Arbitrage with bounded Liquidity

ArXiv ID: 2507.02027 “View on arXiv”

Authors: Christoph Schlegel, Quintus Kilbourn

Abstract

We derive the arbitrage gains or, equivalently, Loss Versus Rebalancing (LVR) for arbitrage between \textit{“two imperfectly liquid”} markets, extending prior work that assumes the existence of an infinitely liquid reference market. Our result highlights that the LVR depends on the relative liquidity and relative trading volume of the two markets between which arbitrage gains are extracted. Our model assumes that trading costs on at least one of the markets is quadratic. This assumption holds well in practice, with the exception of highly liquid major pairs on centralized exchanges, for which we discuss extensions to other cost functions.

Keywords: arbitrage, Liquidity Valuation Adjustment (LVR), quadratic costs, market microstructure, imperfect liquidity, derivatives/options

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper presents advanced mathematical derivations, including stochastic calculus (GBM), quadratic cost approximations, and limit analysis to derive closed-form expressions for LVR. It is theoretically dense but lacks empirical validation, backtests, or implementation details, focusing entirely on model derivation and theoretical extensions.

  flowchart TD
    A["Research Goal: Derive arbitrage gains / LVR for<br>two imperfectly liquid markets (extending<br>reference market assumptions)"] --> B["Methodology: Quadratic Trading Cost Model<br>Assume costs quadratic on at least one market"]

    B --> C["Data/Inputs: Relative Liquidity Parameters<br>Relative Trading Volume between Markets"]

    C --> D["Computational Process: Modeling Arbitrage<br>Between Two Constrained Markets<br>Derive Closed-Form LVR Expression"]

    D --> E["Key Outcome: LVR depends on<br>Relative Liquidity & Trading Volume<br>Calibrated for Imperfect Liquidity"]