On portfolios generated by optimal transport

TL;DR

This paper explores the connections between different types of functionally generated portfolios, including additive and multiplicative, through the lens of optimal transport and information geometry, providing a unified framework and empirical insights.

Contribution

It characterizes all possible functional portfolio constructions that encompass additive and multiplicative forms using divergence functionals and information geometry.

Findings

Additive portfolios relate to Bregman divergence in information geometry.

Unified framework for portfolio generation methods.

Empirical example demonstrating theoretical concepts.

Abstract

First introduced by Fernholz in stochastic portfolio theory, functionally generated portfolio allows its investment performance to be attributed to directly observable and easily interpretable market quantities. In previous works we showed that Fernholz's multiplicatively generated portfolio has deep connections with optimal transport and the information geometry of exponentially concave functions. Recently, Karatzas and Ruf introduced a new additive portfolio generation whose relation with optimal transport was studied by Vervuurt. We show that additively generated portfolio can be interpreted in terms of the well-known dually flat information geometry of Bregman divergence. Moreover, we characterize, in a sense to be made precise, all possible forms of functional portfolio constructions that contain additive and multiplicative generations as special cases. Each construction involves a divergence functional on the unit simplex measuring the market volatility captured, and admits a pathwise decomposition for the portfolio value. We illustrate with an empirical example.