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On Cost-Sensitive Distributionally Robust Log-Optimal Portfolio

Chung-Han Hsieh, Xiao-Rou Yu

Abstract

This paper addresses a novel \emph{cost-sensitive} distributionally robust log-optimal portfolio problem, where the investor faces \emph{ambiguous} return distributions, and a general convex transaction cost model is incorporated. The uncertainty in the return distribution is quantified using the \emph{Wasserstein} metric, which captures distributional ambiguity. We establish conditions that ensure robustly survivable trades for all distributions in the Wasserstein ball under convex transaction costs. By leveraging duality theory, we approximate the infinite-dimensional distributionally robust optimization problem with a finite convex program, enabling computational tractability for mid-sized portfolios. Empirical studies using S\&P 500 data validate our theoretical framework: without transaction costs, the optimal portfolio converges to an equal-weighted allocation, while with transaction costs, the portfolio shifts slightly towards the risk-free asset, reflecting the trade-off between cost considerations and optimal allocation.

On Cost-Sensitive Distributionally Robust Log-Optimal Portfolio

Abstract

This paper addresses a novel \emph{cost-sensitive} distributionally robust log-optimal portfolio problem, where the investor faces \emph{ambiguous} return distributions, and a general convex transaction cost model is incorporated. The uncertainty in the return distribution is quantified using the \emph{Wasserstein} metric, which captures distributional ambiguity. We establish conditions that ensure robustly survivable trades for all distributions in the Wasserstein ball under convex transaction costs. By leveraging duality theory, we approximate the infinite-dimensional distributionally robust optimization problem with a finite convex program, enabling computational tractability for mid-sized portfolios. Empirical studies using S\&P 500 data validate our theoretical framework: without transaction costs, the optimal portfolio converges to an equal-weighted allocation, while with transaction costs, the portfolio shifts slightly towards the risk-free asset, reflecting the trade-off between cost considerations and optimal allocation.

Paper Structure

This paper contains 20 sections, 6 theorems, 31 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Ambiguity Consideration
  3. Distributionally Robust Optimization Approach
  4. Transaction Costs Consideration
  5. Contributions of This Paper
  6. Preliminaries
  7. Notations
  8. Cost-Sensitive Account Dynamics
  9. Wasserstein Ambiguity Set
  10. Robustly Survivable Trades and Feasible Controls
  11. Cost-Sensitive DRO Problem
  12. Theoretical Results
  13. Empirical Studies
  14. Effect of Ambiguity Sizes
  15. Out-Of-Sample Trading Performance
  16. ...and 5 more sections

Key Result

Lemma 2.4

Let $V(0) > 0$ and $w \in \mathcal{W}$ be given. If the weight vector $w$ satisfies $\sum_{i=1}^{m} w_i {\mathcal{X}}_{\min, i} + c(w) > 0,$ then, for any $\varepsilon \geq 0$ and for all distributions $\mathbb{F} \in \mathcal{B}_\varepsilon( \widehat{\mathbb{F}} )$, we have

Figures (7)

  • Figure 1: Historical Adjusted Closing Prices for the 10 Selected Stocks of S&P 500.
  • Figure 2: Effect of Varying Ambiguity Size on Optimal Weights Using Data from January of 2022.
  • Figure 3: Effect of Varying Ambiguity Size on Optimal Weights Using Data from January of 2023.
  • Figure 4: Out-of-Sample Monthly Rebalanced Account Value Trajectories: Different Radius Sizes and Transaction Costs. Note that $\varepsilon = 0$ Corresponds to Classical ELG Strategy, and $\varepsilon > 0$ Corresponds to Robust ELG Strategy with $\varepsilon = 1$ Approximating to an Equal Weights Portfolio.
  • Figure 5: Transaction Cost Effect: Out-of-Sample Monthly Rebalanced Account Value Trajectories.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Remark 2.1
  • Definition 2.2: Wasserstein Metric
  • Remark 2.3
  • Lemma 2.4: Robustly Survivable Trades
  • proof
  • Remark 2.5
  • Remark 2.7
  • Theorem 3.1: Convex Approximation
  • proof
  • Corollary 3.2: Reduction of Semi-Infinite Constraint
  • ...and 12 more