Further Information
We make precise a remarkable connection, first observed by Pal and Wong (2016) and further analysed in the doctoral thesis of Vervuurt (2016), between functionally generated investments and optimal transport, in a model-free discrete-time financial market. A functionally generated portfolio (FGP) computes the investment in each stock through the prism of the super-differential of the logarithm of a concave function (the generating function of the FGP) of the market weight vector. Such portfolios have been shown to outperform the market under suitable conditions. Here, in our pathwise discrete-time scenario, we equate the convex-analytic cyclical monotonicity property characterising super-differentials, with a $c$-cyclical monotonicity property of the unique Monge solution of an appropriately constructed optimal transport problem with cost function $c$, which transfers the market portfolio distribution to the FGP distribution. Using the super-differential characterisation of functional investments, we construct optimal transport problems for both traditional (multiplicative) FGPs, and an ``additive'' modification introduced by Karatzas and Ruf (2017), featuring the same cost function in both cases, which characterise the functional investment. In the multiplicative case, the construction differs from Pal and Wong (2016) and Vervuurt (2016), who used a ``multiplicative'' cyclical monotonicity property, as opposed to the classical cyclical monotonicity property used here.
We establish uniqueness of the solution to the relevant optimal transport problem, elevating the connection observed by Pal and Wong (2016) to an exact equivalence between optimal transport and functional generation. We explore ramifications, including pathwise discrete-time master equations for the evolution of the relative wealth of the investment when using the market portfolio as numeraire. We take the pathwise continuous time limit, assuming continuous paths which admit well-defined quadratic variation, to establish model-free continuous-time master equations for both types of functionally generated investment, providing an alternative derivation to the recent proof of Schied et al (2018) of the master equation for multiplicative FGPs, as well as an extension to the case of additive functionally generated trading strategies.