Optimal execution with deterministically time varying liquidity: well posedness and price manipulation

ArXiv ID: 2410.04867 “View on arXiv”

Authors: Unknown

Abstract

We investigate the well-posedness in the Hadamard sense and the absence of price manipulation in the optimal execution problem within the Almgren-Chriss framework, where the temporary and permanent impact parameters vary deterministically over time. We present sufficient conditions for the existence of a unique solution and provide second-order conditions for the problem, with a particular focus on scenarios where impact parameters change monotonically over time. Additionally, we establish conditions to prevent transaction-triggered price manipulation in the optimal solution, i.e. the occurence of buying and selling in the same trading program. Our findings are supported by numerical analyses that explore various regimes in simple parametric settings for the dynamics of impact parameters.

Keywords: Optimal Execution, Almgren-Chriss Model, Market Impact, Price Manipulation, Transaction Costs, Equities (Algorithmic Trading)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, focusing on well-posedness, existence/uniqueness of solutions, and analytical conditions using calculus of variations and differential equations, while empirical validation is limited to simple numerical experiments without real data or detailed backtesting.

  flowchart TD
    A["Research Goal:<br/>Well-Posedness & Price Manipulation<br/>in Optimal Execution w/ Varying Liquidity"] --> B["Methodology:<br/>Extension of Almgren-Chriss Model<br/>Deterministic Time-Varying Parameters"]
    B --> C{"Analysis 1:<br/>Hadamard Well-Posedness"}
    B --> D{"Analysis 2:<br/>Price Manipulation Conditions"}
    C --> E["Result:<br/>Existence & Uniqueness<br/>Second-Order Conditions"]
    D --> F["Result:<br/>Absence of Transaction-Triggered<br/>Price Manipulation"]
    E --> G["Outcome:<br/>Validated via Numerical Simulations<br/>for Monotonic & Parametric Regimes"]
    F --> G