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

TL;DR

This paper extends stochastic portfolio theory to include stochastic transaction costs, deriving bounds for relative wealth, and empirically demonstrates that diversified portfolios outperform benchmarks even during liquidity crises.

Contribution

It introduces a theoretical framework for portfolios under stochastic costs and validates it with empirical data, showing sustained outperformance during crises.

Findings

Diversity- and entropy-weighted portfolios outperform benchmarks by 3.6 and 2.9 percentage points annually after costs.

Theoretical bounds ensure relative arbitrage persists despite stochastic transaction costs.

Empirical validation using intraday bid-ask spreads confirms the model's order-of-magnitude estimates.

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.