Portfolio Optimization under Transaction Costs with Recursive Preferences

ArXiv ID: 2402.08387 “View on arXiv”

Authors: Unknown

Abstract

The Merton investment-consumption problem is fundamental, both in the field of finance, and in stochastic control. An important extension of the problem adds transaction costs, which is highly relevant from a financial perspective but also challenging from a control perspective because the solution now involves singular control. A further significant extension takes us from additive utility to stochastic differential utility (SDU), which allows time preferences and risk preferences to be disentangled. In this paper, we study this extended version of the Merton problem with proportional transaction costs and Epstein-Zin SDU. We fully characterise all parameter combinations for which the problem is well posed (which may depend on the level of transaction costs) and provide a full verification argument that relies on no additional technical assumptions and uses primal methods only. The case with SDU requires new mathematical techniques as duality methods break down. Even in the special case of (additive) power utility, our arguments are significantly simpler, more elegant and more far-reaching than the ones in the extant literature. This means that we can easily analyse aspects of the problem which previously have been very challenging, including comparative statics, boundary cases which heretofore have required separate treatment and the situation beyond the small transaction cost regime. A key and novel idea is to parametrise consumption and the value function in terms of the shadow fraction of wealth, which may be of much wider applicability.

Keywords: Merton Problem, Stochastic Differential Utility (SDU), Transaction Costs, Singular Control, Stochastic Optimization, Wealth Management / Portfolio Choice

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, focusing on stochastic control, singular control, and the derivation of a full verification argument using primal methods for Epstein-Zin SDU, which requires advanced mathematics (stochastic differential equations, ODEs, free-boundary problems). It lacks empirical components such as backtests, datasets, or statistical metrics, relying instead on theoretical analysis without implementation details for real-world data.

  flowchart TD
    A["Research Goal<br/>Extend Merton Problem<br/>with Transaction Costs &<br/>Recursive Preferences"] --> B["Model Setup<br/>Epstein-Zin SDU &<br/>Proportional Transaction Costs"]
    B --> C["Novel Methodology<br/>Shadow Fraction of Wealth<br/>Parametrization &<br/>Primal Verification Argument"]
    C --> D{"Key Mathematical Process<br/>Solve Characteristic ODEs<br/>& Verify Value Function"}
    D --> E["Computational Analysis<br/>Numerical Solution of<br/>Boundary Conditions &<br/>Comparative Statics"]
    E --> F["Key Findings & Outcomes<br/>1. Complete Parameter Characterization<br/>2. Unified Treatment of Boundary Cases<br/>3. Analytical Insights beyond Small Costs"]