Portfolios, optimal transport and informations geometry
Can we outperform a market index in the presence of volatility? What is the optimal frequency to rebalance a portfolio? We show that these questions can be analyzed using modern ideas in probability and information geometry (geometry in information theory). We quantify market volatility by a logarithmic divergence which is a distance-like quantity analogous to the relative entropy, and in this context portfolio selection (such as the universal portfolio) has a lot in common with nonparametric statistics. Mathematically, the divergence is intimately related to exponentially concave functions and the solution of an optimal transport problem with a logarithmic cost. It induces a rich differential geometric structure with numerous applications. In particular, a dualistic Pythagorean theorem gives insight into the optimal frequency of discrete rebalancing.