A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle
AA recursive algorithm for multivariate risk measures anda set-valued Bellman’s principle
Zachary Feinstein ∗ Birgit Rudloff † July 5, 2016
Abstract
A method for calculating multi-portfolio time consistent multivariate risk measuresin discrete time is presented. Market models for d assets with transaction costs orilliquidity and possible trading constraints are considered on a finite probability space.The set of capital requirements at each time and state is calculated recursively back-wards in time along the event tree. We motivate why the proposed procedure can beseen as a set-valued Bellman’s principle, that might be of independent interest withinthe growing field of set optimization. We give conditions under which the backwardscalculation of the sets reduces to solving a sequence of linear, respectively convex vectoroptimization problems. Numerical examples are given and include superhedging underilliquidity, the set-valued entropic risk measure, and the multi-portfolio time consistentversion of the relaxed worst case risk measure and of the set-valued average value atrisk. Keywords and phrases: dynamic risk measures, transaction costs, set-valued riskmeasures, set optimization, vector optimization, algorithms, dynamic programming,Bellman’s principle
Mathematics Subject Classification (2010):
Multivariate risk measures, also called set-valued risk measures, were introduced in astatic one-period setting by Meddeb, Jouini, Touzi [15] in 2004, and further studiedin [13, 10, 11, 4], to consider risk evaluations of random vectors motivated by marketmodels with transaction costs. Dynamic set-valued risk measures were presented in[7, 9, 8, 2]. The set-valued version of time-consistency, called multi-portfolio timeconsistency, was introduced in [7] and it was shown to be equivalent to a recursiveform for the risk measure. ∗ Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO63108, USA, [email protected]. † Vienna University of Economics and Business, Institute for Statistics and Mathematics, Vienna A-1020,AUT, birgit.rudloff@wu.ac.at. a r X i v : . [ q -f i n . R M ] J u l n the present paper we will show that this recursive form can be seen as a set-valued version of Bellman’s principle. On one hand, it enables us to calculate the valueof a risk measure, that is, the set of all risk compensating initial portfolio holdings,backwards in time. This is in the spirit of dynamic programming. On the other hand,one can show that the principle of optimality holds true: the truncated optimal strategycalculated at time t = 0 is still optimal for the optimization problems appearing at anylater time point t > Consider a discrete time setting T = { , , ..., T } with finite time horizon T and afinite filtered probability space (Ω , F , ( F t ) t ∈ T , P ) with the power set sigma algebra,i.e. F = 2 Ω , and F T = F . Denote by L t = L (Ω , F t , P ; R d ) the linear space of theequivalence classes of F t -measurable functions X : Ω → R d and by L := L T .We write L t, + = { X ∈ L t : X ∈ R d + P -a.s. } for the closed convex cone of R d -valued F t -measurable random vectors with non-negative components. Note that an element X ∈ L t has components X , ..., X d in L t ( R ) = L (Ω , F t , P ; R ). More generally, wedenote by L t ( D t ) those random vectors (or variables) in L t (resp. L t ( R )) that take P -a.s. values in D t .As in [16] and discussed in [24, 17], the portfolios in this paper are in ‘physicalunits’ of an asset rather than the value in a fixed num´eraire. That is, for a portfolio X ∈ L t , the values of X i (for 1 ≤ i ≤ d ) are the number of units of asset i in theportfolio at time t .Let the set of eligible portfolios M ⊆ R d , i.e. those portfolios which can be used tocompensate for the risk of a portfolio, be the same for all times t and states ω ∈ Ω. Let M be a finitely generated linear subspace of R d with M + := M ∩ R d + (cid:54) = { } . Denote M t := L t ( M ) and M t, + = L t ( M + ). Of particular interest, especially when dealing withthe market extension discussed in section 6, is the case where all assets are eligible, .e., M = R d .A conditional risk measure is a function which maps a d -dimensional random vari-able into P ( M t ; M t, + ) := { D ⊆ M t : D = D + M t, + } , a subset of the power set of M t . That is, conditional risk measures map into collections of random vectors in M t .Conditional risk measures were defined in [7] as follows.A function R t : L → P ( M t ; M t, + ) is a conditional risk measure at time t if it is1. M t -translative: ∀ m t ∈ M t : R t ( X + m t ) = R t ( X ) − m t ;2. L + -monotone: Y − X ∈ L + ⇒ R t ( Y ) ⊇ R t ( X );3. finite at zero: ∅ (cid:54) = R t (0) (cid:54) = M t .A conditional risk measure R t is called normalized if R t ( X ) = R t ( X ) + R t (0)holds for every X ∈ L t . It is local if for every X ∈ L and every A ∈ F t it holds1 A R t ( X ) = 1 A R t (1 A X ) , where 1 A is the indicator function. A conditional risk measure at time t is called conditionally convex if for all X, Y ∈ L , and for all λ ∈ L t ([0 , R t ( λX + (1 − λ ) Y ) ⊇ λR t ( X ) + (1 − λ ) R t ( Y ) , it is conditionally positive homogeneous if for all X ∈ L and for all λ ∈ L t ( R ++ ) R t ( λX ) = λR t ( X ) , and it is conditionally coherent if it is conditionally convex and conditionally posi-tive homogeneous. A conditional risk measure R t is closed if graph( R t ) := { ( X, u ) ∈ L × M t : u ∈ R t ( X ) } is closed in the product topology. It is called (conditionally)convex upper continuous if R − t ( D ) := { X ∈ L : R t ( X ) ∩ D (cid:54) = ∅} is closed for any closed (conditionally) convex set D ∈ G ( M t ; M t, − ) := { D ⊆ M t : D =cl co( D + M t, − ) } .A dynamic risk measure is a sequence of conditional risk measures ( R t ) t ∈ T . Itis said to have one of the above properties if R t has this property for every t ∈ T . The acceptance set at time t associated with a conditional risk measure R t is defined by A t = { X ∈ L : 0 ∈ R t ( X ) } . Let us now discuss the economic interpretation of a dynamic risk measure ( R t ) t ∈ T .At time t = 0, one would choose a portfolio u ∈ R ( X ), usually to be as small aspossible, that is, a weakly minimal element of the set R ( X ), to be put aside to make X acceptable at time T according to the acceptance set A . As time progresses to t = 1 and new information become available, one would update this risk compensatingportfolio to keep the overall position acceptable (now according to A ) by choosing aportfolio u ∈ − u + R ( X ), again usually a weakly minimal element of this set. Thiscould mean injecting more capital/assets or extracting them. Now in total u + u ∈ ( X ) has been put aside to compensate the risk of X . This procedure continues untiltime T . Thus, at any time t ∈ T one has put aside a portfolio that compensates forthe risk of X according to the time t risk measure R t .To ensure that updating the risk evaluation is done in a time consistent way, theconcept of time consistency for scalar risk measures was extended to the set-valuedframework in [7] and is called multi-portfolio time consistency. Definition 2.1.
A dynamic risk measure ( R t ) t ∈ T is called multi-portfolio time consis-tent if for all times t ∈ T \{ T } , all portfolios X ∈ L , and all sets Y ⊆ L the implication R t +1 ( X ) ⊆ (cid:91) Y ∈ Y R t +1 ( Y ) ⇒ R t ( X ) ⊆ (cid:91) Y ∈ Y R t ( Y ) (2.1) is satisfied. In [7, theorem 3.4], it was shown that for a normalized dynamic risk measure( R t ) t ∈ T , with R t : L → P ( M t ; M t, + ) for all times t , ( R t ) t ∈ T being multi-portfolio timeconsistent is equivalent to ( R t ) t ∈ T being recursive, that is for every time t ∈ T \{ T } R t ( X ) = (cid:91) Z ∈ R t +1 ( X ) R t,t +1 ( − Z ) =: R t,t +1 ( − R t +1 ( X )) , (2.2)where R t,t +1 : M t +1 → P ( M t ; M t, + ) denotes the stepped risk measure R t,t +1 = R t | M t +1 ,that is the restriction of R t to M t +1 .Given an arbitrary dynamic risk measure ( R t ) t ∈ T on L , one can compose the one-stepped risk measures R t,t +1 backwards in time to obtain a multi-portfolio time con-sistent risk measure ( ˜ R t ) t ∈ T as follows: For all X ∈ L define˜ R T ( X ) = R T ( X ) , (2.3) ∀ t ∈ { , , ..., T − } : ˜ R t ( X ) = (cid:91) Z ∈ ˜ R t +1 ( X ) R t,t +1 ( − Z ) . (2.4)Then, ( ˜ R t ) t ∈ T is multi-portfolio time consistent and satisfies the properties of M t -translativity and monotonicity of a dynamic risk measures, but may fail to be finiteat zero. Additionally, if ( R t ) t ∈ T is (conditionally) convex, (conditionally) coherent, or(conditionally) convex upper continuous and (conditionally) convex, then ( ˜ R t ) t ∈ T hasthe same property, see proposition 3.11 in [7] and proposition 5.1 in [9]. Example 2.2.
The set-valued average value at risk was introduced and computed in [14]in the static setting. In [7] the definition was extended to the dynamic framework asfollows. For parameters λ t ∈ L t , where < λ ti < and X ∈ L define AV @ R λt ( X ) = { diag ( λ t ) − E [ Z | F t ] − z : Z ∈ L + , X + Z − z ∈ L + , z ∈ L t } ∩ M t . ( AV @ R λt ) t ∈ T is a normalized closed conditionally coherent dynamic risk measure. Themulti-portfolio time consistent version (cid:94) AV @ R λt was discussed in [9], with the dualrepresentation deduced for the case M = R d . Since we are only considering a finiteprobability space, we can immediately conclude (as in [14]) that the dynamic averagevalue at risk is a polyhedral risk measure. A set-valued Bellman’s principle
We now want to answer the question if it is possible to use the nested formulation (2.2),or, more generally, the backward composition (2.3), (2.4) to explicitly calculate the set R t ( X ), respectively the multi-portfolio time consistent version ˜ R t ( X ), backwards intime. If this is possible it would justify calling this procedure a set-valued Bell-man’s principle , yielding a dynamic programming method for set-valued functions.This would be an interesting insight in itself within the field of set-optimization withapplications beyond the one considered here.Recall that we assumed a finite sample space Ω with the power set sigma algebra,i.e. F = 2 Ω . We define Ω t as the set of atoms in F t . For any ω t ∈ Ω t we denote thesuccessor nodes by succ( ω t ) = { ω t +1 ∈ Ω t +1 : ω t +1 ⊆ ω t } . We use the convention that for an F t -measurable random variable u , we denote by u ( ω t ) the value of u at node ω t , that is u ( ω t ) := u ( ω ) for some ω ∈ ω t . Further, wedenote by R t ( X )[ ω t ] := { u ( ω t ) : u ∈ R t ( X ) } the collection of projections of elementsof R t ( X ) onto ω t . Though R t ( X ) is a collection of random variables rather than arandom set, R t ( X )[ · ] is a random set.In order to study a possible calculation of a set of random variables ˜ R t ( X ) back-wards in time on a finite event tree, one first needs to check if one can calculate theset ˜ R t ( X ) ω t -wise at each node. That is, we wish to verify that u ∈ ˜ R t ( X ) ⇐⇒ u ( ω t ) ∈ ˜ R t ( X )[ ω t ] ∀ ω t ∈ Ω t . This consideration is not a concern when dealing with scalar risk measures, but in theset-valued case certain conditions are needed to ensure it. Since we work on a finiteprobability space, for an ω t -wise approach to (2.3), (2.4) one will need ( ˜ R t ) t ∈ T to have F t -decomposable images, i.e.,˜ R t ( X ) = 1 A ˜ R t ( X ) + 1 A c ˜ R t ( X ) ∀ X ∈ L ∀ A ∈ F t , which is satisfied if ( ˜ R t ) t ∈ T has conditionally convex images. Furthermore, for themulti-portfolio time consistent version to only depend on the possible future (successor)states, one will need ( R t ) t ∈ T to be local. Remark 3.1.
Assuming ( R t ) t ∈ T to be conditionally convex implies both, ( R t ) t ∈ T be-ing local (see proposition 2.8 in [7]), as well as ( ˜ R t ) t ∈ T being conditionally convex(proposition 3.11 in [7]) and thus having conditionally convex images.In order to implement the set optimization problem, we wish to reframe the back-ward composition (2.3), (2.4) as a vector optimization problem. For this reason werequire that ( ˜ R t ) t ∈ T has closed images as well. Remark 3.2.
It is challenging to ensure that ( ˜ R t ) t ∈ T has closed images. Let us givethree examples, where ( ˜ R t ) t ∈ T is closed (and thus has closed images): a) if the dynamicrisk measure ( R t ) t ∈ T is convex upper continuous and convex (see proposition 5.1 in [9]),or b) if the dynamic risk measure ( R t ) t ∈ T is conditionally convex upper continuous andconditionally convex, or c) if the dynamic risk measure ( R t ) t ∈ T is polyhedral (thatis if graph( R t ) is a convex polyhedron, i.e. the intersection of finitely many closedhalf-spaces). f ( ˜ R t ) t ∈ T defined in (2.3), (2.4) is conditionally convex, but does not already haveclosed images, one needs to consider its closed-valued version, i.e.¯ R T ( X ) := cl( R T ( X )) (3.1) ∀ t ∈ { , , ..., T − } : ¯ R t ( X ) := cl (cid:91) Z ∈ ¯ R t +1 ( X ) R t ( − Z ) (3.2)for all portfolios X ∈ L . We will show that ( ˜ R t ) t ∈ T can be approximated for arbitrarilysmall δ > R t ) t ∈ T that admits an ω t -wise representation. The approximation isunderstood in the following sense. ( ¯ R t ) t ∈ T is called an approximation of ( ˜ R t ) t ∈ T if¯ R t ( X ) + δm ⊆ ˜ R t ( X ) ⊆ ¯ R t ( X )for any time t , for every portfolio X ∈ L , for any approximation tolerance δ > m ∈ int( M + ) in the subspace topology, i.e. with the topology τ M := { A ∩ M : A ∈ τ } where τ is the topology on R d . Thus, if ( ˜ R t ) t ∈ T does not have closed images, we willuse construction (3.1), (3.2) to calculate an approximation ( ¯ R t ) t ∈ T of ( ˜ R t ) t ∈ T . Theorem 3.3.
Let ( R t ) t ∈ T be a conditionally convex dynamic risk measure. Let ( ˜ R t ) t ∈ T denote its multi-portfolio time consistent version as defined in (2.3) , (2.4) .Then, we can calculate an approximation ( ¯ R t ) t ∈ T of ( ˜ R t ) t ∈ T in an ω t -wise manner by ¯ R T ( X )[ ω T ] = cl( R T ( X )[ ω T ]) , ∀ ω T ∈ Ω T ; (3.3)¯ R t ( X )[ ω t ] = cl (cid:91) (cid:8) R t,t +1 ( − Z )[ ω t ] : Z ( ω t +1 ) ∈ ¯ R t +1 ( X )[ ω t +1 ] ∀ ω t +1 ∈ succ( ω t ) (cid:9) (3.4) for every state ω t ∈ Ω t and all times t ∈ T \{ T } .Proof. Define ( ¯ R t ) t ∈ T by equations (3.1), (3.2). Clearly, ( ¯ R t ) t ∈ T has closed images. Onecan show that ( ¯ R t ) t ∈ T is conditionally convex by backward induction. The assertionis clearly true for ¯ R T . Now assume ¯ R t +1 is conditionally convex and let X, Y ∈ L and λ ∈ L t ([0 , λ ¯ R t ( X ) + (1 − λ ) ¯ R t ( Y ) = λ cl (cid:91) Z X ∈ ¯ R t +1 ( X ) R t ( − Z X ) + (1 − λ ) cl (cid:91) Z Y ∈ ¯ R t +1 ( Y ) R t ( − Z Y ) ⊆ cl cl (cid:91) Z X ∈ ¯ R t +1 ( X ) λR t ( − Z X ) + cl (cid:91) Z Y ∈ ¯ R t +1 ( Y ) (1 − λ ) R t ( − Z Y ) = cl (cid:91) Z X ∈ ¯ R t +1 ( X ) Z Y ∈ ¯ R t +1 ( Y ) [ λR t ( − Z X ) + (1 − λ ) R t ( − Z Y )] ⊆ cl (cid:91) Z X ∈ ¯ R t +1 ( X ) Z Y ∈ ¯ R t +1 ( Y ) R t ( − ( λZ X + (1 − λ ) Z Y ))= cl (cid:91) Z ∈ λ ¯ R t +1 ( X )+(1 − λ ) ¯ R t +1 ( Y ) R t ( − Z ) ⊆ cl (cid:91) Z ∈ ¯ R t +1 ( λX +(1 − λ ) Y ) R t ( − Z ) = ¯ R t ( λX + (1 − λ ) Y ) . ince the probability space is assumed to be finite, Ω t is by definition the finest partitionof Ω in F t and 1 ω t u ∈ ω t ¯ R t ( X ) if and only if u ( ω t ) ∈ ¯ R t ( X )[ ω t ]. Since ( ¯ R t ) t ∈ T hasconditionally convex images and the probability space is finite, it follows that u ∈ ¯ R t ( X )if and only if u ( ω t ) ∈ ¯ R t ( X )[ ω t ] for every ω t ∈ Ω t . Thus, one can calculate ¯ R t ( X ) ω t -wise. Therefore the terminal condition (3.3) holds trivially. Let t ∈ T \{ T } and ω t ∈ Ω t ,using (3.2) and the local property for R t (see [7, proposition 2.8]) it follows that1 ω t ¯ R t ( X ) = 1 ω t cl (cid:91) Z ∈ ¯ R t +1 ( X ) R t ( − Z )= cl (cid:91) Z ∈ ¯ R t +1 ( X ) ω t R t (1 ω t ( − Z )) = cl (cid:91) Z ∈ ωt ¯ R t +1 ( X ) ω t R t ( − Z ) . Note that Z ∈ ω t ¯ R t +1 ( X ) if and only if Z ( ω t +1 ) ∈ ¯ R t +1 ( X )[ ω t +1 ] for every ω t +1 ∈ succ( ω t ) and Z ( ω t +1 ) = 0 otherwise. But as we only need to consider 1 ω t Z by R t local,the only constraints on Z are imposed by ω t +1 ∈ succ( ω t ). Thus, (3.4) follows.Finally we will show that ¯ R t is an approximation of ˜ R t . By definition, ¯ R T ( X ) :=cl( ˜ R T ( X )) for all X ∈ L , which implies that ¯ R T ( X ) + δ T m ⊆ ˜ R T ( X ) ⊆ ¯ R T ( X ) forany portfolio X ∈ L , for any δ T >
0, and any m ∈ int( M + ). For the proof by induction,assume that ¯ R t +1 is an approximation of ˜ R t +1 . Then for t ≤ T − R t ( X ) = (cid:91) Z ∈ ˜ R t +1 ( X ) R t ( − Z ) ⊆ cl (cid:91) ¯ Z ∈ ¯ R t +1 ( X ) R t ( − ¯ Z ) = ¯ R t ( X ) ⊆ cl (cid:91) Z ∈ ˜ R t +1 ( X ) R t ( − Z + δ t +1 m ) = cl (cid:91) Z ∈ ˜ R t +1 ( X ) R t ( − Z ) − δ t +1 m ⊆ (cid:91) Z ∈ ˜ R t +1 ( X ) R t ( − Z ) − ( (cid:15) + δ t +1 ) m = ˜ R t ( X ) − ( (cid:15) + δ t +1 ) m for any (cid:15), δ t +1 >
0, and any m ∈ int( M + ). The first inclusion on the second linefollows from the induction hypothesis. The first inclusion on the third line followssince cl( ˜ R t ( X )) ⊆ ˜ R t ( X ) − (cid:15)m for any (cid:15) > X ∈ L . Denote δ t := (cid:15) + δ t +1 > t and any δ t > R t ( X ) + δ t m ⊆ ˜ R t ( X ) ⊆ ¯ R t ( X ).In the recursion (3.4), the calculation is dependent on R t,t +1 ( − Z )[ ω t ] which is bylocality of R t equal to R t,t +1 ( − ω t Z )[ ω t ], i.e. R t,t +1 ( − Z )[ ω t ] only depends on the partof Z that can be attained from state ω t . For a local risk measure, we can thereforedefine R t,t +1 ( · )[ ω t ] on M t +1 [ ω t ] := L ( ∪ succ( ω t ) , succ( ω t ) , P ( ·| ω t ); M ) (the equivalenceclass of F t +1 -measurable random variables in the eligible space M starting from node ω t ) by R t,t +1 ( Z )[ ω t ] := R t,t +1 ( ˆ Z )[ ω t ] for Z ∈ M t +1 [ ω t ] and some ˆ Z ∈ M t +1 with Z ( ω t +1 ) = ˆ Z ( ω t +1 ) for every ω t +1 ∈ succ( ω t ). Remark 3.4.
If ( ˜ R t ) t ∈ T defined in (2.3), (2.4) does have closed images already, ( ˜ R t ) t ∈ T coincides with ( ¯ R t ) t ∈ T and thus can be calculated in an ω t -wise manner by (3.3), (3.4).Observe that (3.2) is a set-valued optimization problem in the complete lattice G ( M t ; M t, + ) := { D ⊆ M t : D = cl co( D + M t, + ) } , and (3.4) is a set-valued optimizationproblem in the complete lattice G ( M ; M + ). As such, while the initial proposed problem(3.2) requires a lattice in sets of random vectors, the ω t -wise representation allows us o consider the lattice in sets of real-valued vectors instead. That is, the objectivefunction R t,t +1 at node ω t is a set-valued function that is minimized over the constraintset ¯ Z t := { Z ∈ M t +1 [ ω t ] : Z ( ω t +1 ) ∈ ¯ R t +1 ( X )[ ω t +1 ] ∀ ω t +1 ∈ succ( ω t ) } :¯ R t ( X )[ ω t ] = cl (cid:91) Z ∈ ¯ Z t R t,t +1 ( − Z )[ ω t ] = inf Z ∈ ¯ Z t R t,t +1 ( − Z )[ ω t ] . (3.5)Recall from [19] that the (lattice) infimum over a function f : M t +1 [ ω t ] → G ( M ; M + ) isgiven by inf X ∈X f ( X ) = cl co (cid:83) X ∈X f ( X ) for X ⊆ M t +1 [ ω t ], where the convex hull infront of the union can be dropped here as ¯ R t ( X )[ ω t ] is convex by theorem 3.3. Usingthe idea of [25], one can transform the set-valued problem (3.5) into a linear vectoroptimization problem that can be solved e.g. by Benson’s algorithm if the risk measure( R t ) t ∈ T is polyhedral, i.e., the graph of R t is a convex polyhedron. More generally, if( R t ) t ∈ T is the upper image of a convex vector optimization problem, one can transformthe set-valued problem (3.5) into a convex vector optimization problem that can beapproximately solved by the algorithms discussed in [21]. In both cases, one uses thatthe value of the set-optimization problem (3.5) can be written as the value of a vectoroptimization problem ¯ R t ( X )[ ω t ] = inf ( Z,Y ) ∈Z t Φ t ( Z, Y ) (3.6)for the linear vector-valued function Φ t ( Z, Y ) = { Y } , feasible set Z t = { ( Z, Y ) ∈ M t +1 [ ω t ] × M : Y ∈ R t,t +1 ( − Z )[ ω t ] , Z ( ω t +1 ) ∈ ¯ R t +1 ( X )[ ω t +1 ] ∀ ω t +1 ∈ succ( ω t ) } and ordering cone M + . Let Φ t ( Z t ) = { Φ t ( Z ) : Z ∈ Z t } denote the image of the feasibleset. The set cl(Φ t ( Z t ) + M + ) is called the upper image of the vector optimizationproblem (3.6). We now discuss the constraints by looking at two cases: the polyhedralcase and the convex case. Recall that a risk measure R t is polyhedral if its graph is a convex polyhedron, i.e.the intersection of finitely many closed half-spaces. It is also equivalent to R t havinga polyhedral acceptance set. For a polyhedral risk measure ( R t ) t ∈ T , problem (3.6) is alinear vector optimization problem. Proposition 4.1.
If the dynamic risk measure ( R t ) t ∈ T is conditionally convex andpolyhedral, its multi-portfolio time consistent version ( ˜ R t ) t ∈ T , defined in (2.3) , (2.4) ,can be calculated ω t -wise, where in each node ω t ∈ Ω t , t ∈ T \{ T } , the linear vectoroptimization problem (3.6) has to be solved.Proof. The ω t -wise representation follows from theorem 3.3. Now, let us show thatproblem (3.6) is a linear vector optimization problem. By ( R t ) t ∈ T polyhedral and since t,t +1 [ ω t ] maps into G ( M ; M + ), R t,t +1 ( − Z )[ ω t ] is the upper image of a linear vectoroptimization problem (see remark 5.1 in [12]), thus R t,t +1 ( − Z )[ ω t ] = { P t ( x ) + M + : B t x ≥ b t } (4.1)for a vector x = ( Z, z ) T that might include some auxiliary variable z , and for ma-trices P t and B t and vectors b t of appropriate dimensions. Then, the constraints Y ∈ R t,t +1 ( − Z )[ ω t ] can be equivalently written as( − Z, Y ) ∈ graph R t,t +1 [ ω t ] ⇐⇒ ˆ M T ( Y − P t ( x )) ≥ , B t x ≥ b t , where the matrix ˆ M contains the generating vectors of the positive dual M ++ of theordering cone M + . Thus, these constraints are linear. To obtain linearity of the otherconstraints, note that ( R t ) t ∈ T polyhedral implies ( R t ) t ∈ T closed (by definition) and( ˜ R t ) t ∈ T closed. To see the last implication observe that R t is polyhedral if and onlyif A t is a polyhedron. The acceptance set of ˜ R t is given by ˜ A t = A t,t +1 + ˜ A t +1 , see[7, corollary 3.14]. A t,t +1 is a polyhedron since R t,t +1 is polyhedral, by backwardsrecursion we assume ˜ A t +1 is a polyhedron, and the sum of polyhedra is a polyhedron.Therefore ˜ A t is a polyhedron, which is equivalent to ˜ R t being polyhedral (and thusclosed as well). Thus, in the polyhedral case, ( ¯ R t ) t ∈ T coincides with ( ˜ R t ) t ∈ T . Thelinearity of the constraints Z ∈ ˜ R t +1 ( X ) follow by induction. The constraints from theterminal condition Z ( ω T ) ∈ ˜ R T ( X )[ ω T ] = R T ( X )[ ω T ] are linear for all ω T ∈ Ω T by R T polyhedral. Thus, let us assume the constraints Z ( ω t +1 ) ∈ ˜ R t +1 ( X )[ ω t +1 ] are linearfor all ω t +1 ∈ succ( ω t ) for a given node ω t ∈ Ω t , then we need to show that ˜ R t ( X )[ ω t ]is polyhedral. Since Ω is assumed to be finite, problem (3.6) is clearly a linear vectoroptimization problem. By (3.5), (3.6) and R t,t +1 ( − Z )[ ω t ] mapping into G ( M ; M + ), wehave ˜ R t ( X )[ ω t ] = Φ t ( Z t ) + M + , which is for finite Ω closed and polyhedral. Thus,˜ R t ( X )[ ω t ] is the upper image of the linear vector optimization problem (3.6).Thus, if one assumes ( R t ) t ∈ T to be a conditionally convex and polyhedral riskmeasure, ˜ R t ( X )[ ω t ] is for every t and ω t the upper image of a linear vector optimizationproblem, which is polyhedral and can be calculated by Benson’s algorithm (see [12]).Then, the set ˜ R t ( X ) can be calculated backwards in time, by solving at each node alinear vector optimization problem. Note that at time t , the problems for each ω t ∈ Ω t can be calculated in parallel instead of sequentially which reduces computational time.Several examples will be discussed in section 7.Benson’s algorithm is an appropriate tool to solve the vector optimization problem(3.6) as it takes advantage of the fact that the dimension dim( M ) of the image space isusually significantly smaller than the dimension d × | succ( ω t ) | + d + | z | of the pre-imagespace, where | succ( ω t ) | denotes the number of successor nodes of ω t and | z | denotesthe dimension of the auxiliary variables in (4.1).In practice (especially if M is higher dimensional), when the number of verticesof the set ˜ R t ( X )[ ω t ] is very high, one would calculate an approximation of ˜ R t ( X )[ ω t ]having fewer vertices, see remark 4.10 in [12]. Then, for the backward recursion, onewould need to know how the approximation errors accumulate over time. This will bediscussed in propositions 4.4 and 4.7 below in a more general framework. Example 4.2.
The relaxed worst case risk measure was introduced in the static frame-work in example 5.2 of [12]. The idea behind the relaxed worst case risk measure is to odify the worst case risk measure so that portfolios with “small” negative componentscan still be acceptable.In this paper we consider the dynamic extension of such a risk measure. By defi-nition it is a polyhedral and conditionally convex, but not conditionally coherent, riskmeasure; the acceptance set at time t is given by A RW Ct = ( − ε + L + ) ∩ L ( G ) for some level ε ∈ R d + and some finitely generated convex cone G ⊇ R d + and G (cid:54) = R d .Note that if G = R d + or ε = 0 then the relaxed worst case risk measure is equivalent tothe worst case risk measure.Let ( R t ) t ∈ T be the relaxed worst case risk measure with polyhedral acceptance set A RW Ct . Then, by proposition 4.1 one can calculate its multi-portfolio time consistentversion ( ˜ R t ) t ∈ T ω t -wise, where in each node ω t ∈ Ω t , t ∈ T \{ T } , the linear vectoroptimization problem (3.6) has to be solved. It should be noted that in general R t (cid:54) = ˜ R t ,i.e. the relaxed worst case risk measure is not multi-portfolio time consistent. As the upper image of a convex vector optimization problem can only be calculated bya polyhedral approximation yielding an inner as well as an outer approximation withrespect to some error level (cid:15) (see e.g. [6, 21]), we introduce approximations of sets,respectively functions. Throughout, we fix a parameter m ∈ int M + . Definition 4.3.
Given a set S ∈ P ( M ; M + ) and an error level (cid:15) > , we call a set S (cid:15) ∈ P ( M ; M + ) an approximation of S , if S (cid:15) + (cid:15)m ⊆ S ⊆ S (cid:15) . Given a set-valued function F : L → P ( M t ; M t, + ) and an error level (cid:15) > , we call thefunction F (cid:15) : L → P ( M t ; M t, + ) an approximation of F if F (cid:15) ( X ) + (cid:15)m ⊆ F ( X ) ⊆ F (cid:15) ( X ) for every X ∈ L. In the convex case one can in general only approximately calculate the constraintset ¯ R t +1 ( X ) in the backward recursion (3.4), respectively (3.5). Let us study therobustness of the set-optimization problem (3.5) to this perturbation of the constraints. Proposition 4.4.
Let (cid:15) > . Let ¯ R (cid:15)t +1 ( X )[ ω t +1 ] be an (cid:15) -approximation of ¯ R t +1 ( X )[ ω t +1 ] for each ω t +1 ∈ succ( ω t ) , then ¯ R (cid:15)t ( X )[ ω t ] defined by ¯ R (cid:15)t ( X )[ ω t ] := cl (cid:91) Z ∈ ¯ Z (cid:15)t R t,t +1 ( − Z )[ ω t ] , (4.2) with ¯ Z (cid:15)t := { Z ∈ M t +1 [ ω t ] : Z ( ω t +1 ) ∈ ¯ R (cid:15)t +1 ( X )[ ω t +1 ] ∀ ω t +1 ∈ succ( ω t ) } , is an (cid:15) -approximation of ¯ R t ( X )[ ω t ] defined in (3.4) . roof. The assumption implies ¯ Z t ⊆ ¯ Z (cid:15)t ⊆ ¯ Z t − (cid:15)m . This together with (3.5) andtransitivity of R t yields¯ R t ( X )[ ω t ] = cl (cid:91) Z ∈ ¯ Z t R t,t +1 ( − Z )[ ω t ] ⊆ cl (cid:91) Z ∈ ¯ Z (cid:15)t R t,t +1 ( − Z )[ ω t ] = ¯ R (cid:15)t ( X )[ ω t ] ⊆ cl (cid:91) Z + (cid:15)m ∈ ¯ Z t R t,t +1 ( − Z )[ ω t ] = cl (cid:91) Z ∈ ¯ Z t R t,t +1 ( − Z )[ ω t ] − (cid:15)m = ¯ R t ( X )[ ω t ] − (cid:15)m. Thus, ¯ R (cid:15)t ( X )[ ω t ] + (cid:15)m ⊆ ¯ R t ( X )[ ω t ] ⊆ ¯ R (cid:15)t ( X )[ ω t ], i.e. ¯ R (cid:15)t ( X )[ ω t ] is a (cid:15) -approximationof ¯ R t ( X )[ ω t ].Next, we discuss under which conditions problem (4.2) is a convex vector optimiza-tion problem and which additional assumptions are necessary to apply the algorithmproposed in [21] to calculate a polyhedral approximation of the upper image of thisconvex vector optimization problem. Assumption 4.5. a) Let the objective function in (4.2) be of the form R t,t +1 ( Z )[ ω t ] = { Ψ t ( Z, z )+ M + : g t ( Z, z ) ≤ } for an M + -convex vector function Ψ t , a component-wise convex vector function g t and a vector z , all of appropriate and finite dimen-sions.b) Let the function Ψ t in a) be continuous, and let the feasible set X := { ( Z, z ) : g t ( Z, z ) ≤ } satisfy int X (cid:54) = ∅ .Assumption 4.5 a) means that the closure of R t,t +1 ( Z )[ ω t ] is itself the upper imageof a convex vector optimization problem. Proposition 4.6.
Let the objective function R t,t +1 ( Z )[ ω t ] in (4.2) satisfy assump-tion 4.5 a) and let ¯ R (cid:15)t +1 ( X )[ ω t +1 ] be a polyhedron for each ω t +1 ∈ succ( ω t ) . Then,problem (4.2) is a convex vector optimization problem.Proof. Similar to (3.6), the set-valued problem (4.2) can be written as a vector opti-mization problem inf ( Z,Y ) ∈Z (cid:15)t Φ t ( Z, Y ) , by setting Φ t ( Z, Y ) = { Y } (which is a linear vector function), defining the feasible set as Z (cid:15)t = { ( Z, Y ) ∈ M t +1 [ ω t ] × M : Y ∈ R t,t +1 ( − Z )[ ω t ] , Z ( ω t +1 ) ∈ ¯ R (cid:15)t +1 ( X )[ ω t +1 ] ∀ ω t +1 ∈ succ( ω t ) } and using M + as the ordering cone. The constraints Z ( ω t +1 ) ∈ ¯ R (cid:15)t +1 ( X )[ ω t +1 ]are by assumption linear. Under assumption 4.5 a), the constraints Y ∈ R t,t +1 ( − Z )[ ω t ]in (3.6) can be equivalently written as( Y, − Z ) ∈ graph R t,t +1 [ ω t ] ⇐⇒ ˆ M T (Ψ t ( − Z, z ) − Y ) ≤ , g t ( − Z, z ) ≤ , where the matrix ˆ M contains the generating vectors of M ++ . ˆ M T (Ψ t ( − Z, z ) − Y ) is acomponent-wise convex vector function since Ψ t is a M + -convex vector function. Thus,these are convex constraints and (4.2) is a convex vector optimization problem.The additional assumptions 4.5 b) are necessary to ensure that problem (4.2) canbe (approximately) solved by the algorithms presented in [21]. In detail, under as-sumptions 4.5 and if the feasible set X := { ( Z, z ) : g t ( Z, z ) ≤ } is compact, [21, heorems 4.9 and 4.14] state that the algorithms in [21] provide an approximation ofthe upper image of (4.2), i.e. a polyhedral approximation of ¯ R t ( X ), if they termi-nate. However, the compactness assumption is typically not satisfied in the setting ofrisk measures. In that case [21, remark 3 in section 4.3] shows that the algorithmspresented in [21] still return an approximation of the upper image of (4.2) as long asall the scalar optimization problems within the algorithm can be solved and the algo-rithm terminates. In the example of the set-valued entropic risk measure consideredin section 4.9, this will indeed be the case.Since in general problem (4.2) can only be solved approximately (e.g. by the algo-rithms in [21]), one also need to study how the approximation errors made at differenttime points accumulate over time. Proposition 4.7.
Let (cid:15), γ > . If ¯ R (cid:15),γt ( X )[ ω t ] is a γ -approximation of ¯ R (cid:15)t ( X )[ ω t ] defined in (4.2) , then ¯ R (cid:15),γt ( X )[ ω t ] is an ( (cid:15) + γ ) -approximation of ¯ R t ( X )[ ω t ] defined in (3.4) .Proof. ¯ R (cid:15),γt ( X )[ ω t ] being a γ -approximation of ¯ R (cid:15)t ( X )[ ω t ] means¯ R (cid:15),γt ( X )[ ω t ] + γm ⊆ ¯ R (cid:15)t ( X )[ ω t ] ⊆ ¯ R (cid:15),γt ( X )[ ω t ] . Proposition 4.4 shows that ¯ R (cid:15)t ( X )[ ω t ] is an (cid:15) -approximation of ¯ R t ( X )[ ω t ], i.e.¯ R (cid:15)t ( X )[ ω t ] + (cid:15)m ⊆ ¯ R t ( X )[ ω t ] ⊆ ¯ R (cid:15)t ( X )[ ω t ] . Both chains of inclusions yield¯ R (cid:15),γt ( X )[ ω t ] + ( (cid:15) + γ ) m ⊆ ¯ R t ( X )[ ω t ] ⊆ ¯ R (cid:15),γt ( X )[ ω t ] . We are now ready to prove the main result of this section. Recall that the aim wasto (approximately) calculate the multi-portfolio time consistent risk measure ( ˜ R t ) t ∈ T backwards in time in the spirit of a set-valued Bellman’s principle. We will see that( ˜ R t ) t ∈ T can be obtained by solving at each node backwards in time a convex vectoroptimization problem. In practice, these problems can only be approximately solved.But we are able to determine the overall approximation error, when the approximationerror at each node is chosen to be (cid:15) >
0. One could of course also vary this error levelat different nodes or different time points and obtain corresponding results.
Proposition 4.8.
Let ( R t ) t ∈ T be a conditionally convex dynamic risk measure sat-isfying assumption 4.5. Let (cid:15) > . Then for any time t and given X ∈ L , we canfind a [( T − t + 1) (cid:15) + δ ] -approximation of the multi-portfolio time consistent version ( ˜ R t ( X )) t ∈ T defined in (2.3) , (2.4) , by calculating backwards in time at each node ω t ∈ Ω t an (cid:15) -approximation of the upper image of the convex vector optimization problem (4.2) .Here δ > can be chosen arbitrarily small.Proof. Assumption 4.5 and the local property of ( R t ) t ∈ T imply that all the assumptionsof theorem 3.3 are satisfied, thus a δ -approximation ( ¯ R t ) t ∈ T of ( ˜ R t ) t ∈ T can be calculated ω t -wise for arbitrarily small δ > t = T one obtains an (cid:15) -approximation ¯ R (cid:15)T ( X ) of ¯ R T ( X ) = cl( ˜ R T ( X )) =cl( R T ( X )) by calculating an (cid:15) -approximation of the upper image of the convex vector ptimization problem R T ( X )[ ω T ] = { Ψ T ( X, z ) + M + : g T ( X, z ) ≤ } (see assump-tion 4.5 a)) at each node ω T ∈ Ω T . ¯ R (cid:15)T ( X ) is by construction polyhedral (see thealgorithms in [21]) and is the input for problem (4.2) at time t = T −
1, which isby proposition 4.6 then a convex vector optimization problem. Its solution would byproposition 4.4 yield an (cid:15) -approximation ¯ R (cid:15)T − ( X ) of ¯ R T − ( X ), but one can in generalonly calculate an (cid:15) -solution. This (cid:15) -solution yields an (cid:15) -approximation of ¯ R (cid:15)T − ( X ),which is by proposition 4.7 a 2 (cid:15) -approximation of ¯ R T − ( X ).Going backwards like this yields for any t a ( T − t + 1) (cid:15) -approximation of ¯ R t ( X ),which is by theorem 3.3 and the logic of adding up approximation errors as in proposi-tion 4.7 a ( T − t + 1) (cid:15) + δ -approximation of the multi-portfolio time consistent version˜ R t ( X ) for arbitrarily small δ > Example 4.9.
The set-valued entropic risk measure was studied in [1] in a singleperiod static framework. The dynamic version was discussed in [9]. As in the scalarcase, the entropic risk measure is intimately related to the exponential utility function.Consider risk aversion parameters λ t ∈ L t ( R d ++ ) , C t ∈ G ( L t ; L t, + ) with ∈ C t and C t ∩ L t ( R d −− ) = ∅ . The dynamic entropic risk measure is defined by R entt ( X ; λ t , C t ) := { u ∈ M t : E [ u t ( X + u ) | F t ] ∈ C t } (4.3) for every X ∈ L where u t ( x ) = ( u t, ( x ) , ..., u t,d ( x d )) T for any x ∈ R d and u t,i ( z ) = − e − λtiz λ ti for z ∈ R and i = 1 , ..., d .An approximate calculation of the static entropic risk measure was shown in [21] viasolving a convex vector optimization problem. With the method presented in section 3we are able to compute an approximation ( ¯ R entt ) t ∈ T of the multi-portfolio time consistentversion ( ˜ R entt ) t ∈ T by backward composition for a general space of eligible portfolios M t ,(stochastic) risk aversion parameters λ t , and polyhedral parameters C t . It was provenin [9] that the entropic risk measure is c.u.c. and multi-portfolio time consistent in thecase that M = R d , constant λ ∈ R d ++ , and C t = L t, + , i.e. R entt = ˜ R entt = ¯ R entt .From the definition of the entropic risk measure with C t polyhedral, it is clear that R entt satisfies assumption 4.5. Thus, by proposition 4.8 one can calculate an approxi-mation of the multi-portfolio time consistent version ( ˜ R entt ( X )) t ∈ T by calculating back-wards in time at each node ω t ∈ Ω t an approximation of the upper image of the convexvector optimization problem (4.2) using the algorithms presented in [21]. The risk measure ( ˜ R t ) t ∈ T , while constructed backwards in time, has a nice financialinterpretation involving portfolio injections made as time progresses, that is an inter-pretation forwards in time: For every choice of a risk compensating portfolio holding Z ∈ ˜ R ( X ) at time t = 0, there exists, by equation (2.4), a sequence of portfolioholdings ( Z t ) t ∈ T \{ } such that Z t ∈ ˜ R t ( X ) (5.1)and Z t − ∈ R t − ( − Z t ) . (5.2) nclusion (5.1) means Z , Z , ..., Z T are the risk compensating portfolio holdings attimes 0 , , ..., T . An intuitive interpretation of (5.2) can be obtained by the followingreformulation. Defining the portfolio injections (respectively withdrawals - if negative)( u t ) t ∈ T that are needed to update the risk compensating portfolio holdings by u t = Z t − Z t − (with u = Z ), the two conditions on ( Z t ) t ∈ T can be rewritten in terms of( u t ) t ∈ T as follows u t ∈ ˜ R t (cid:32) X + t − (cid:88) s =0 u s (cid:33) for every time t , and 0 ∈ R t,t +1 ( − u t +1 ) , (5.3)for t ∈ T \{ T } . Inclusion (5.3) means the risk of the portfolio injection needed at time t + 1 is acceptable at time t with respect to the one-period risk measure R t,t +1 . Thisgives the main interpretation of the backward composition of ( R t ) t ∈ T . At each one-period step the original measure ( R t ) t ∈ T is used, but it is used in a time consistent wayin the sense of Bellman.One can observe Bellman’s principle of optimality: The at t truncated optimalsolution ( Z s ) Ts = t obtained at time 0 from (2.4) and a given Z ∈ ˜ R ( X ) is still optimalat any later time point t ∈ T . To see that, note that for the risk compensatingportfolio holding Z t ∈ ˜ R t ( X ), ( Z s ) Ts = t satisfies the conditions Z s ∈ ˜ R s ( X ) and Z s − ∈ R s − ( − Z s ), s ∈ { t, ..., T } from (2.4).Let us now explain on how to compute ( ˜ R t ( X )) t ∈ T and how to obtain for a given Z ∈ ˜ R ( X ) at time t = 0 a sequence ( Z t ) t ∈ T \{ } of risk compensating portfolio hold-ings on the realizing path. ( ˜ R t ) t ∈ T can be calculated with the approach discussed insections 4.1 and 4.2. Benson’s algorithm also calculates a solution of the linear vectoroptimization problems in the sense of definition 2.20 in [19], respectively, an (cid:15) -solutionin the sense of definition 3.3 in [21] for a convex vector optimization problem. Thesefinite solution sets are then used to calculate the sequence of risk compensating port-folio holdings ( Z t ) t ∈ T , respectively the injection/withdrawal strategy ( u t ) t ∈ T , forwardsin time on the realizing path by solving an additional linear program, specified in thefollowing, at each point in time. Let ¯ X t [ ω t ] = { ( Z it +1 [ ω t ] , Y it [ ω t ]) : i = 1 , ..., n, n ∈ N } ⊆ M t +1 [ ω t ] × M be the ( (cid:15) -)solution set to the vector optimization problem (3.6).Let us first explain the method in case of a linear vector optimization problem: Forany Z in the risk measure ˜ R ( X ), there exists a convex combination of elements of thesolution on the efficient frontier (the collection of nondominated vectors with ordering M + ) such that Z ≥ (cid:80) ni =1 λ ∗ i Y i . This coefficient vector λ ∗ ∈ R n + can be found bysolving any linear optimization problem of the formmin λ ∈ R n + c T (cid:0) Y , · · · , Y n (cid:1) λ subject to (cid:0) Y , · · · , Y n (cid:1) λ ≤ Z , (cid:126) T λ = 1 (5.4)with c ∈ R d + \{ } . The coefficient vector λ ∗ ∈ R n + can then be used to define Z ∗ := (cid:80) ni =1 λ ∗ i Y i on the efficient frontier of ˜ R ( X ). Notice that Z ∗ = Z if Z is already onthe efficient frontier. Additionally, the next time step full capital requirement is givenby Z := (cid:80) ni =1 λ ∗ i Z i , which might not be on the efficient frontier of ˜ R ( X ). This processis repeated through the event tree forwards in time. The choice of cost vector c (oralternatively a nonlinear cost function) determines the possible liquidation/withdrawalstrategy akin to that discussed for the superhedging risk measure in [20]. n the case of a convex vector optimization problem, one can calculate only a poly-hedral approximation (e.g.with error level ˜ (cid:15) = ( T + 1) (cid:15) + δ as in proposition 4.8) ˜ R ˜ (cid:15) ( X )of ˜ R ( X ). Thus, when choosing the initial capital, one would pick a minimal capitalfrom the calculated inner approximation, and not the true set, i.e. u ∈ ˜ R ˜ (cid:15) ( X ) + ˜ (cid:15)m .Noting that an (cid:15) -solution of problem (3.6) provides a solution to the linear vectoroptimization problem whose upper image is the inner approximation, the same proce-dure as in the linear case can be applied, just replacing ˜ R ( X ) by its inner polyhedralapproximation. One obtains an ˜ (cid:15) t -optimal strategy of risk compensating portfolios( Z t ) t ∈ T , respectively portfolio injections ( u t ) t ∈ T , with ˜ (cid:15) t = ( T − t + 1) (cid:15) + δ when usingthe same error level (cid:15) > Market extensions are considered when one is not only interested in putting a ‘capitalrequirement’ u ∈ R t ( X ) at time t aside and holding it until time T to make X risk neu-tral, but in exploiting the trading opportunities at the market to minimize the amountof capital needed for risk compensation. For the definition of the market extension be-low, we will set M = R d , i.e. we consider the full space of eligible assets. A justificationfor that comes from a mathematical as well as an interpretational aspect, which will bedetailed in remark 6.4 below. But one can already understand that choice by realizingthat the role of M comes mainly from a regulatory point of view. A regulator mightonly allow capital requirements to be made in certain currencies for example, and thesecapital requirements are held until time T . But the market extension is more linked tointernal risk measurement and management as one is exploring trading opportunitiesin possibly all assets, and thus will hold at any time t a portfolio in possibly all assets,so there is no need in restricting the capital requirements to be made in certain assetsonly.The market extension ( R mart ) t ∈ T of a dynamic risk measure ( R t ) t ∈ T is given by R mart ( X ) := (cid:91) k ∈ K t R t ( X − k )for some K t ⊆ L modeling the set of attainable claims. When K t ⊆ L t then itimmediately follows that R mart ( X ) = R t ( X ) + K t . (6.1)Let us give a few examples, all are special cases of the set-valued portfolios introducedin [4]. Example 6.1.
In a market with proportional transaction costs, trading is modeled bya sequence of solvency cones ( K t ) t ∈ T , see [16, 24, 17]. K t is a solvency cone at time t if it is an F t -measurable cone such that for every ω ∈ Ω , K t [ ω ] is a closed convexcone with R d + ⊆ K t [ ω ] (cid:40) R d . K t is generated by the bid and ask prices between anytwo assets at time t . In a market with proportional transaction costs, one would set K t = L t ( K t ) (see [7]). Example 6.2.
More generally, in markets with illiquidity (convex transaction costs)as in [22], trading is modeled by a sequence of convex solvency regions ( K t ) t ∈ T . K t is a convex solvency region at time t if it is an F t -measurable set such that for very ω ∈ Ω , K t [ ω ] is a closed convex set with R d + ⊆ K t [ ω ] (cid:40) R d . Then, one would set K t = L t ( K t ) . Example 6.3.
One could also incorporate trading constraints on the size of trans-actions by considering convex random sets D t (not necessarily mapping into the closedconvex upper sets G ( R d , R d + ) ) as follows. Given t ∈ T , let D t : Ω → R d (with R d denot-ing the power set of R d ) be an F t -measurable function such that D t [ ω ] is a closed convexset and K t [ ω ] ∩ D t [ ω ] (cid:54) = ∅ for every ω ∈ Ω . Then, one would set K t = L t ( K t ∩ D t ) . We now want to consider the market extended multi-portfolio time consistent ver-sion of ( R t ) t ∈ T with respect to K t ⊆ L t . Different possibilities arise and will be brieflydiscussed. First note that the market extension of the multi-portfolio time consistentversion ( ˜ R t ) t ∈ T , given by ( ˜ R t ( · ) + K t ) t ∈ T , is not multi-portfolio time consistent, thusthe solutions do not satisfy Bellman’s principle as detailed in section 5 and there is nogood economic interpretation. Therefore, only carefully alternating a market exten-sion step (6.1) and a backward recursion step, that is applying (6.1) in each step of thebackward recursion will yield the desired results and interpretations. As such we willdefine ˜ R mart ( X ) := (cid:91) Z ∈ ˜ R mart +1 ( X ) ( R t ( − Z ) + K t ) = (cid:91) k ∈ K t (cid:91) Z ∈ ˜ R mart +1 ( X − k ) R t ( − Z ) . (6.2)This is the multi-portfolio time consistent version of the market extension, i.e., thebackward composition of ( R mart ) t ∈ T . The obtained risk measure ( ˜ R mart ) t ∈ T does indeedsatisfy Bellman’s principle and we will give the economic interpretation of the solutionsbelow. Equation (6.2) shows that the two operations ‘market extension’ and ‘multi-portfolio time consistent version’ are interchangeable at any single time point t within arecursion step under the assumption M = R d . However, as noted previously, the marketextension of the multi-portfolio time consistent version, ( ˜ R t ( · ) + K t ) t ∈ T , is not equal tothe multi-portfolio time consistent version of the market extension, ( ˜ R mart ) t ∈ T , definedin (6.2). Thus, intrinsic to the definition of ( ˜ R mart ) t ∈ T is that both the market extensionand backward recursion (independent of the order of operations) are computed at atime point t + 1 and used as the input for the backward recursion in the next timepoint t .In analogy to section 5 for the regulator risk measure, one can obtain a nice finan-cial interpretation of the market extended composed risk measure ( ˜ R mart ) t ∈ T involvingportfolio injections and trades made forwards in time. In addition to the sequenceof portfolios holdings ( Z t ) t ∈ T one obtained for the regulator risk measure, there addi-tionally exists by equation (6.2) a sequence of trades ( k t ) t ∈ T such that Z t ∈ ˜ R mart ( X ), k t ∈ K t , and Z t − k t ∈ R t ( − Z t +1 ) for every choice of a risk compensating portfolio hold-ing Z ∈ ˜ R mar ( X ) at time t = 0. That means Z , Z , ..., Z T are the risk compensatingportfolio holdings before trades at times 0 , , ..., T and Z − k , Z − k , ..., Z T − k T arethe risk compensating portfolio holdings after the trades at times 0 , , ..., T . Equiva-lently, the portfolio injections (respectively withdrawals - if negative) ( u t ) t ∈ T , neededto update the risk compensating portfolio holdings and defined by u t = Z t − Z t − + k t − (with u = Z ), satisfy u t ∈ ˜ R mart (cid:32) X + t − (cid:88) s =0 ( u s − k s ) − k t (cid:33) or every time t , and 0 ∈ R t ( − u t +1 ) , for t ∈ T \{ T } . This gives the same interpretation as with the composed regulator riskmeasure discussed in section 5: The portfolio injections of the next time period t + 1are random, but acceptable with respect to the one-period risk measure R t,t +1 .Let us now discuss how to calculate ˜ R mart ( X ). Assume K t is closed and condi-tionally convex for all times t , then ( ˜ R mart ) t ∈ T can be calculated with the approachdiscussed in sections 4.1 and 4.2 by adding K t [ ω t ] to the vector optimization problem(3.6) at each time t . If K t = L t ( K t ) for a solvency cone K t for all times t (as inexample 6.1), the set-valued risk measure can also be computed directly by replacingthe ordering cone M + = R d + with K t [ ω t ] in problem (3.6).As with the ‘regulator risk measure’ considered in the previous sections, the marketextension ˜ R mart ( X ) might not be closed, but an arbitrarily close approximation is givenby its closed-valued variant¯ R mart ( X ) := cl (cid:91) Z ∈ ¯ R mart +1 ( X ) ( R t ( − Z ) + K t ) . Solving at each node backwards in time the vector optimization problem with ob-jective Φ t ( Z, Y + k ) and feasible region ( Z, Y, k ) ∈ Z t × K t [ ω t ] (with ¯ R t +1 replacedby ¯ R mart +1 ) yields ( ¯ R mart ) t ∈ T . Benson’s algorithm yields the solution set ¯ X mart [ ω t ] ⊆ L t +1 [ ω t ] × R d × K t [ ω t ] (with L t +1 [ ω t ] = L ( ∪ succ( ω t ) , succ( ω t ) , P ( ·| ω t ); R d )), which canbe used to calculate the sequence ( Z t ) t ∈ T \{ } and now additionally the sequence oftrades ( k t ) t ∈ T forwards in time on the realizing path for a given Z ∈ ¯ R mar ( X ) (orits inner approximation) at time t = 0. Utilizing (5.4), we can find a convex com-bination of the Y elements of the solution to describe any portfolio on the efficientfrontier of the risk measure, respectively of the inner approximation of it in the con-vex, non-polyhedral case. The same convex coefficients are then used for both thetrading strategy and the next time step full capital requirements, which are then usedas the starting value in the next period. Remark 6.4.
Let us comment on the choice of M = R d in this subsection. Aftercalculating the risk measure ( ˜ R mart ) t ∈ T , it is of course possible to choose a subspaceof eligible portfolios M and choose the capital requirements to be in that space (if˜ R mart ( X ) ∩ M t is non-empty).However, if the subspace M (cid:54) = R d were to be chosen first and used for the recur-sive computation, then the last equality in (6.2) would no longer hold. This wouldcause several problems as the market extended multi-portfolio time consistent version,i.e. ˜ R mar,Mt ( X ) := (cid:83) k ∈ K t (cid:83) Z ∈ ˜ R mart +1 ( X − k ) R t ( − Z ) ∩ M t , while retaining the capital in-jection interpretation given above, is in general not multi-portfolio time consistent.Furthermore, it would be more restrictive than the approach proposed above as itholds ˜ R mar,Mt ( X ) ⊆ ˜ R mart ( X ) ∩ M t .On the other hand, the multi-portfolio time consistent version of the market ex-tension, while being multi-portfolio time consistent, does not admit a good economicinterpretation for M (cid:54) = R d .All of these problems disappear if M = R d . For this, and the motivation given atthe beginning of this subsection, we suggest to use M = R d when considering marketextensions and if needed one can choose u t ∈ ˜ R mart ( X ) ∩ M t afterwards. xample 6.5. We can use the algorithm from section 3 to compute the set of super-hedging prices under either a conical or convex market model. Dual representationsfor the set of superhedging portfolios are considered in e.g. [16, 24, 17, 7] under pro-portional transaction costs; and in e.g. [22, 9] under convex transaction costs. Wecalculate the set of superhedging portfolios by computing the market-compatible versionof the worst case risk measure. Let ( R t ) t ∈ T be the worst case risk measure, that is R t : L → P ( L t ; L t, + ) with R t ( X ) = { u ∈ L t : X + u ∈ L + } . The worst case risk measure ( R t ) t ∈ T is conditionally convex and polyhedral with R t,t +1 ( − Z )[ ω t ] in (4.1) given as the upper image of a linear vector optimization problem. By proposi-tion 4.1 one can calculate ( R t ) t ∈ T ω t -wise since the worst case risk measure is multi-portfolio time consistent.Consider the market extension of the worst case risk measure, where trading ismodeled by a sequence of solvency regions ( K t ) t ∈ T . The multi-portfolio time consistentmarket extension ( ˜ R mart ) t ∈ T with K t = L t ( K t ) is nothing else than the superhedgingrisk measure. In particular, for a given claim X ∈ L , the set SHP t ( X ) := ˜ R mart ( − X ) is the set of superhedging portfolios of X .Under proportional transaction costs, modeled by a sequence of solvency cones ( K t ) t ∈ T , by proposition 4.1 and the discussion in section 6, the set of superhedgingportfolios SHP t ( X ) of X can be calculated backwards in time by solving a sequence oflinear vector optimization problems (3.6) with ordering cone K t [ ω t ] . This backwardsrecursive algorithm is exactly the one proposed in [20], see also [23], which could beobtained as the simple structure of the worst case risk measure ( R t ) t ∈ T yields a greatsimplification to the recursive structure (2.2) , respectively (2.3) , (2.4) . Note that thissimplification is specific to that example, which means that the method in [20, 23] can-not be generalized to other risk measures, whereas the approach discussed in this paperis widely applicable. In this section we will apply the algorithms for the recursive calculation of polyhedraland conditionally convex risk measures presented in section 3. Specifically, we willconsider the superhedging risk measure, relaxed worst case risk measure, average valueat risk, and entropic risk measure.We consider a multi-dimensional tree that approximates the d − P are given by correlated geometric Brownian motions: dS it = S it ( µ i dt + σ i dW it ) , i = 1 , ..., d − W i and W j with correlation ρ ij ∈ [ − , n d − (recombining) branches forany natural number n ≥ ν ∈ R ++ (instead of 2 d − branches and ν = 1 from the binomial model resented in [18]). That is, since every asset can rise or fall, we consider the set ofpossible up-down scenarios given by E = (cid:26) ( w , ..., w d − ) T : w i ∈ (cid:26) − ν, − ν + 2 νn − , ..., ν − νn − , ν (cid:27) ∀ i = 1 , ..., d − (cid:27) . We note that in the situation with only a single risky asset, n = 2, and ν = 1, thisreduces to the Cox-Ross-Rubinstein binomial tree model. To calculate the (conditional)probabilities of reaching a successor node, we partition the space R d − into n d − boxesin such a way that each element of E resides in a unique box, then the probability ofrising or falling by level e ∈ E is given by the probability of the (multivariate) normaldistribution over the box containing e .For simplicity, we additionally assume that the proportional transaction costs areconstant for each of the risky assets, given by γ = ( γ , ..., γ d − ) T ∈ R d − (possibly 0).Thus the bid and ask prices are given by ( S bt ) i = S it (1 − γ i ) and ( S at ) i = S it (1 + γ i )respectively for every i = 1 , ..., d −
1. In the case that γ i = 0 then the bid-ask spreadis 0 for the i th risky asset.Assume the existence of a risk-free asset with dynamics ( B t ) t ∈ T and no bid-askspread, i.e. B bt = B at at all times t . Further, we consider the case where cash (i.e. therisk-free asset) is an intermediary for all transactions. That is, the exchange betweenany two assets is done via cash and not directly. Under proportional transactioncosts, the above simplifying assumptions ensure that the solvency cone K t at time t isgenerated by the columns of the matrix (cid:32) (cid:16) S at B t (cid:17) T − (cid:16) S bt B t (cid:17) T − I ( d − × ( d − I ( d − × ( d − (cid:33) where I ( d − × ( d − denotes the identity matrix with d − K t at time t by the dual equations: k ∈ K t if K + t ( k ) := k + θ t, (cid:104) − exp (cid:16) − S bt k θ t, (cid:17)(cid:105) θ t, (cid:104) − exp (cid:16) − k S at θ t, (cid:17)(cid:105) + k ≥ . In the above equations we specify a parameter θ t ∈ L t ( R ) which defines the max-imum number of risky asset that can be bought with the riskless asset θ t, and themaximum number of the riskless asset that can be bought with the risky asset θ t, attime t . That is, beginning from the 0 portfolio, it would not be possible to attain morethan θ t, units of the risk-free asset or θ t, units of the risky asset. For simplicity, weassume that the trading strategy chosen does not impact the future market. Thus, atrade at time t does not affect the market at time t + 1. Example 7.1.
Consider a market with proportional transaction costs and two assets(a risk-free bond and a risky asset). We choose as a market model a recombining treewith 25 branches and T = 9 time steps over a one year time horizon. Further considerthe maximal possible rise or fall in the Brownian motion to be given by ν = 2 . Considerthe market with high proportional transaction costs, defined by γ = 30% . et the risk-free rate of return be . Let the drift for the risky asset be µ = 12 . and the volatility given by σ = 0 . . Consider the initial value of the risky asset to be S = $100 (measured in the risk-free asset).Consider the superhedging risk measure and the average value at risk with constantparameter λ = (30% , T on the terminal payoff X of an at the money Europeanput option, i.e. with strike price $100 . Running the polyhedral algorithm presented inthis paper, the efficient frontier of the time superhedging risk measure and composedmarket extended average value at risk (cid:94) AV @ R λ,mar ( X ) is given by figure 1. The circlesin figure 1 denote the vertices of the risk measures. It is clear that the superhedgingrisk measure provides a more conservative set of risk compensating portfolios. Notethat the deviation from a line to the efficient frontier of the superhedging risk measureis due entirely to the transaction costs, which is in contrast to the average value of riskas the corners there are not solely determined by the transaction costs. −80 −60 −40 −20 0 20 40 60 80 100−0.500.51 Risk−free asset R i sky a ss e t SuperhedgingAverage Value at Risk
Figure 1: Example 7.1: The superhedging risk measure and composed average value at riskunder high proportional transaction costs
Example 7.2.
Consider a market with proportional transaction costs and three assets(risk-free bond and two correlated risky assets). We will approximate the market witha binomial tree model with T = 20 time steps over a one year time horizon. Considera market with proportional transaction costs defined by γ = 5% .Let the risk-free rate of return be . Let the drift for the risky assets be given by µ = 15% and µ = 30% . Let the volatility for the risky assets be given by σ = 0 . and σ = 1 . Let the correlation be given by ρ = 0 . . Consider the initial value of therisky assets to be S = ($1 , $1) T (measured in the risk-free asset).Let X be the terminal payoff of an outperformance option with strike price K =$1 . , i.e. X = (cid:16) − KI { max( S aT ) ≥ K } , I { ( S aT ) ≥ ( S aT ) , ( S aT ) ≥ K } , I { ( S aT ) ≥ ( S aT ) , ( S aT ) ≥ K } (cid:17) T .Consider the relaxed worst case risk measure with constant parameters ε i = . for i = 0 , , and G is the convex cone generated by the vectors (1 , − . , − . T , − . , , − . T , and ( − . , − . , T . The market extended multi-portfolio time con-sistent version of the relaxed worst case risk measure can be calculated via the polyhedralalgorithm presented in this paper; a contour plot of the efficient frontier of the time risk measure ˜ R mar ( X ) is given by figure 2. Notice that, as the capital in the risk-lessasset increases the level of capital necessary in the risky assets decreases. However,increasing capital in one risky asset cannot totally offset a decrease in capital in theother risky asset, as evidenced by the curvature of the contour lines. − − − − − − Risky asset 1 R i sky a ss e t −20 −15 −10 −5 0 5 10 15 20−20−15−10−505101520 Figure 2: Example 7.2: Contour plot of the efficient frontier for the composed relaxed worstcase risk measure with 3 assets at differing levels of capital in the risk-less asset
Example 7.3.
Consider a market with convex transaction costs and two assets (risk-free bond and a risky asset). We consider the 2 time step Cox-Ross-Rubinstein model.We consider a market with proportional transaction costs given by γ = 5% and convextransaction costs given by θ t = (500 , T almost surely for each time point.Let the risk-free rate of return be . Let the drift for the risky asset be µ = 12 . and the volatility given by σ = 0 . . Consider the initial value of the risky asset to be S = $1 (measured in the risk-free asset).Let X be the terminal payoff from an out of the money binary option paying out $10 with strike price $1 . . We are able to compute an approximation of the efficientfrontier at time for the superhedging and entropic risk measures by running the convexalgorithm presented in this text. We consider two cases for the entropic risk measure,each with constant parameters λ ti = 10% for each asset and time and C t = L t ( C ) where:1. C = cone((1 , T , (0 , T ) be the convex cone generated by the vectors (1 , T and (0 , T , i.e. the restrictive entropic risk measure; and2. C = cone((1 , − . T , ( − . , T ) .Running the convex algorithm presented in this paper, with the approximation errorat time t = 0 is given by (cid:15) < . in all cases, the efficient frontier of the time omposed market extended risk measures is shown in figures 3, 4, and 5. The firstplot, figure 3 shows that at a large enough scale the different risk measures all appearidentical. The curvature of the risk measures is also very evident at this size, theasymptotic behavior at − in each asset is due to the choice of θ . In figure 4, wecan see discrepancies between the superhedging risk measure and the two (composed)entropic risk measures. However both (composed) entropic risk measures still appear tocoincide. In the final plot, figure 5, the distinction between the least restrictive entropicrisk measure is pronounced at this zoomed-in level of detail. The superhedging riskmeasure provides the most conservative estimate, in line with the theory. −1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1000010002000300040005000600070008000900010000 Risk−free asset R i sky a ss e t SuperhedgingEntropic: C = [[1;0],[0;1]]Entropic: C = [[1;−0.90],[−0.90;1]]
Figure 3: Example 7.3: Convex superhedging and entropic risk measures under proportionaland convex transaction costs, zoomed out view22
150 −100 −50 0 50 100 150 200−150−100−50050100150200 Risk−free asset R i sky a ss e t SuperhedgingEntropic: C = [[1;0],[0;1]]Entropic: C = [[1;−0.90],[−0.90;1]]
Figure 4: Example 7.3: Convex superhedging price and entropic risk measures under pro-portional and convex transaction costs, mid-sized view −3 −2 −1 0 1 2 3 4 5−3−2−1012345 Risk−free asset R i sky a ss e t SuperhedgingEntropic: C = [[1;0],[0;1]]Entropic: C = [[1;−0.90],[−0.90;1]]
Figure 5: Example 7.3: Convex superhedging price and entropic risk measures under pro-portional and convex transaction costs, near 0
References [1] Cagin Ararat, Andreas H. Hamel, and Birgit Rudloff. Set-valued shortfall anddivergence risk measures.
Submitted for publication , 2014.
2] Imen Ben Tahar and Emmanuel L´epinette. Vector-valued coherent risk mea-sure processes.
International Journal of Theoretical and Applied Finance ,17(2):1450011, 2014.[3] Harold Benson. An outer approximation algorithm for generating all efficientextreme points in the outcome set of a multiple objective linear programmingproblem.
Journal of Global Optimization , 13(1):1–24, 1998.[4] Ignacio Cascos and Ilya Molchanov. Multivariate risk measures: a constructiveapproach based on selections.
Mathematical Finance , 2014.[5] Matthias Ehrgott, Andreas L¨ohne, and Lizhen Shao. A dual variant of Benson’s“outer approximation algorithm” for multiple objective linear programming.
Jour-nal of Global Optimization , 52(4):757–778, 2012.[6] Matthias Ehrgott, Lizhen Shao, and Anita Sch¨obel. An approximation algorithmfor convex multi-objective programming problems.
Journal of Global Optimiza-tion , 50(3):397–416, 2011.[7] Zachary Feinstein and Birgit Rudloff. Time consistency of dynamic risk measuresin markets with transaction costs.
Quantitative Finance , 13(9):1473–1489, 2013.[8] Zachary Feinstein and Birgit Rudloff. A comparison of techniques for dynamicmultivariate risk measures. In A. Hamel, F. Heyde, A. L¨ohne, B. Rudloff, andC. Schrage, editors,
Set Optimization and Applications in Finance , PROMS series,pages 3–41. Springer, 2015.[9] Zachary Feinstein and Birgit Rudloff. Multi-portfolio time consistency for set-valued convex and coherent risk measures.
Finance and Stochastics , 19(1):67–107,2015.[10] Andreas H. Hamel and Frank Heyde. Duality for set-valued measures of risk.
SIAM Journal on Financial Mathematics , 1(1):66–95, 2010.[11] Andreas H. Hamel, Frank Heyde, and Birgit Rudloff. Set-valued risk measures forconical market models.
Mathematics and Financial Economics , 5(1):1–28, 2011.[12] Andreas H. Hamel, Andreas L¨ohne, and Birgit Rudloff. Benson type algorithmsfor linear vector optimization and applications.
Journal of Global Optimization ,59(4):811–836, 2014.[13] Andreas H. Hamel and Birgit Rudloff. Continuity and finite-valuedness of set-valued risk measures. In C. Tammer and F. Heyde, editors,
Festschrift in Celebra-tion of Prof. Dr. Wilfried Grecksch’s 60th Birthday , pages 46–64. Shaker Verlag,2008.[14] Andreas H. Hamel, Birgit Rudloff, and Mihaela Yankova. Set-valued average valueat risk and its computation.
Mathematics and Financial Economics , 7(2):229–246,2013.[15] Elyes Jouini, Moncef Meddeb, and Nizar Touzi. Vector-valued coherent risk mea-sures.
Finance and Stochastics , 8(4):531–552, 2004.[16] Yuri M. Kabanov. Hedging and liquidation under transaction costs in currencymarkets.
Finance and Stochastics , 3(2):237–248, 1999.[17] Yuri M. Kabanov and Mher Safarian.
Markets with Transaction Costs: Mathe-matical Theory . Springer Finance. Springer, 2009.
18] Ralf Korn and Stafanie M¨uller. The decoupling approach to binomial pricing ofmulti-asset options.
Journal of Computational Finance , 12(3):1–30, 2009.[19] Andreas L¨ohne.
Vector Optimization With Infimum and Supremum . Vector Op-timization. Springer, 2011.[20] Andreas L¨ohne and Birgit Rudloff. An algorithm for calculating the set of su-perhedging portfolios in markets with transaction costs.
International Journal ofTheoretical and Applied Finance , 17(2):1450012, 2014.[21] Andreas L¨ohne, Birgit Rudloff, and Firdevs Ulus. Primal and dual approximationalgorithms for convex vector optimization problems.
Journal of Global Optimiza-tion , 60(4):713–736, 2014.[22] Teemu Pennanen and Irina Penner. Hedging of claims with physical delivery underconvex transaction costs.
SIAM Journal on Financial Mathematics , 1(1):158–178,2010.[23] Alet Roux and Tomasz Zastawniak. American and Bermudan options in currencymarkets under proportional transaction costs.
Acta Applicandae Mathematicae ,pages 1–39, 2015.[24] Walter Schachermayer. The fundamental theorem of asset pricing under propor-tional transaction costs in finite discrete time.
Mathematical Finance , 14(1):19–48,2004.[25] Carola Schrage and Andreas L¨ohne. An algorithm to solve polyhedral convex setoptimization problems.
Optimization , 62(1):131–141, 2013., 62(1):131–141, 2013.