On the generalization of Esscher and variance premiums modified for the elliptical family of distributions

Esscher premiums, Esscher transforms and "exponential tilting" [Wang, S., 2002. A Set of New Methods and Tools for Enterprise Risk Capital Management and Portfolio Optimization. 2002 CAS Summer Forum, Dynamic Financial Analysis Discussion papers] are regarded as convenient tools in risk measurement and portfolio allocation. The main component of these measures is the variance-covariance structure of the multivariate distribution, which makes them especially attractive for multivariate normal portfolios, since the latter are uniquely determined by their variance-covariance structure. However, if the distribution deviates from the normal by having, for example, heavy tailed marginals, the allocation methods based on Esscher transforms fail to reflect this deviation even if the distribution still preserves the same variance-covariance structure as normal.