Trading Horizon and Execution Schedule under Concave Impact and Hurst Scaling

01 ABSTRACT

Classical execution models typically assume linear impact, diffusive prices, and an exogenous horizon. This paper selects the execution horizon by maximizing the Sharpe ratio of the trade's realized P&L under concave impact and Hurst-scaled price fluctuations, and derives the optimal schedule shape for a given window. The authors show that an interior optimal horizon exists when two key quantities are both positive; otherwise the order should be executed at the fastest or slowest admissible pace, and the schedule under a Hurst-scaled envelope is front-loaded and tapers smoothly.

02 KEY FINDINGS

1. The execution horizon is chosen to maximize the Sharpe ratio of realized P&L, with price fluctuations and concave impact jointly affecting cost and risk.
2. An interior optimal horizon exists only when two quantities are both positive: impact persistence elasticity and net urgency rate; otherwise boundary speeds are optimal.
3. For a given window, the optimal schedule satisfies φ ∝ (λ − w(t))_+^{1/β}, has a closed form under a Hurst-scaled envelope, and saves envelope cost relative to TWAP.
4. In the interior regime, the first-order condition involves no impact coefficient, order size, or volume, and the objective is flat, implying a wide range of near-optimal windows.