Model-free Portfolio Theory: A Rough Path Approach

TL;DR

This paper introduces a model-free, rough path-based approach to stochastic portfolio theory, enabling analysis of more general portfolios without probabilistic assumptions, and demonstrating asymptotic growth rate equivalences.

Contribution

It develops a novel rough path framework for model-free stochastic portfolio theory, extending the class of portfolios analyzed and establishing new asymptotic growth rate results.

Findings

Pathwise formula for relative wealth process

Asymptotic growth rate of generalized Cover's portfolio matches the best retrospective portfolio

Log-optimal portfolios in ergodic diffusions share growth rates with Cover's portfolio

Abstract

Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory (SPT). Our approach allows to handle significantly more general portfolios compared to previous model-free approaches based on F{\"o}llmer integration. Without the assumption of any underlying probabilistic model, we prove a pathwise formula for the relative wealth process which reduces in the special case of functionally generated portfolios to a pathwise version of the so-called master formula of classical SPT. We show that the appropriately scaled asymptotic growth rate of a far reaching generalization of Cover's universal portfolio based on controlled paths coincides with that of the best retrospectively chosen portfolio within this class. We provide several novel results concerning rough integration, and highlight the advantages of the rough path approach by showing that (non-functionally generated) log-optimal portfolios in an ergodic It{\^o} diffusion setting have the same asymptotic growth rate as Cover's universal portfolio and the best retrospectively chosen one.