Convex ordering criteria for Levy processes

Modelling financial and insurance time series with Levy processes or with exponential Levy processes is a relevant actual practice and an active area of research. It allows qualitatively and quantitatively good adaptation to the empirical statistical properties of asset returns. Due to model incompleteness it is a problem of considerable interest to determine the dependence of option prices in these models on the choice of pricing measures and to establish nontrivial price bounds. In this paper we review and extend ordering results of stochastic and convex type for this class of models. We also extend the ordering results to processes with independent increments (PII) and present several examples and applications as to alpha-stable processes, NIG-processes, GH-distributions, and others. Criteria are given for the Levy measures which imply corresponding comparison results for European type options in (exponential) Levy models.