Model-free portfolio theory and its functional master formula

TL;DR

This paper develops two pathwise Itô calculus-based versions of the master formula in Fernholz's stochastic portfolio theory, enabling analysis of portfolios generated from functions depending on current market states and entire market paths, with empirical validation.

Contribution

It introduces two novel pathwise Itô calculus frameworks for the master formula, expanding Fernholz's stochastic portfolio theory to path-dependent and state-dependent portfolio generation.

Findings

The formulas work with empirical market data.

They accommodate path-dependent portfolio generating functions.

The methods are validated through multiple examples.

Abstract

We use pathwise It\^o calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of F\"ollmer's pathwise It\^o calculus and works for portfolios generated from functions that may depend on the current states of the market portfolio and an additional path of finite variation. The second version is formulated within the functional pathwise It\^o calculus of Dupire (2009) and Cont \& Fourni\'e (2010) and allows for portfolio-generating functionals that may depend additionally on the entire path of the market portfolio. Our results are illustrated by several examples and shown to work on empirical market data.