# Pathwise functional portfolio generation and optimal transport

## Michael Monoyios (University of Oxford)

### Wednesday, October 10, 2018 4:00 pmMONT 313 (Storrs)

We describe a precise connection, first observed by Pal and Wong, between functionally generated investments and optimal transport, in a model-free discrete-time financial market. A functionally generated portfolio (FGP), developed in continuous time Ito markets by Fernholz in Stochastic Portfolio Theory, compute the investment in each stock through the prism of the super-differential of a concave function of the market weight vector. Such portfolios have been shown to outperform the market under suitable conditions. Here, in a pathwise discrete-time scenario, we equate a convex-analytic cyclical monotonicity property characterizing super-differentials, with a $c$-cyclical monotonicity property of the unique Monge (deterministic) solution of an appropriately constructed optimal transport problem with cost function $c$, which transfers the market portfolio distribution to the functionally generated portfolio distribution. By establishing uniqueness of the solution to the optimal transport problem, we raise the connection observed by Pal and Wong to an exact equivalence between optimal transport and functional generation. This is established for both the traditional (multiplicative) functionally generated portfolios, and an `additive'' modification introduced by Karatzas and Ruf. Ramifications include pathwise discrete-time master equations for the evolution of the relative wealth of the investment when using the market portfolio as the numeraire. We take the pathwise continuous time limit, assuming continuous paths which admit well-defined quadratic variation, to establish the model-free continuous-time master equation for both types of functionally generated investment, providing an alternative derivation to the recent proof of Schied et al of the master equation for multiplicative FGPs, as well as an extension to the case of additive functionally generated trading strategies. (Joint work with Alexander Vervuurt).