Characterizing Path-Independent Fees: A Route to Zero Impermanent Loss in CPMMs

TL;DR

This paper characterizes fee structures in CPMMs that ensure path independence, enabling zero impermanent loss for specific initial states and aiding protocol design.

Contribution

It derives a class of fee functions dependent only on the pool invariant and constructs a parametric family achieving zero impermanent loss for certain states.

Findings

Path-independent fee functions depend only on the pool invariant.

A parametric family of fee functions can achieve zero impermanent loss for specific initial states.

No universal fee function can eliminate impermanent loss across all states.

Abstract

Constant Product Market Makers use fees that are typically fixed proportions of trade size. When these fees are automatically reinvested into the pool, as in Uniswap~V2 and some designs of Uniswap V4, the final state after a trade can depend on how the trade is split into smaller transactions. This path dependence complicates the risk assessment for liquidity providers and affects composability guarantees. We characterize the functional class of fee structures that ensure path independence: the combined fee factor must depend only on the current pool invariant k=xy. For this class, we derive a system of ordinary differential equations governing pool dynamics and obtain a closed-form integral exchange formula. Within this class, we construct a parametric family of fee functions that achieve zero Impermanent Loss for a given initial pool state, and prove that no universal fee function can eliminate Impermanent Loss for all initial states simultaneously. We analyze implications for arbitrage windows and slippage, and validate our theory through controlled simulations. Our framework provides protocol designers with a principled approach to fee optimization that aligns liquidity provider and trader incentives while preserving composability.