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.

翻译：去中心化金融（DeFi）是区块链技术中快速演进的领域，通过Web3应用实现了对金融服务模式的变革性突破。借助智能合约，DeFi允许开发者构建灵活且创新的金融工具。在众多流动性最高的DeFi基础协议中，去中心化交易所（DEX）的兑换协议（如Uniswap、Curve和Balancer）实现了快速的代币间兑换。然而，新型兑换机制也引入了市场低效性，可能被套利者系统性利用。本文聚焦于基于自动做市商（AMM）的兑换协议，其储备量乘积保持不变作为不变量。我们分析了套利者与AMM流动性池之间的交互机制，并基于实证池配置构建了数学模型。利用该模型，我们推导出流动性提供者（LP）与套利者联合收益的边界，提出一种估算无常损失（IL）发生前预期区块数的方法，并获得在单个区块内维持无常收益（IG）区间所需的最低池手续费下界。所提出的框架通过量化共生盈利区间，扩展了现有LP风险评估方法，为手续费选择提供了原则性依据，使LP与套利者的激励趋于一致，从而增强市场稳定性。