Maximum‐loss, minimum‐win and the Esscher pricing principle

Maximum‐loss (Max‐loss) was recently introduced as a valuation functional in the context of systematic stress testing. The basic idea is to value a (financial) random variable by its worst case expectation, where the most unfavourable probability measure–the ‘worst case distribution’–lies within a given Kullback–Leibler radius around a previously estimated distribution. The article gives an overview of the properties of this measure and analyses relations to other risk and acceptability measures and to the well‐known Esscher pricing principle, used in insurance mathematics and option pricing. The main part of the article focuses then on optimal decision‐making–in particular related to portfolio optimization–with Max‐loss as the objective function to be minimized. A simple algorithm for dealing with the resulting saddle point problem is introduced and analysed.