The num eraire portfolio, asymmetric information and entropy

We consider simple models of nancial markets with less and better informed investors described by a smaller and a larger ltration on a general stochastic basis that describes the market dynamics, including continuous and jump components. We study the relation between dierent forms of non-existence of arbitrage and the characteristics of the stochastic basis under the dierent ltrations. This is achieved through the analysis of the properties of the num eraire portfolio. Furthermore, we focus on the problem of calculating the additional logarithmic utility of the better informed investor in terms of the Shannon entropy of his additional information. The information drift, i.e. the drift to eliminate in order to preserve the martingale property in the larger ltration turns out to be the crucial quantity needed to tackle these problems. We show that the expected logarithmic utility increment due to better information equals its Shannon entropy also in case of a pure jump basis with jumps that are quadratically hedgeable, and so extend a similar result known for bases consisting of continuous semimartingales. An example illustrates that the equality may not persist if both continuous and jump components are present in the underlying.