Uniswap v3: Modeling Impermanent Loss and Fee Limits

This paper takes a close look at Uniswap V3 to make its rules and mechanisms clear, starting from the open source code. It carefully models how impermanent loss for liquidity providers evolves without assuming anything about trades or liquidity changes. The authors introduce the idea of a liquidity curve, show how payoffs at maturity can be built from those curves and outside tokens, and suggest this can be used to synthesize concave payoffs. Finally, they analyze the long-run behavior of collected swap fees under a mild market assumption, linking fee value to an integral of option-like prices as a small price-change threshold goes to zero.

What the study examined

This work revisits the protocol rules of Uniswap V3 using the project’s open source code as a starting point. The authors set out to build a clear, unambiguous description of how the system works and then use that foundation to study several quantitative questions that matter to users.

The study focused on three technical areas: the evolution of impermanent loss for liquidity providers without assuming anything about incoming trades or liquidity events; a new concept called a liquidity curve that links initial liquidity placement to payoffs at maturity; and an asymptotic analysis of collected swap fees under a mild assumption about how pool price tracks a latent market price.

Key findings

The authors present a detailed account of impermanent loss that traces how a provider’s position changes over time, explicitly avoiding common simplifications such as assuming constant liquidity. This gives a precise description of losses and gains tied to price moves and liquidity adjustments.

They introduce liquidity curves as a way to describe how concentrated liquidity can be allocated. For any given curve, the paper shows how to compute the resulting payoff at a future date after fees, and conversely demonstrates that any concave payoff can be replicated by choosing an initial liquidity curve together with holdings of tokens outside the pool. This observation opens a path to using the platform to synthesize option-like payoffs.

On collected fees, the authors perform an asymptotic analysis that removes common simplifying hypotheses. Under a mild assumption that the pool price follows a latent price process closely whenever the latter moves by a small percentage, they let that percentage threshold shrink to zero. In this limit, the accumulated fees are shown to match an integral expression involving call and put prices, making a direct link between fees earned and option-style valuations.

Why it matters

By grounding analysis in the protocol code and avoiding common approximations, the paper gives a firmer basis for understanding risks and returns for liquidity providers. The detailed impermanent loss model helps clarify what a provider experiences over time when prices move and liquidity is adjusted.

The liquidity curve concept provides a practical lens for thinking about how to design positions that produce desired payoffs, and the link to concave payoff replication suggests new ways to use the platform for structured payoff design. The asymptotic fee result connects earned fees to more familiar option prices, offering a theoretical bridge between decentralized exchange earnings and classical financial instruments.

Together, these contributions aim to improve quantitative understanding of the protocol and to suggest new possibilities for designing positions and interpreting fee income in market terms.