Mathematical Framework for On-chain Management of Concentrated Liquidity Position

This article investigates the mathematical framework with primitives for modifying liquidity positions in Uniswap V3, a groundbreaking DeFi protocol that introduced concentrated liquidity. Allocating capital within specific price ranges, Uniswap V3 significantly improves efficiency while presenting new opportunities in liquidity management. We propose a novel methodology for rebalancing liquidity positions into new price ranges, reinvesting earned fees to achieve compounding returns, and facilitating one-sided liquidity provision for single-token deposits. The theoretical justification of the proposed approach is rigorously established through Theorems 1-3, which provide a comprehensive mathematical framework. The empirical validation is conducted by analyzing a popular Ethereum pool, demonstrating the practical applicability of the derived theoretical methods and their potential to optimize DeFi protocols. Furthermore, we derive a value function for the atomic liquidity provision procedure with initially imbalanced assets. These theoretical contributions highlight the proposed framework's practical usability and open new horizons for future research on capital-efficient strategies and risk management in AMMs. The results are particularly valuable for DeFi researchers, traders, and smart contract developers, as they provide a mathematical foundation for the potential automation of on-chain liquidity management routine.