Functionally Generated Portfolios Under Stochastic Transaction Costs: Theory and Empirical Evidence

2025-07-12

AI Summary

Classical Stochastic Portfolio Theory (SPT) neglects transaction costs and fails to capture market frictions under liquidity shocks. Method: We extend SPT to a continuous-time framework featuring state-dependent, stochastic proportional transaction costs. We derive a closed-form lower bound on cost-adjusted relative wealth and establish sufficient conditions for the existence of relative arbitrage under stochastic costs. Using an associated Itô diffusion model, we implement Milstein-based numerical simulations and conduct empirical validation using high-frequency CRSP small-cap stock data alongside limit-order-book–based proxy transaction costs. Contribution/Results: Under realistic cost constraints, diversified-weighted and entropy-weighted portfolios still generate annualized excess returns of 3.6% and 2.9%, respectively. These findings demonstrate that the proposed theoretical framework robustly accommodates real-world market frictions and retains practical efficacy in portfolio construction.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Assuming frictionless trading, classical stochastic portfolio theory (SPT) provides relative arbitrage strategies. However, the costs associated with real-world execution are state-dependent, volatile, and under increasing stress during liquidity shocks. Using an Ito diffusion that may be connected with asset prices, we extend SPT to a continuous-time equity market with proportional, stochastic transaction costs. We derive closed-form lower bounds on cost-adjusted relative wealth for a large class of functionally generated portfolios; these bounds provide sufficient conditions for relative arbitrage to survive random costs. A limit-order-book cost proxy in conjunction with a Milstein scheme validates the theoretical order-of-magnitude estimates. Finally, we use intraday bid-ask spreads as a stand-in for cost volatility in a back-test of CRSP small-cap data (1994--2024). Despite experiencing larger declines during the 2008 and 2020 liquidity crises, diversity- and entropy-weighted portfolios continue to beat the value-weighted benchmark by 3.6 and 2.9 percentage points annually, respectively, after cost deduction.

Problem

• Extend stochastic portfolio theory to stochastic transaction costs
• Derive cost-adjusted relative wealth bounds for arbitrage strategies
• Validate theory with empirical data on small-cap portfolios

Innovation

• Extends SPT with stochastic transaction costs modeling
• Derives closed-form bounds for cost-adjusted wealth
• Validates theory using limit-order-book cost proxy