In this paper, the market extension of set-valued risk measures for models with proportional transaction costs is linked with set-valued risk minimization problems. As a particular example, the set-valued average value at risk (AV@R) is defined and its market extension and corresponding risk minimization problems are studied. We show that for a finite probability space the calculation of the values of AV@R reduces to linear vector optimization problems which can be solved using known algorithms. The formulation of AV@R as a linear vector optimization problem is an extension of the corresponding scalar result by Rockafellar and Uryasev .
New versions of the set-valued average value at risk for multivariate risks are introduced by generalizing the well-known certainty equivalent representation to the set-valued case. The first "regulator" version is independent from any market model whereas the second version, called the market extension, takes trading opportunities into account. Essential properties of both versions are proven and an algorithmic approach is provided which admits to compute the values of both version over finite probability spaces. Several examples illustrate various features of the theoretical constructions .
R
In this paper, the market extension of set-valued risk measures for models with proportional transaction costs is linked with set-valued risk minimization problems. As a particular example,
New versions of
R
(AV@R) is defined and its market extension and corresponding risk minimization problems are studied. We show that for a finite probability space the calculation of
for multivariate risks are introduced by generalizing the well-known certainty equivalent representation to the set-valued case. The first "regulator" version is independent from any market model whereas the second version, called the market extension, takes trading opportunities into account. Essential properties of both versions are proven and an algorithmic approach is provided which admits to compute
R
AV@R reduces to linear vector optimization problems which can be solved using known algorithms. The formulation of AV@R as a linear vector optimization problem is an extension of the corresponding scalar result by Rockafellar and Uryasev
both version over finite probability spaces. Several examples illustrate various features of the theoretical constructions