Generalizations of Functionally Generated Portfolios with Applications to Statistical Arbitrage

TL;DR

This paper extends the theory of functionally generated portfolios by allowing more flexible numeraires and dynamic generating functions, maintaining analytical tractability for applications like statistical arbitrage and risk management.

Contribution

It introduces generalized FGPs with stochastic dynamic generating functions and arbitrary positive wealth numeraires, preserving the master equation's usefulness.

Findings

Generalized FGPs retain analytical tractability.

Applications demonstrated in statistical arbitrage and risk immunization.

Flexible framework for scenario analysis and mirror portfolios.

Abstract

The theory of functionally generated portfolios (FGPs) is an aspect of the continuous-time, continuous-path Stochastic Portfolio Theory of Robert Fernholz. FGPs have been formulated to yield a master equation - a description of their return relative to a passive (buy-and-hold) benchmark portfolio serving as the num\'eraire. This description has proven to be analytically very useful, as it is both pathwise and free of stochastic integrals. Here we generalize the class of FGPs in several ways: (1) the num\'eraire may be any strictly positive wealth process, not necessarily the market portfolio or even a passive portfolio; (2) generating functions may be stochastically dynamic, adjusting to changing market conditions through an auxiliary continuous-path stochastic argument of finite variation. These generalizations do not forfeit the important tractability properties of the associated master equation. We show how these generalizations can be usefully applied to scenario analysis, statistical arbitrage, portfolio risk immunization, and the theory of mirror portfolios.