Optimal Hedge Ratio for Delta-Neutral Liquidity Provision under Liquidation Constraints

Paper: arXiv 2603.19716

Authors: Atsushi Hane

Abstract

We study the problem of optimally hedging the price exposure of liquidity positions in constant-product automated market makers (AMMs) when the hedge is funded by collateralized borrowing. A liquidity provider (LP) who borrows tokens to construct a delta-neutral position faces a trade-off: higher hedge ratios reduce price exposure but increase liquidation risk through tighter collateral utilization. We model token prices as correlated geometric Brownian motions and derive the hedge ratio h that maximizes risk-adjusted return subject to a liquidation-probability constraint expressed via a first-passage-time bound. The unconstrained optimum h* admits a closed-form expression, but at h* the liquidation probability is prohibitively high. The practical optimum h** = min(h*, h_bar(alpha)) is determined by the binding liquidation constraint h_bar(alpha), which we evaluate analytically via the first-passage-time formula and confirm with Monte Carlo simulation. Simulations calibrated to on-chain data validate the analytical results, demonstrate robustness across realistic parameter ranges, and show that the optimal hedge ratio lies between 50% and 70% for typical DeFi lending conditions. Practical guidelines for rebalancing frequency and position sizing are also provided.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper tackles a practical DeFi problem with solid mathematical rigor, utilizing stochastic calculus and first-passage-time models to solve for optimal hedging under liquidation constraints. The analytical findings are effectively validated through Monte Carlo simulations calibrated to real on-chain data, striking a good balance between theoretical modeling and practical implementation.

Research Flowchart

  flowchart TD
    A["Research Question: Optimal Delta-Neutral<br>LP Hedging under Liquidation Constraints"] --> B["Inputs: Token Price Dynamics (GBM)<br>& On-Chain DeFi Lending Parameters"]
    B --> C["Derive Unconstrained Hedge Ratio (h*)<br>& First-Passage-Time Liquidation Bound"]
    C --> D["Evaluate Binding Liquidation Constraint h̄(α)<br>via Analytics & Monte Carlo Simulation"]
    D --> E["Practical Optimum: h** = min(h*, h̄(α))"]
    E --> F["Key Outcome: Optimal Hedge Ratio is 50%-70%<br>with Rebalancing & Sizing Guidelines"]