Convex ordering of Esscher and minimal entropy martingale measures for discrete time models

We recall and extend some sufficient conditions for the convex comparison of martingale measures in a one period setting, based on the elasticity of the pricing kernel. We show that the minimal entropy martingale measure (MEMM) and the Esscher martingale measure are comparable in the convex order, and which one is dominating depends on the sign of the risk premium on the underlying. If it is positive, then the MEMM gives a lower price to each convex payoff. We show how the comparison result can be extended to the multiperiod i.i.d. case and discuss the problems related to the general, non i.i.d. case, proving a Lemma that links one period comparison with multi period comparison under more general assumptions. © Springer-Verlag Italia 2012.