On Solving Robust Log-Optimal Portfolio: A Supporting Hyperplane Approximation Approach

TL;DR

This paper introduces a hyperplane approximation method to efficiently solve distributionally robust log-optimal portfolio problems, accommodating practical trading constraints and validated through empirical stock data analysis.

Contribution

It proposes a linear programming reformulation of a complex robust portfolio optimization problem using supporting hyperplanes, enabling efficient computation and practical application.

Findings

The method effectively handles ambiguity in return distributions.

The approach accommodates transaction costs, leverage, shorting, and diversification.

Empirical results demonstrate the approach's practical viability with historical stock data.

Abstract

A {log-optimal} portfolio is any portfolio that maximizes the expected logarithmic growth (ELG) of an investor's wealth. This maximization problem typically assumes that the information of the true distribution of returns is known to the trader in advance. However, in practice, the return distributions are indeed {ambiguous}; i.e., the true distribution is unknown to the trader or it is partially known at best. To this end, a {distributional robust log-optimal portfolio problem} formulation arises naturally. While the problem formulation takes into account the ambiguity on return distributions, the problem needs not to be tractable in general. To address this, in this paper, we propose a {supporting hyperplane approximation} approach that allows us to reformulate a class of distributional robust log-optimal portfolio problems into a linear program, which can be solved very efficiently. Our framework is flexible enough to allow {transaction costs}, {leverage and shorting}, {survival trades}, and {diversification considerations}. In addition, given an acceptable approximation error, an efficient algorithm for rapidly calculating the optimal number of hyperplanes is provided. Some empirical studies using historical stock price data are also provided to support our theory.