Optimization of relative arbitrage

TL;DR

This paper investigates the conditions under which relative arbitrages can be constructed using functionally generated portfolios in stochastic portfolio theory, revealing limitations when replacing the market portfolio with others like equal or entropy weighted portfolios.

Contribution

It demonstrates that relative arbitrages are not possible with certain alternative portfolios under the same conditions, and introduces a shaped-constrained optimization approach for these portfolios.

Findings

Relative arbitrages cannot be constructed with equal or entropy weighted portfolios under certain market conditions.

Introduces a shaped-constrained optimization problem for functionally generated portfolios.

Provides theoretical insights into portfolio construction limitations.

Abstract

In stochastic portfolio theory, a relative arbitrage is an equity portfolio which is guaranteed to outperform a benchmark portfolio over a finite horizon. When the market is diverse and sufficiently volatile, and the benchmark is the market or a buy-and-hold portfolio, functionally generated portfolios introduced by Fernholz provide a systematic way of constructing relative arbitrages. In this paper we show that if the market portfolio is replaced by the equal or entropy weighted portfolio among many others, no relative arbitrages can be constructed under the same conditions using functionally generated portfolios. We also introduce and study a shaped-constrained optimization problem for functionally generated portfolios in the spirit of maximum likelihood estimation of a log-concave density.