Optimal payoff profiles for law-invariant risk measures

Alexander Schied

Abstract

Motivated by the problem of finding a risk-minimizing hedging strategy, we discuss a Neyman-Pearson type problem for law-invariant risk measures. Similar problems were studied by Huber and Strassen (1973) and by Cvitanic and Karatzas (2001), who focus on general existence results and the interpretation of solutions in terms of least favourable pairs of (probability) measures. Here, we use different methods and a more specific setting in order to obtain more explicit formulas for solutions. Special emphasis is given to the class of quantile-based coherent risk measures that were introduced by Kusuoka (2001). We also discuss a class of risk measures that are derived from robust utility functionals.