Mathematical Finance and Probability Seminars (Since covid these events are taking place online.)

Jouini et al. (Finance & Stochastics 8, 2004) proposed the concept of set–valued coherent risk measures in order to incorporate market frictions like transaction costs, liquidity bounds etc. in the evaluation of the risk of a portfolio consisting of d greater than or equal to 1 assets.

We extend this concept to general risk measures with values in the set of all subsets of a finite dimensional space being set–valued cash invariant and monotone functions. For the general case, we establish primal representation results in terms of acceptance sets. For the convex case, we give a complete duality theory parallel to the scalar case including a penalty function representation. This theory is based on extensions of Convex Analysis to set–valued functions which are also new – including definitions of Fenchel conjugates for set–valued convex functions and a corresponding biconjugation theorem.

A list of examples is given along with "standarized" procedures (primal and dual) how a known scalar risk measure can be transformed into a set–valued one. In many cases, there is more than one generalization (more or less risk averse) of one scalar risk measure. We shall present a set–valued counterpart for (negative) expectation, set–valued VaRs and CVaRs, generalizations of (negative) essential infimum and set–valued entropy measures. ( Slides)

Speaker: Andreas H Hamel, Princeton University