On Frequency-Based Optimal Portfolio with Transaction Costs

Paper: arXiv 2301.02754

Authors: Chung-Han Hsieh, Yi-Shan Wong

Abstract

The aim of this paper is to investigate the impact of rebalancing frequency and transaction costs on the log-optimal portfolio, which is a portfolio that maximizes the expected logarithmic growth rate of an investor’s wealth. We prove that the frequency-dependent log-optimal portfolio problem with costs is equivalent to a concave program and provide a version of the dominance theorem with costs to determine when an investor should invest all available funds in a particular asset. Then, we show that transaction costs may cause a bankruptcy issue for the frequency-dependent log-optimal portfolio. To address this issue, we approximate the problem to obtain a quadratic concave program and derive necessary and sufficient optimality conditions. Additionally, we prove a version of the two-fund theorem, which states that any convex combination of two optimal weights from the optimality conditions is still optimal. We test our proposed methods using both intraday and daily price data. Finally, we extend our empirical studies to an online trading scenario by implementing a sliding window approach. This approach enables us to solve a sequence of concave programs rather than a potentially computational complex stochastic dynamic programming problem.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides strong mathematical contributions by reformulating the log-optimal portfolio problem with transaction costs into a concave program and proving optimality conditions such as a version of the two-fund theorem. It validates these theoretical extensions using both intraday and daily price datasets, as well as an online sliding window implementation.

Research Flowchart

  flowchart TD
    A[Goal: Maximize Log-Optimal Portfolio Growth with Transaction Costs & Rebalancing Frequency] --> B[Formulate Problem as Concave Program & Identify Bankruptcy Issue]
    B --> C[Input Data: Intraday & Daily Price Data]
    C --> D[Approximate Problem to Quadratic Concave Program & Derive Optimality Conditions]
    D --> E[Implement Online Trading via Sliding Window Approach]
    E --> F[Outcomes: Proved Dominance & Two-Fund Theorems, Solved Bankruptcy Issue Efficiently]