of supermodular transfers allows us to derive concrete ordering criteria for the maximal risks.


Introduction
In recent years a lot of eort has been undertaken to base the evaluation of risk bounds for the joint portfolio S = d i=1 X i of a risk vector X = (X 1 , . . . , X d ) on reliable information on the marginals F i of X i and on the joint dependence structure of X . Considering law-invariant convex risk measures Ψ it is wellknown that Ψ is consistent with respect to the convex order, i.e. S 1 ≤ cx S 2 =⇒ Ψ(S 1 ) ≤ Ψ(S 2 ) (1) assuming generally that S i ∈ L 1 (P ) are integrable and dened on a non-atomic measure space (Ω, A, P ) . Thus it is sucient to determine (sharp) upper bounds w.r.t. ≤ cx in order to determine (sharp) upper risk bounds for In the case that there is only marginal information but no further dependence information on the risk vector X available, an upper bound for the joint portfolio S = d i=1 X i in convex order is given by the comonotonic sum S For many applications, the comonotonic upper bound Ψ(S c ) of the risk Ψ(S) is too wide to be useful. Therefore, in recent years various approaches have been investigated to introduce additional dependence information and structural information in order to tighten the risk bounds.
A promising approach in this direction, the partially specied risk factor models, have been introduced in Bernard et al. (2017). It is assumed in this approach that the risk vector X is described by a factor model for functions f i , where Z is a systemic risk factor and ε i are individual risk factors. It is assumed that the joint distributions H i of (X i , Z) , 1 ≤ i ≤ d , are known. The joint distributions of (ε i ) and Z however are not specied, in contrast to the usual independence assumption in factor models. This means that both the copulas C Xi,Z of (X i , Z) and the marginal distributions of X i ∼ F i and Z ∼ G are known, but the dependence structure of (X 1 , . . . , X d )|Z = z is not specied.
The common systemic risk factor Z however can be used to reduce the dependence uncertainty (DU). It has been shown in Bernard et al. (2017, Proposition 3.2) that in the partially specied risk factor model a sharp upper bound in convex order is given by the conditionally comonotonic sum, i.e. for U ∼ U (0, 1) independent of Z holds Furthermore, S c Z is an improvement of the comonotonic sum S c , i.e.
S c Z ≤ cx S c . (4) In this paper, we assume that Z is a real-valued random variable. Then, the upper bound S c Z depends only on the specied marginals F i and G and on the bivariate copulas C i = C Xi,Z ∈ C 2 , where C d denotes the set of d-copulas. The conditionally comonotonic sum S c Z thus solves the optimization problem where the max is w.r.t. convex order ≤ cx .
In the following, we investigate how the solution in (5) varies in dependence on the constraints C i . More generally, we aim to determine criteria for copula classes S i ⊂ C 2 of bivariate copulas and classes F i of univariate distribution functions such that a solution of the maximization problem exists and can be determined for all F i ∈ F i and for all continuous distribution functions G . Equivalently, maximization problem 6 can be formulated as (7) for classes S i of copulas, transformation functions f i ∈ G i = {F −1 i |F i ∈ F i } and continuous distribution functions G .
After the problem formulation and motivation we introduce in Section 2 the upper product of bivariate copulas which describes the dependence structure of conditionally comonotonic random vectors. We develop several tools for approximation of these products. In particular, we deal with the approximation by n-grid copulas. In Section 3 we reduce ordering properties of the portfolio sums in partially specied factor models by approximation to ordering properties on n-grid models. As a basic new tool, we introduce the ordering ≤ ∂∆ of sign changes of the derivatives of the copulas. In our main result, Theorem 3.10, we show that the ≤ ∂∆ -ordering is sucient for ordering upper products and thus for ordering risk bounds in factor models. For the ordering of the n-grid copulas we make essential use of the ordering results by mass transfer theory as developed in Müller (2013). The partially quite technical proofs are deferred to the appendix. In Section 4, we give an application to nancial data. We improve the standard DU interval for the Average Value-at-Risk of a portfolio of European options on dierent assets by up to 30% .
2 The upper product of bivariate copulas A d-copula is a distribution function on the d-dimensional unit cube [0, 1] d with uniform univariate margins. Due to Sklar's Theorem, the distribution of a random vector can be separated in its univariate margins and a copula which completely describes the dependence structure of the random vector. Some specic copulas that we need in the following are the copulas M d ∈ C d , Π d ∈ C d and W 2 ∈ C 2 which model comonotonicity, independence and countermonotonicity, respectively. For an introduction to copulas, we refer to Nelsen (2006).
Since the univariate margins are xed, a solution of (5) and (6) only depends on the copula of (X 1 , . . . , X d ) . Varying the solution in dependence on the constraints C i ∈ C 2 motivates to introduce upper products of bivariate copulas which are the copulas of conditionally comonotonic distributions.
In the case that C t = Π 2 for all t , where Π 2 denotes the bivariate independence copula, there is a correspondence of the C-product with Markov processes (see Darsow et al. (1992, Theorem 3.2 and Theorem 3.3). For our purposes, we are interested in a d-dimensional extension of the case that C t = M 2 for all t . An extension of (8) to the case of d-fold products as needed in the partially specied factor model is given as follows.
where ∂ 2 denotes the rst partial derivative with respect to the second argument.
Since F Ui|Z=t can be considered as a distribution function for all t , Lebesgue's dierential theorem shows that ∂ 2 A i (u i , t) = F Ui|Z=t (u i ) , and ∂ 2 A i ( · , t) can also be considered as a distribution function for almost all t . From Sklar's Theorem it follows that C t (∂ 2 C 1 ( · , t), . . . , ∂ 2 C d ( · , t)) denes a distribution function for almost all t . Thus also the mixture * C (C 1 , . . . , C d ) denes a distribution function. Since ∂ 2 C i (1, t) = 1 for all t , * C (C 1 , . . . , C d ) has uniform margins and thus is a d-copula.
Note that copulas are almost surely partially dierentiable (see Nelsen (2006, Theorem 2.2.7)) and the integral is dened as a Lebesgue-integral.
Remark 2.2 Let (X 1 , . . . , X d , Z) be a (d + 1)-dimensional random vector such that F Z is continuous. Then, from Sklar's Theorem, the transformation formula and Proposition 2.1 it follows that

Denition of the upper product and elementary properties
For application to risk bounds in partially specied risk factor models we consider the special case of the * C -product with C = M d := {M d } 0≤t≤1 leading to the notion of the upper product. Due to Proposition 2.1 the operator in the following denition is well-dened. Denition 2.3 (Upper product) The upper product of bivariate copulas The following proposition gives some elementary properties of the upper product. Point (i) explains the choice of the name upper product. Point (ii) explains that the upper product describes the case of conditionally comonotonic copulas and thus gives the connection to risk bounds in partially specied factor models (see also Remark 2.5 (a)).

Proposition 2.4 For
. . , A d , D ∈ C 2 and for a random vector (U 1 , . . . , U d ) on (Ω, A, P ) holds: (iii) In general, the upper product is neither commutative nor associative.
where the fourth equality holds with an argument as in (9). Hence, it holds V d = U . The reverse direction follows from the equations in (10).
Assume without loss of generality that A 1 = A 2 . Due to the continuity of copulas there exist (v 1 , v 2 ) ∈ (0, 1) 2 and ε > 0 such that Then, the assertion follows from (iv).
(viii): Due to (ii) assume that (U 1 , . . . , U d ) ∼ i A i and Z ∼ U (0, 1) such that (U 1 , . . . , U d )|Z = t is comonotonic for all t . Then, we obtain where the third equality holds due to the conditional comonotonicity.
Remark 2.5 (a) From Proposition 2.4 (ii) and Sklar's Theorem it follows that for Z ∼ U (0, 1) the upper product describes the dependence structure of the solution of (5), i.e.
for X i ∼ F i and C Xi,Z = C i . More generally, applying the transformation formula yields that (11) holds true for all Z with continuous distribution function G = F Z .
(b) The continuity of G is decisive for (11). Assume for example that G follows a Dirac distribution. Then, any arbitrary copula C i describes the dependence structure of (X i , Z) , Z ∼ G , and hence, knowledge of C Xi,Z is no information. Thus, the worst case distribution in (5) must be given through the comonotonic random vector X c (which coincides with X c Z in this case). But from Proposition 2.4 (v) we obtain that i C i = M d if not all C i coincide. Let λ be the Lebesgue measure on B([0, 1]) . Denote by T the set of measurable transformations T : ((0, 1), B((0, 1)), λ) → ((0, 1), B((0, 1)), λ) that are measure preserving, i.e. T * λ = λ , where T * λ(A) := λ(T −1 (A)) for all A ∈ B((0, 1)) denotes the distribution of the image of λ under T . Let T P be the set of all T ∈ T such that T is bijective and its inverse T −1 is measure preserving. Then, elements of T P are denoted shues, see Durante and Sánchez (2012).
The following statement shows that the upper product is invariant under joint shues of the factor variable.
Let µ C be the probability measure induced by C and denote by K C the corresponding Markov kernel such that µ C ( ds, dt) = K C ( ds, t) dt . Then, from the disintegration theorem it follows that ∂ 2 C(u, s) = K C ([0, u], s) almost surely. Denote by (g 1 , g 2 ) ∈ T × T the measure-preserving decomposition of C according to Kolesárová et al. (2008, Theorem 3.1) such that C g1,g2 = C . Then, for all (u, v) ∈ [0, 1] 2 holds where the second equality is true because T is λ-preserving, the third equality holds by the transformation formula, the fth equality holds due to the disintegration theorem. The sixth equality holds because C = C g1,g2 . From Kolesárová et al. (2008, Theorem 3.1) we also get that C g1,T •g2 denes a copula because T • g 2 is measure preserving. This proves the rst statement.
Since S T (C i ) ∈ C 2 for all i , the upper product S T (C i ) is well-dened.
Hence, the second statement follows from

Approximation of upper products of copulas
The ordering properties developed in this paper depend strongly on the approximation of the upper products by upper products of discrete grid copulas. In the second part of this section we derive this kind of approximations. In the rst part of this section we give some continuity results.
The upper product of copulas depends on the partial derivatives of its arguments. So, approximating the upper product also means approximating the partial derivatives. As we show in the following example uniform convergence of (D i n ) n ⊂ C 2 is not sucient for uniform convergence of ( i D i n ) n .
Example 2.7 Let (T n ) n∈N ⊂ T P be a shue-of-min approximation of Π 2 , i.e.
S Tn (M 2 ) → Π 2 pointwise (and thus from ArzelàAscoli's Theorem also uniform), see Mikusinski et al. (1992, Theorem 3.1). Since S T (Π 2 ) = Π 2 for all T ∈ T P , it follows that where the last equality follows from Proposition 2.6. Thus uniform convergence of (D i n ) n does not imply in general (uniform) convergence of the upper products.
To establish continuity properties of upper products we consider the follow-ing metrics on C 2 (see Trutschnig (2011, Lemma 4)).
Let d sup be the supremum metric on C d . Then, the following continuity result holds true.
Proposition 2.8 Let D be one of the metrics D 1 , D 2 , and D ∞ . Then, the is continuous in each place and also jointly continuous.
Proof: Since the metrics D 1 , D 2 , and D ∞ are equivalent (see Trutschnig (2011, Theorem) The assertion follows from ArzelàAscoli's Theorem with the equicontinuity of the set of copulas.
For n ∈ N and d ≥ 1 denote by uniformly to the cell. Let C n be the cumulative distribution function associated with β n , i.e.
Then, it holds that C n is a copula for all n , C n (u) = C(u) for all u ∈ G d n,0 and C n → C uniformly. The sequence (C n ) n is called the checkerboard approximation of C and C n is the n-checkerboard copula of C .
Proof: Dening ∂-convergence as in Mikusi«ski and Taylor (2010, Denition 3) it is shown in Trutschnig (2011, p. 695) that the topology of ∂-convergence is strictly ner than the topology of D 1 . Then, the statement follows from Proposition 2.8 with the ∂-convergence of the checkerboard approximations as shown in Mikusi«ski and Taylor (2010, Theorem 5).
Similar results hold also true for checkmin approximations and Bernstein approximations of copulas (see Mikusi«ski and Taylor (2010, Theorem 6 and Theorem 7)) In the following, we make essential use of discrete approximations of the upper product by so-called grid copulas.
Denition 2.10 For d ∈ N , a (signed) n-grid d-copula (shortly grid copula) D is the (signed) distribution function of a (signed) probability distribution on G d n,0 with uniform univariate margins, i.e. for all i = 1, . . . , d holds D(u) = k n , for all k = 0, . . . , n , if u i = k n and u j = 1 for all j = i . Denote by C d,n (C s d,n ) the set of all (signed) d-dimensional n-grid copulas.
An 1 n -scaled doubly stochastic matrix is dened as an n × n-matrix with nonnegative entries and row resp. column sums equal to 1 n . By an signed 1 n -scaled doubly stochastic matrix we mean an 1 n -scaled doubly stochastic matrix where also negative entries are allowed.
The following statement is immediate.
Lemma 2.11 There is a one-to-one correspondence between the set of (signed) n-grid 2-copulas and the set of (signed) 1 n -scaled doubly stochastic matrices.
Note that also bivariate n-checkerboard copulas can be represented by 1 n -scaled doubly stochastic matrices.
For a bivariate (signed) n-grid copula E ∈ C 2,n (∈ C s 2,n ) let e , dened through be its corresponding (signed) probability mass function, where ∆ i n denotes the dierence operator of length 1 n with respect to the i-th variable, i.e. ∆ i n g(u) := g(u) − g((u − 1 n e i ) ∨ 0) for u ∈ G d n,0 and e i being the unit vector with value 1 in the i-th component. Further, dene its corresponding (signed) 1 n -scaled doubly stochastic matrix (e kl ) 1≤k,l≤n by For every copula D ∈ C d denote by G n (D) its canonical n-grid copula dened Dene the upper product A version for signed grid copulas is dened analogously.
We show that the upper product of bivariate copulas can be uniformly approximated by the upper product of the corresponding grid copula approximations in the extended version given by (14).
Proposition 2.12 (Grid copula approximation of the upper product) It can be shown that D n is a copula for all n . We need to show that for all u ∈ [0, 1] d with the equicontinuity of (D n ) n∈N . The proof of the convergence in (15) is given in the appendix.
3 Ordering risk bounds for X i in partially specied factor models To solve maximization problem (6) for suitable sets S i we aim to order solutions of the maximization problem (5) w.r.t. ≤ cx for all marginal distributions F i and in dependence on the constraints C i . We rst demonstrate that the usual ordering conditions (like supermodular ordering) for the constraints C i ∈ C 2 do not imply ordering of the upper product i C i . We are, therefore, led to introduce a new type of orderings dened by the sign changes of the copula derivatives. The main result in this paper, Theorem 3.10, states that these new ordering conditions imply the desired ordering properties of the upper products.
It turns out that the supermodular ordering ≤ sm of random vectors is sucient for convex ordering of the sums independent of the marginal distributions whereas the weaker concordance ordering ≤ c may lack this property. For an overview on stochastic orderings, see Müller and Stoyan (2002, Example 3.9.7) and Shaked and Shanthikumar (2007). Hence, the aim is to nd conditions on A necessary condition for (16) is the lower orthant ordering, i.e.
Ordering the constraints with respect to the supermodular ordering is not sufcient to obtain (16) as the following example illustrates.
Example 3.1 (a) The upper product is not componentwise increasing w.r.t.
the supermodular ordering, i.e. A < sm B for A, B ∈ C 2 does not imply C∨A < sm C∨B for all C ∈ C 2 , because C = A yields C∨A = M 2 > sm C∨B using Proposition 2.4(v).
Note that also a pointwise ordering of the integrands in (17) is not possible.
This demands to obtain ordering criteria for the whole integral. The identity motivates the following lemma.
Lemma 3.2 Let f, g : [0, 1] → R be integrable functions with the properties that Then it holds that where h − resp. h + denotes the negative resp. positive part of a function h . Further, every change of the sign sequence in (ii) or in (iii) produces a change of the inequality signs in (19).
Hence, it follows that using Condition (i), and because f ≤ g on (0, s) the inequality holds true due If the sign sequence in condition (iii) is (+, −) , then the statement follows from the above one by changing the roles of f and g . The other cases follow by symmetry.
On the basis of the previous lemma, we introduce a new ordering on C 2 and show in the sequel that this ordering provides supermodular ordering criteria for the upper product of bivariate copulas. Denition 3.3 (Sign sequence ordering of derivative dierences) 4. Analogously, dene the symmetric sign sequence relation of derivative dif- For bivariate grid copulas, the relations ≤ ∂∆ and ≤ s∂∆ are dened in the same way.
The ≤ ∂∆ -relation is a relation that is strictly stronger than the ≤ sm -relation.
It can easily be veried that the reverse directions in the following result do not hold.
because the integrand has no (−, +)-sign change in t and the integral vanishes for v = 1 .

Example 3.5 (a) Elliptical copulas: Let
Assume that the radial part R has a continuous distribution function. Then the copula C Xi,Z of (X i , Z) is uniquely determined. Assume that −1 < ρ 1 < ρ 2 < 1 . Then, from Cambanis et al. (1981, Corollary 5) where the last equivalence holds because ρ 1 < ρ 2 . Hence, we obtain where F R ± = F Xi = F Z is the distribution function of R ± := RU (1) . But this means that C X1,Z ≤ ∂∆ C X2,Z .
In the following, we show that the ≤ ∂∆ -ordering of the constraints implies the ≤ sm -ordering of the upper product if we substitute the greatest or smallest element in the ≤ ∂∆ -increasing sequence of constraints, see Theorem 3.10. For the proof, we approximate the upper product by grid-copulas and use the lower orthant ordering result given in the following proposition.
The upper orthant ordering follows analogously with Proposition 2.4 (viii).
To show the ≤ sm -ordering of the upper product it suces to order the grid copula approximations w.r.t. ≤ sm as the following result states.
The grid-copula approximations dene distributions with nite support. But the supermodular ordering of distributions with nite support has been characterized by supermodular transfers in Müller (2013, Theorem 2.5.4). It is clear that this result also holds for nite signed distributions with nite support: Proposition 3.8 Let µ and ν be nite signed distributions on G d n . Then, µ ≤ sm ν if and only if there exist a nite number m ∈ N 0 , weights q i > 0 and points The signed measures η i are called supermodular transfers and are indicated by i.e. mass of size q i is transferred from x i and y i to x i ∧ y i and x i ∨ y i .
The following result states that (21) even holds w.r.t. the ≤ sm -ordering in the case of grid copulas. The technical proof is given in the appendix.
Proposition 3.9 Let A 1 , . . . , A d , B 1 , B 2 ∈ C 2,n be bivariate grid copulas such Then, there exists a nite sequence (E i ) 0≤i≤m of signed probability distribution functions on G 2 n such that It follows that Now, we can formulate the main result of this article which provides some important properties of the ≤ ∂∆ -ordering. In contrast to the ≤ sm -ordering on C 2 (see Example 3.1(b)), the ≤ ∂∆ -ordering on C 2 is sucient for the supermodular ordering of the upper product.
Theorem 3.10 Let A 1 , . . . , A d , B 1 , B 2 ∈ C 2 be bivariate copulas such that ei- Then, it holds that Proof: Assume that (i) holds. Then, we obtain G n (A j ) ≤ ∂∆ G n (B i ) and G n (B 1 ) ≤ s∂∆ G n (B 2 ) for all 1 ≤ j ≤ d , i = 1, 2 and n ∈ N . Thus, the statement follows from Proposition 3.9 (v) and Proposition 3.7.
If (ii) holds, then the statement follows from (i) and Proposition 2.6 with T (t) = 1 − t .
It can be shown analogously that (22) can be generalized to for every δ ∈ N 0 . Applying (23) repeatedly, we obtain together with Proposition 3.4 the following corollary.
Corollary 3.11 Let C 1 , . . . , C d ∈ C 2 be bivariate copulas such that C 1 ≤ ∂∆ Remark 3.12 (a) Theorem 3.10 and Corollary 3.11 indicate: The closer the elements are together w.r.t the ≤ ∂∆ -ordering the greater is their upper product w.r.t. the supermodular ordering. Note that we only modify the most extreme elements keeping the others xed.
(b) Corollary 3.11 is a generalization of Ansari and Rüschendorf (2016, Corollary 3 and Proposition 6) to general classes of copulas and to the supermodular ordering.
Coming back to the comparison of solutions of (5) w.r.t. the constraints C i we get the following result.
Corollary 3.13 Let W 1 , . . . , W d , Z be real random variables such that the sequence of copulas (C Wi,Z ) 1≤i≤d is ≤ ∂∆ -increasing. Assume that Z has a continuous distribution function. Let X i := g i (W i ) for g i increasing. Then, for Proof: This follows from Remark 2.2, Proposition 2.4 (i) and Corollary 3.11.
(b) Let (C γ ) γ∈R ⊂ C 2 be a ≤ ∂∆ -increasing family of bivariate copulas. Denote by F 1 the set of all univariate distribution functions and by F ↑ the set of all increasing functions. As a consequence of Corollary 3.13 we obtain solutions of maximization problem (6) resp. (7) for

Application
As application we consider a portfolio Σ t := 6 i=1 Y i t of calls and puts on dierent assets. More specically, let Y i t := (S i t − K i ) + be calls for i = 1, 2, 3 and Y i t := (K i − S i t ) + be puts for i = 4, 5, 6 , on assets S i t for dierent strikes K i > 0 , where (S i t ) t≥0 denotes the asset price process of Allianz (i = 1), Daimler (i = 2), Siemens (i = 3), Deutsche Bank (i = 4), SAP (i = 5) resp. Adidas (i = 6). For times to maturity T = 15 trading days resp. T = 50 trading days resp. T = 100 trading days , we aim to get improved risk bounds (w.r.t the standard comonotonic risk bound) for Σ T applying Corollary 3.13 where daily historical data are given. Denote by (S 0 t ) t≥0 the risk factor process which is the DAX in our case.
We model S t = (S 0 describes the dependence structure of (ξ i 1 , ξ 0 1 ) by subfamily of a ≤ ∂∆ -increasing family of copulas (see Example 3.5) which can be chosen arbitrarily.
For the estimation of the distribution of S i T , we distinguish between the following two specications of Assumption (I): 1. (a) Each (S i t ) t≥0 , i = 0, . . . , 7 , follows a geometric Brownian motion, i.e.
For the estimation of upper bounds for the time T -increments (ξ 1 1 , . . . , ξ 7 1 ) in supermodular ordering, we specify Assumption (III) as follows: 3. For xed ν ∈ (2, ∞] , the dependence structure of (ξ i 1 , ξ 0 1 ) is described by a family (C ρ ν ) ρ∈I i of t-copulas with unknown correlation parameter ρ ∈ I i and ν degrees of freedom for some intervals I i ⊂ [−1, 1] (which we specify later), i.e. C ξ i For the estimation of the intervals I i , we use the i.i.d. assumption in Assumption (II) to determine (one-sided) condence intervals for the correlation of (ξ i 1 , ξ 0 1 ) from historical log-return data.
Compared to the basic assumptions underlying multivariate exponential Lévy models the above assumptions are quite weak. The dependence structure among the components is not uniquely determined. For larger values of T (which we consider in this application), the set of historical data is too small to determine the unknown correlation parameter reliably. Thus, we need to solve maximization problem (6) instead of maximization problem (5).
Such solutions lead to improved risk bounds for the portfolio Σ T given the observed starting values (S 1 0 , . . . , S 7 0 ) and constraints S i . We speak about Model Gauss if S T is modeled by Assumptions (1a),(II) and (3) and about Model NIG if S T is modeled by Assumptions (1b),(II) and (3).
Note that for ν → ∞ the t-copula passes into a Gaussian copula. In contrast to Gaussian copulas, t-copulas exhibit tail-dependencies with equal coecients of lower resp. upper tail dependence where t ν denotes the standard univariate Student's t-distribution function with ν degrees of freedom (see Demarta and McNeil (2005)).

Application to real market data
As data set, we take the daily adjusted close data from yahoo nance from 23/04/2008 to 20/04/2018. It contains the values of 2540 trading days for 7 assets (with some missing data) which we denote by (s 0 k , s 1 k , . . . , s 6 k ) 1≤k≤2540 , see Denote by (x i k ) k , x i k := log s i 2540−t k−1 − log s i 2540−t k , the historical time Tlog-returns of the i-th asset (see Figure 1 in the case T = 1). Hence, for T = 15 resp. T = 50 resp. T = 100 , the sequence (x i k , x 0 k ) k consists of 169 resp. 50 resp.  i between the T -days log-returns of the i-th underlying asset and the DAX estimated from log-return data (x i k , x 0 k ) k over T days for T = 15 , T = 50 resp. T = 100 trading days; ρ T i denotes the lower bound of the 95%-condence interval forρ T i under a bivariate normality assumption.
25 pairs of data. Table 2 shows the empirical correlationρ T i of (x i k , x 0 k ) k (which estimates the correlation of (ξ i 1 , ξ i 0 )) and a lower bound ρ T i for the one-sided 95%-condence interval forρ T i . This justies a determination of I i := [ρ T i , 1] for the unspecied intervals I i in Assumption (3).
Since Y i T is an increasing resp. decreasing transformation of S i T we can choose where ρ T := min i=1,2,3 ρ T i and ρ T := max i=4,5,6 −ρ T i . Now, Corollary 3.13 and (2) yield are the quantile functions of the calls resp. puts Y i T . Further, (f i (Z, ε)) i given , 5, 6 is the conditionally on Z comonotonic random vector for random variables Z, ε ∼ U (0, 1) that are independent. Note that the distribution function of (f (ρ, ν, Z, ε), Z) is the tcopula with correlation ρ and ν degrees of freedom (see Aas et al. (2009)).
As a consequence of (1) and (26) we obtain More specically, let Ψ be the Average Value-at-Risk at level λ (also known as Expected Shortfall ) dened by It is well-known that AVaR λ is a convex, law-invariant risk measure. In Tables 3, (4) resp. (5), we compare the improved risk bound AVaR λ (Σ c T,Z,(ηi),ν ) given by (27) with the standard comonotonic bound AVaR λ (Σ c T ) in Models Gauss and NIG (7million simulated points) for dierent λ and ν and for T = 15 resp. T = 50 resp. T = 100 trading days.
We observe that both the improved and the standard portfolio risk bounds AVaR λ (Σ c T,ρ T ,ρ T ,ν ) resp. AVaR λ (Σ c T ) depend for high levels λ on the model for the univariate margins of the summands and their tails. The fatter tails of the NIG distribution yield higher risks. But for larger times T to maturity, we see that the dierences are less signicant. This can be explained by the fact that the parameters δ i = T δ i and α i = α i (see Table 1) of L i T are quite large for large T and thus F L i T is approximately normal with variance δ i /α i (see Barndor-Nielsen (1978, p.153)). In our application, Model NIG ts the data better than Model Gauss (see Figure 2). In contrast, for levels λ ≤ 0.95 the results in this application nearly coincide for Models Gauss and NIG.
Further, we observe that the improvement of the risk bounds depends on    Table 5: Comparison of the improved risk bound AVaR λ (Σ c 100,ρ 100 ,ρ 100 ,ν ) with the standard comonotonic risk bound AVaR λ (Σ c 100 ) for AVaR λ (Σ 100 ) in Model Gauss resp. NIG for T = 100 trading days for dierent levels λ , for dierent ν and for xed ρ 100 = .6703 and ρ 100 = −0.2905 . The relative DU-improvement given by 1 − the degree of freedom ν of the constraining t-copula families S i . The smaller the parameter ν the higher is the tail-dependence of the (t-)copula of see (25). This means that extreme tail events occur more often simultaneously in the components which leads to higher risks. The empirical data exhibit taildependencies, see Figure 3. Thus, a t-copula with degree of freedom ν not too large should be preferred to a Gaussian copula in this application.
We see that the improvement of the standard DU-interval [EΣ T , AVaR λ (Σ c T )] is largest for T = 15 trading days (about 20% to 30%) and smallest for T = 100 trading days (about 10% to 15%). A large improvement means a small correlation parameter for (28) which is achieved if T is small, i.e. M ρ 15 , ρ 15 = 0.0795 < M ρ 50 , ρ 50 = 0.2697 < M ρ 100 , ρ 100 = 0.5154 , because in this case the underlying data sets (x i k ) k are larger such that the lower bounds ρ T i for the 95%-condence intervals forρ T i are larger. Thus, the intervals I i could be chosen tighter for smaller T .

Appendix
Proof of the convergence in (15)    where both inequalities hold true due to Jensen's inequality.
Proof of Proposition 3.9: Denote by b ι = (b ι kl ) 1≤k,l≤n the corresponding 1 nscaled doubly stochastic matrix of B ι , ι = 1, 2 . Consider the following algorithm that constructs the sequence (E i ) 1≤i≤m adjusting in each step the 1 n -scaled doubly stochastic matrix b 1 to b 2 by a simple supermodular transfer that preserves the ≤ ∂∆ -relation with respect to each A j .
The last equality holds because the lines of (e i κι ) are for κ > k , i.e. u < v * , already adjusted to the lines of (b 2 κι ) . For u > v * we obtain from (33) that ∆ 2 n B 2 (u, t)−∆ 2 n E i+1 (u, t) = ∆ 2 n B 2 (u, t)− ∆ 2 n E i (u, t) where the latter has no (−, +)-sign change as assumed.
Fourthly, we observe from the proof of (34) that there exists a nite i ∈ N such that mass in the lines of (e i kl ) kl has been adjusted to (c kl ) kl for all k = n, . . . , 2 . Then, since both e i and b 2 are (signed) 1 n -scaled doubly stochastic also e i 1l = b 2 1l holds. Thus, it is sucient to stop the algorithm setting m := i if k = 2 and e i kl = b 2 kl for all 1 ≤ l ≤ n . This proves E m = B 2 . (i): For each i ∈ {0, . . . , m} it follows by construction that n ι=1 e i ικ = 1 for all κ and n κ=1 e i ικ = 1 for all ι . Note that elements of e i can get negative. Thus P E i denes a signed probability measure on G 2 n for all i . Since 0 = P E i+1 − P E i (x) = ±η for exactly 4 points x ∈ G 2 f (t 1 ) ≤ g(t 1 ) < 0 , but f (t 2 ) = g(t 2 ) > 0 which is a contradiction to the assumption that f has no (−, +)-sign change. If t 2 = t * , then f (t 2 ) ≥ g(t 2 ) > 0 and f (t 1 ) = g(t 1 ) < 0 , which again is a contradiction. The second case in (38) follows analogously. This completes the prove of (iii).