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

TL;DR

This paper derives an optimal hedge ratio for liquidity providers in AMMs that balances risk and liquidation constraints, validated through analytical formulas and simulations, with practical guidelines for DeFi applications.

Contribution

It introduces a closed-form solution for the optimal hedge ratio considering liquidation risk, combining analytical modeling with Monte Carlo validation in DeFi contexts.

Findings

Optimal hedge ratio between 50% and 70% for typical DeFi conditions

Analytical formulas accurately predict liquidation probabilities

Robust results across various parameter settings

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.