This paper is concerned with the utility-based risk of a financial position in a multi-asset market with frictions. Risk is quantified by set-valued risk measures, and market frictions are modeled by conical/convex random solvency regions representing proportional transaction costs or illiquidity effects, and convex random sets representing trading constraints. First, with a general set-valued risk measure, the effect of having trading opportunities on the risk measure is considered, and a corresponding dual representation theorem is given. Then, assuming individual utility functions for the assets, utility-based shortfall and divergence risk measures are defined , which form two classes of  set-valued convex risk measures . Minimal penalty functions are computed in terms of the vector versions of the well-known divergence functionals (generalized relative entropy). As special cases, set-valued versions of the entropic risk measure and the average value at risk are obtained. The general results on the effect of market frictions are applied to the utility-based framework and conditions concerning applicability are presented .
Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued versions of the entropic risk measure and the average value at risk . As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved .
R
This paper is concerned with the
Risk measures for multivariate financial positions are studied in a
R
risk of a financial position in a multi-asset market with frictions. Risk is quantified by set-valued risk measures, and market frictions are modeled by conical/convex random solvency regions representing proportional transaction costs or illiquidity effects, and convex random sets representing trading constraints. First, with a general set-valued risk measure, the effect of having trading opportunities on the risk measure is considered, and a corresponding dual representation theorem is given. Then, assuming individual utility functions for the assets, utility-based
framework. Under a certain incomplete preference relation,
R
, which form
as the optimal values of specific set minimization problems. The dual relationship between these
A
multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include
D
convex risk measures . Minimal penalty functions are computed in terms of the vector versions of the well-known divergence functionals (generalized relative entropy). As special cases, set-valued
R
are obtained. The general results on the effect of market frictions are applied to the utility-based framework and conditions concerning applicability are presented
. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved