Optimal Exit Time for Liquidity Providers in Automated Market Makers

Paper: arXiv 2509.06510

Authors: Philippe Bergault, Sébastien Bieber, Leandro Sánchez-Betancourt

Abstract

We study the problem of optimal liquidity withdrawal for a representative liquidity provider (LP) in an automated market maker (AMM). LPs earn fees from trading activity but are exposed to impermanent loss (IL) due to price fluctuations. While existing work has focused on static provision and exogenous exit strategies, we characterise the optimal exit time as the solution to a stochastic control problem with an endogenous stopping time. Mathematically, the LP’s value function is shown to satisfy a Hamilton-Jacobi-Bellman quasi-variational inequality, for which we establish uniqueness in the viscosity sense. To solve the problem numerically, we develop two complementary approaches: a Euler scheme based on operator splitting and a Longstaff-Schwartz regression method. Calibrated simulations highlight how the LP’s optimal exit strategy depends on the oracle price volatility, fee levels, and the behaviour of arbitrageurs and noise traders. Our results show that while arbitrage generates both fees and IL, the LP’s optimal decision balances these opposing effects based on the pool state variables and price misalignments. Lastly, we find the optimal fee level for the representative LP when they play the exit strategy we derived. This work contributes to a deeper understanding of dynamic liquidity provision in AMMs and provides insights into the sustainability of passive LP strategies under different market regimes.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical theory, including Hamilton-Jacobi-Bellman quasi-variational inequalities and viscosity solutions, indicating high math complexity. It also demonstrates empirical rigor through calibrated simulations, two numerical methods (Euler scheme and Longstaff-Schwartz), and detailed sensitivity analyses on fee levels, volatility, and trader behavior.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Find optimal exit time for LPs<br>in AMMs given IL and fees"] --> B["Model Formulation<br>Stochastic Control Problem<br>Endogenous Stopping Time"]
  B --> C{"Data & Calibration"}
  C --> C1["Oracle Price Volatility"]
  C --> C2["Fee Levels"]
  C --> C3["Trader Behavior<br>Arbitrageurs & Noise"]
  B --> D["Computational Methods"]
  D --> D1["HJB QVIE<br>Operator Splitting"]
  D --> D2["Longstaff-Schwartz<br>Regression"]
  D --> E["Simulation & Analysis"]
  E --> F["Key Findings & Outcomes"]
  F --> F1["Optimal exit balances<br>fees vs. impermanent loss"]
  F --> F2["Strategy depends on<br>pool state & misalignment"]
  F --> F3["Derived optimal fee level<br>for the representative LP"]

Cited References (114)

1. Price-aware automated market makers: Models beyond brownian prices and static liquidity — → 2405.03496
2. Automated market making: the case of pegged assets — → 2411.08145
3. Arbitrageurs’ profits, LVR, and sandwich attacks: batch trading as an AMM design response (2023) — link
4. An economic model of a decentralized exchange with concentrated liquidity (2025)
5. An analysis of Uniswap markets (2020)
6. The replicating portfolio of a constant product market (2020)
7. The replicating portfolio of a constant product market with bounded liquidity (2021)
8. Lectures on BSDEs, stochastic control, and stochastic differential games with financial applications (2016)

18 of 114 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.