The Omniscient, yet Lazy, Investor

ArXiv ID: 2510.24467 “View on arXiv”

Authors: Stanisław M. S. Halkiewicz

Abstract

We formalize the paradox of an omniscient yet lazy investor - a perfectly informed agent who trades infrequently due to execution or computational frictions. Starting from a deterministic geometric construction, we derive a closed-form expected profit function linking trading frequency, execution cost, and path roughness. We prove existence and uniqueness of the optimal trading frequency and show that this optimum can be interpreted through the fractal dimension of the price path. A stochastic extension under fractional Brownian motion provides analytical expressions for the optimal interval and comparative statics with respect to the Hurst exponent. Empirical illustrations on equity data confirm the theoretical scaling behavior.

Keywords: fractional Brownian motion, fractal dimension, optimal trading frequency, execution cost, Hurst exponent, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced mathematics including fractional Brownian motion, fractal dimension proofs, and closed-form optimizations, while also providing empirical illustrations on equity data that connect theory to real-world scaling behavior.

  flowchart TD
    A["Research Goal<br>Formalize 'Omniscient, yet Lazy' Investor<br>Optimize Trading under Friction"] --> B["Methodology: Deterministic Model"]
    B --> C["Derive Closed-Form Expected Profit<br>Linking Frequency, Cost, & Path Roughness"]
    C --> D["Prove Existence & Uniqueness<br>of Optimal Trading Frequency"]
    D --> E["Methodology: Stochastic Extension"]
    E --> F["Apply Fractional Brownian Motion<br>with Hurst Exponent H"]
    F --> G["Computational Process<br>Analytical Expressions & Empirical Scaling"]
    G --> H["Key Outcomes<br>Optimal Interval in terms of H & Fractal Dimension<br>Validated on Equity Data"]