Let η = (η1,…,ηn) be a positive random vector. If its coordinates ηiand ηjare exchangeable, i.e. the distribution of η is invariant with respect to the swap πijof itsith andjth coordinates, then Ef(η) = Ef(πijη) for all integrable functionsf. In this paper we study integrable random vectors that satisfy this identity for a particular family of functionsf, namely those which can be written as the positive part of the scalar product 〈u, η〉 with varying weightsu. In finance such functions represent payoffs from exchange options with η being the random part of price changes, while from the geometric point of view they determine the support function of the so-called zonoid of η. If the expected values of such payoffs are πij-invariant, we say that η isij-swap-invariant. A full characterisation of the swap-invariance property and its relationship to the symmetries of expected payoffs of basket options are obtained. The first of these results relies on a characterisation theorem for integrable positive random vectors with equal zonoids. Particular attention is devoted to the case of asset prices driven by Lévy processes. Based on this, concrete semi-static hedging techniques for multi-asset barrier options, such as weighted barrier swap options, weighted barrier quanto-swap options, or certain weighted barrier spread options, are suggested.
An invariance property of quadratic forms in random vectors with a selection distribution, with application to sample variogram and covariogram estimators