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

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.

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

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.
Paper Structure (53 sections, 6 theorems, 39 equations, 4 figures, 13 tables)

This paper contains 53 sections, 6 theorems, 39 equations, 4 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Impermanent loss.
  4. Delta-neutral strategies.
  5. Liquidation risk in DeFi lending.
  6. First passage time.
  7. Model
  8. Price dynamics
  9. Constant-product AMM
  10. Hedging via borrowing
  11. Portfolio value
  12. Loan-to-value constraint
  13. Optimal Hedge Ratio
  14. Objective function
  15. Lognormal moment lemma
  16. ...and 38 more sections

Key Result

Lemma 1

Let $X_T = \ln(S_T^A/S_0^A)$ and $Y_T = \ln(S_T^B/S_0^B)$. Under zero-drift GBM ($\mu_A = \mu_B = 0$), the joint moment generating function is for all $a, b \in \mathbb{R}$.

Figures (4)

  • Figure 1: Sharpe ratio (left axis) and liquidation probability (right axis, green) as functions of the hedge ratio $h$. The optimal $h^{**} = 60\%$ (on the SR + tx basis; raw SR is marginally higher at $h = 0.65$) balances risk-adjusted return against liquidation risk.
  • Figure 2: Expected ROE (left axis) and standard deviation of ROE (right axis) as functions of the hedge ratio. The variance reduction from hedging is substantial up to $h \approx 0.70$, after which liquidation losses cause both return and risk to deteriorate.
  • Figure 3: Sharpe ratio vs. hedge ratio for different correlation values $\rho$. The optimal $h^{**}$ remains between 60% and 70% regardless of correlation.
  • Figure 4: Sharpe ratio vs. hedge ratio for different borrow rates $r_B$. Higher borrow costs reduce overall Sharpe and shift the optimal hedge ratio slightly downward.

Theorems & Definitions (18)

  • Remark 1: Accrued interest
  • Remark 2: Dollar vs. ROE-based Sharpe ratio
  • Lemma 1
  • proof
  • Proposition 2: Expected P&L
  • proof
  • Remark 3
  • Proposition 3: Variance decomposition
  • proof
  • Remark 4
  • ...and 8 more