Current work introduces a theoretically grounded framework for estimating returns of liquidity providers (LPs) in automated market makers (AMM), with a focus on Uniswap V3 (Concentrated Liquidity AMM) and Uniswap V2 (Constant Product AMM). We explicitly model returns as the sum of fee income and impermanent loss, deriving theoretical mathematical expectations for both components under Geometric Brownian Motion price dynamics. While existing models often estimate fees as a simple function of historical trading volume or arbitrage dynamics entirely, our approach bridges market microstructure theory and decentralised finance mechanics to provide a practical valuable tool. The proposed framework decomposes fee income into ordinary user transactions and arbitrage flows, explicitly accounting for position activation probability within user-defined price ranges under stochastic price dynamics. To address real-world complexities – such as variable swap sizes and fluctuating pool liquidity – we introduce interpretable correction factors that adapt to market conditions. Validating through extensive Monte-Carlo simulations with realistic market microstructure, our model demonstrates robustness across diverse volatility regimes and pool configurations. For practitioners, the proposed framework enables LPs to optimise price ranges, compare fee tiers and predict returns under different market conditions. For researchers, it establishes a foundational methodology for analysing liquidity provision in AMMs that connects theoretical price dynamics to impermanent loss and fee generation. We further incorporate operational costs (blockchain gas fees) into the profitability analysis, demonstrating how transaction costs shift optimal position widths and rebalancing thresholds. The calibration constants governing the model are validated through regime-dependent estimation on historical on-chain data and out-of-sample testing, establishing measurable robustness. This work advances the understanding of sustainable liquidity provision in decentralised exchanges and offers practical recommendations for rational capital allocation in DeFi protocols.
VIasov et al. (Wed,) studied this question.