Optimal Exit Time for Liquidity Providers in Automated Market Makers

2025-09-08

AI Summary

This paper investigates the optimal exit timing problem for liquidity providers (LPs) in automated market makers (AMMs), balancing fee revenue against impermanent loss risk. Methodologically, it formulates liquidity withdrawal as a stochastic control problem with endogenous stopping time, driven by price deviation and state variables, and derives the optimal stopping strategy. Theoretically, it establishes uniqueness of the value function in the viscosity solution sense. Numerically, the problem is solved via an operator-splitting Euler scheme combined with the Longstaff–Schwartz least-squares Monte Carlo method, calibrated to empirical market parameters. Results show that the optimal exit strategy critically depends on oracle volatility, fee rates, and arbitrageur responsiveness—thereby characterizing the sustainability boundary of passive liquidity provision across heterogeneous market regimes.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Data transparency and provenance

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. 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.

Problem

• Optimal liquidity withdrawal timing for automated market makers
• Balancing fee earnings against impermanent loss risks
• Solving stochastic control with endogenous stopping time

Innovation

• Stochastic control with endogenous stopping time
• Hamilton-Jacobi-Bellman quasi-variational inequality solution
• Euler scheme and Longstaff-Schwartz regression methods