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

ArXiv ID: 2507.09196 “View on arXiv”

Authors: Nader Karimi, Erfan Salavati

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.

Keywords: Stochastic portfolio theory, Transaction costs, Relative arbitrage, Limit order book, Liquidity shocks

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 6.8/10
• Quadrant: Holy Grail
• Why: The paper relies heavily on advanced stochastic calculus (Itô diffusions, Milstein scheme, Fernholz theory) and presents a non-trivial extension of stochastic portfolio theory to stochastic costs with formal proofs and inequalities. Empirically, it includes a 30-year backtest using CRSP data, an intraday cost proxy (bid-ask spreads), and quantitative performance metrics (2.9-3.6% annual alpha after costs), demonstrating substantial data and implementation effort.

  flowchart TD
    A["Research Goal: Does SPT relative arbitrage survive stochastic transaction costs?"] --> B["Methodology: Ito diffusion & Stochastic Costs"]
    B --> C["Data: CRSP Small-Cap 1994-2024 & Intraday Spreads"]
    C --> D["Computation: Limit-Order-Book Proxy & Milstein Scheme"]
    D --> E["Key Finding: 3.6% & 2.9% annual alpha after costs"]