From Impermanent Loss to Sustainable Gain: Quantifying Profitability Zones for Liquidity Providers on DEX

2026-04-30

AI Summary

This study addresses the lack of a quantitative framework for analyzing impermanent loss risk faced by liquidity providers in automated market maker (AMM) protocols and the associated profit distribution with arbitrageurs. By constructing an empirical pool–based mathematical model that integrates on-chain data and probabilistic analysis, this work is the first to characterize the symbiotic profit region shared by both parties, derive their joint profit bounds, and quantify the probability and duration of entering the impermanent loss regime. Key contributions include a target-probability–based method for computing the lower bound of trading fees, along with estimates of the expected number of blocks before impermanent loss occurs and the minimum fee rate required to sustain positive returns. These results provide theoretical foundations for incentive alignment and market stability in AMM-based decentralized exchanges.

Abstract

Decentralized Finance (DeFi) is a rapidly evolving segment of blockchain technology that enables a transformative approach to financial services through Web3 applications. By leveraging smart contracts, DeFi allows developers to build flexible and innovative financial instruments. Among the most prominent DeFi primitives by liquidity are decentralized exchange~(DEX) swap protocols~(such as Uniswap, Curve, and Balancer) that facilitate fast token-to-token exchanges. However, new exchange mechanisms also introduce new market inefficiencies that can be systematically exploited by arbitrageurs. This paper focuses on swap protocols based on the Automated Market Maker~(AMM), where the product of reserves is preserved as an invariant. We analyze the interaction between arbitrageurs and AMM liquidity pools and develop a mathematical model grounded in empirical pool configurations. Using this model, we derive bounds on the joint revenue of liquidity providers~(LPs) and arbitrageurs, propose a method to estimate the expected number of blocks until the occurrence of Impermanent Loss~(IL), and obtain a lower bound on the pool fee required to achieve a fixed target probability of staying in the Impermanent Gain (IG) zone within a block. The proposed framework extends existing LP risk-assessment methodologies by quantifying symbiotic profitability zones, providing a principled basis for fee selection that aligns LP-arbitrageur incentives and enhances market stability.

Problem

• Impermanent Loss
• Liquidity Providers
• Automated Market Maker
• Decentralized Exchange
• Arbitrageurs

Innovation

• Automated Market Maker
• Impermanent Loss
• Liquidity Provider
• Arbitrageur
• Profitability Zone