Paper: arXiv 2610.23749

Authors: Dnyanesh Kulkarni, Revant Nayar

Abstract

Classical execution models assume linear impact, diffusive prices, and an exogenous horizon. We study execution under concave power-law impact and Hurst-scaled price fluctuations, and make two contributions. First, we choose the execution horizon by maximizing the Sharpe ratio of the trade’s realized P&L. Price moves during the execution window are part of that P&L, so executing more slowly reduces its variance, and because impact is concave it can also lower cost per share. Urgency comes only from alpha realized while the order is being worked. The problem is governed by the signs of two quantities: (i) the duration elasticity of impact $a$, which under a structural prior equals the impact-persistence gap $β- H$ and can take either sign; and (ii) the net urgency $ν$, the rate of alpha realization net of exposure relief. When $ν$ keeps its sign over the window, an interior optimum exists, to leading order in $T/t_f$, only when both are positive; otherwise the order should be worked at the fastest or slowest admissible pace. In the interior regime the first-order condition involves no impact coefficient, order size or volume, and the objective is flat: for a large-cap example, the window within 1% of maximal Sharpe spans more than an order of magnitude. Second, given the window, we derive the schedule shape by convex programming under an envelope cost. For any increasing per-share cost of delay $w(t)$, the KKT solution is $ϕ\propto (λ- w(t))_+^{1/β}$. For a Hurst-scaled envelope $w \propto t^H$, the normalized schedule depends only on $(β, H)$, has a closed form via the incomplete beta function, is front-loaded, and tapers smoothly to zero, saving 10-17% of envelope cost relative to a constant-rate (TWAP) schedule of equal duration. Whether it uses the whole window depends on a single dimensionless robustness level, for which we give a calibration.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical and novel approach to optimal execution, incorporating advanced concepts like concave impact and Hurst scaling. While it provides a ‘worked example’ and discusses calibration, the empirical validation and explicit reproducibility details (like code/data links) are not fully evident in the provided excerpt, suggesting a need for further empirical work to match its theoretical depth.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimize Trading Execution under Concave Impact & Hurst Scaling] --> B{Key Methodologies};

    B --> C[1. Optimal Execution Horizon Selection];
    C --> D[2. Optimal Schedule Shape Derivation];

    D --> E[Data/Inputs: Concave Power-Law Impact Model, Hurst-Scaled Price Fluctuations, Duration Elasticity of Impact 'a', Net Urgency 'ν', Per-share Cost of Delay w(t), Robustness Level];

    E --> F[Computational Processes: Maximize Sharpe Ratio, Convex Programming (KKT Solution)];

    F --> G[Key Outcomes:
        - Optimal Horizon based on signs of 'a' and 'ν'
        - Schedule φ ∝ (λ - w(t))_+^(1/β)
        - Front-loaded, smoothly tapering schedule via Incomplete Beta Function
        - 10-17% Cost Saving vs. TWAP
        - Calibration for Robustness Level
    ];