💻 computer science

From Impermanent Loss to Sustainable Gain: Quantifying Profitability Zones for Liquidity Providers on DEX

This paper presents a mathematical model to quantify the symbiotic profitability between liquidity providers and arbitrageurs in Automated Market Maker (AMM) decentralized exchanges, deriving bounds on joint revenue and proposing fee structures to minimize Impermanent Loss while enhancing market stability.

Original authors: Ignat Melnikov, Roman Vlasov, Vladimir Gorgadze, Andrey Seoev, Yury Yanovich

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ignat Melnikov, Roman Vlasov, Vladimir Gorgadze, Andrey Seoev, Yury Yanovich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Tug-of-War on a Tightrope

Imagine a Decentralized Exchange (DEX) like a giant, automated lemonade stand run by a computer program (a "Smart Contract").

The Liquidity Providers (LPs) are the people who put the lemons and sugar into the stand so customers can buy lemonade. They hope to earn a small fee on every cup sold.
The Arbitrageurs are like professional shoppers who run between this lemonade stand and a nearby supermarket. If the supermarket sells lemons for $1 and the stand sells them for $1.10, the shopper buys at the stand and sells at the supermarket to make a quick profit.

The Problem:
Usually, when these professional shoppers (arbitrageurs) swoop in to fix price differences, they leave the lemonade stand in a slightly worse state than before. The stand owner (the LP) loses some value because the mix of lemons and sugar they have left over isn't as valuable as if they had just kept the ingredients in their own pantry. This loss is called "Impermanent Loss."

For a long time, people thought: "Arbitrageurs are bad for lemonade stand owners. They steal value."

The New Discovery:
This paper argues that this isn't always true. Sometimes, the relationship is a win-win. If the price difference between the stand and the supermarket is small, the shopper can still make a profit, but the fee they pay to the stand is enough to cover the loss the stand owner suffers. In this specific "Goldilocks zone," both the shopper and the stand owner walk away happy.

The authors call this the "Impermanent Gain" zone.

The Three Zones of the Lemonade Stand

The researchers mapped out three specific scenarios based on how different the prices are between the stand and the supermarket:

The "Do Nothing" Zone (No-Arbitrage):
- What happens: The prices are almost identical.
- Result: The professional shoppers don't bother coming because the profit isn't worth the gas fee (transaction cost). The lemonade stand owner sits quietly, neither gaining nor losing.
The "Sweet Spot" (Impermanent Gain Zone):
- What happens: There is a small price difference. The shoppers come in to make a small profit.
- Result: The shoppers pay a small fee to the stand. This fee is larger than the value the stand loses by changing its inventory.
- Analogy: It's like a neighbor paying you $5 to borrow your lawnmower for an hour. Even though your mower gets a little dirty (a small loss), the $5 you got more than covers the wear and tear. You both win.
The "Danger Zone" (Impermanent Loss Zone):
- What happens: The price difference is huge. The shoppers come in and make a massive trade.
- Result: The fee they pay is tiny compared to how much the stand's inventory value drops.
- Analogy: A giant corporation buys your entire stock of lemons at a low price and sells them high. You get a tiny fee, but you are left with a pile of sugar that is now worth much less than before. You lose money.

The Secret Sauce: The Fee

The paper asks a crucial question: How do we keep the lemonade stand in the "Sweet Spot" and out of the "Danger Zone"?

The answer is the Fee.

The researchers built a mathematical model to figure out exactly how high the fee needs to be to act as a safety net.

If the fee is too low (or zero): The shoppers come in constantly, even for tiny price differences. They make a lot of money, but the stand owner gets crushed by the constant inventory changes (Impermanent Loss).
If the fee is just right: It acts like a speed bump. It stops the shoppers from making tiny, risky trades that hurt the stand, but it still allows them to make trades that are profitable for everyone.

The Real-World Experiment

To prove this wasn't just math on paper, the authors set up a real experiment on the Polygon blockchain (a type of digital ledger).

Setup: They created two private lemonade stands (pools) with the same ingredients.
- Stand A: Charged a tiny fee (0.03%).
- Stand B: Charged zero fees.
The Twist: They controlled who could shop there. Only their own robot shopper was allowed to trade, so no outside competitors could interfere.

The Results:

Stand A (With Fee): The robot made fewer trips, but 76% of those trips were in the "Sweet Spot." The stand owner barely lost any value, and the robot still made a good profit. The total wealth created by both of them together was higher.
Stand B (Zero Fee): The robot made 50% more trips and made more profit for itself. However, the stand owner suffered a significant loss (Impermanent Loss) because the robot was trading too aggressively. The total wealth created by the system was actually lower.

The Takeaway

The paper concludes that fees aren't just a way to make money; they are a coordination tool.

Zero fees might look attractive because they encourage more trading, but they often lead to a "race to the bottom" where the traders win and the liquidity providers lose.
A small, calculated fee acts like a traffic light. It slows down the "toxic" trades that hurt the system and encourages trades where everyone benefits.

By using their mathematical model, the authors showed that if you set the fee correctly based on how volatile the market is, you can create a stable environment where liquidity providers can earn money without constantly fearing they will lose their investment to arbitrageurs. It turns a risky game of tug-of-war into a cooperative partnership.

1. Problem Statement

Decentralized Finance (DeFi) relies heavily on Automated Market Makers (AMMs) like Uniswap and Balancer, where Liquidity Providers (LPs) deposit assets into pools to facilitate trading. A primary risk for LPs is Impermanent Loss (IL), which occurs when the value of assets in the pool falls below the value of simply holding those assets due to price divergence.

The Conventional View: Arbitrageurs are typically viewed as the primary cause of IL. They exploit price discrepancies between the DEX and external markets (CEX), rebalancing the pool in a way that often leaves LPs with a less valuable asset mix.
The Gap: While recent research introduced the concept of Impermanent Gain (IG)—where fees earned exceed IL—the precise mathematical boundaries of this "win-win" zone were not fully characterized. Furthermore, there was a lack of a probabilistic framework to help LPs quantify the risk of IL based on market volatility and protocol fees.
The Core Tension: Trading fees (LP revenue) are generated predominantly by arbitrage activity. The same trades that cause IL also generate the fees intended to compensate for it. The paper seeks to mathematically define the conditions where this symbiotic relationship results in net profit for LPs.

2. Methodology

The authors developed a quantitative framework combining game-theoretic optimization, stochastic modeling, and empirical validation.

A. Joint Profitability Zones

The authors conceptualize the relationship between the external market price ( $p_{cex}$ ) and the DEX pool's marginal price ( $p_{dex}$ ) into three distinct zones based on the price ratio $R = p_{cex}/p_{dex}$ :

No-Arbitrage Zone: Price discrepancies are too small to cover transaction costs; no trades occur.
Impermanent Gain (IG) Zone: Moderate price discrepancies trigger arbitrage. Crucially, the trade volume is small enough that the fees collected by LPs exceed the IL caused by the rebalancing. Both the arbitrageur and LPs profit.
Impermanent Loss (IL) Zone: Large price discrepancies lead to large trade volumes. The resulting IL outweighs the fee income, causing a net loss for LPs.

B. Mathematical Modeling

Arbitrageur Optimization: The paper models the arbitrageur's profit maximization problem for Constant Function Market Makers (CFMMs). It derives closed-form solutions for optimal trade sizes ( $dx^*$ ) for Uniswap V2 and Balancer, accounting for token fees ( $\phi$ ).
IG Zone Derivation: By comparing the arbitrageur's optimal trade size against the threshold where LPs begin to suffer net loss, the authors derive closed-form analytical bounds for the price ratio interval $[1/\tau, \tau]$ $[1/ τ, τ]$ where IG occurs.
- For Uniswap V2, the boundary $\tau$ is derived explicitly based on fee parameters $\gamma_1, \gamma_2$ (where $\gamma = 1-\phi$ ).
Stochastic Risk Assessment: The external price dynamics are modeled as a Geometric Brownian Motion (GBM). This allows the calculation of the probability ( $P_{IL}$ ) that a price movement within a single block time ( $\Delta t$ ) will cross from the IG zone into the IL zone.
Fee Optimization: The framework inverts the risk model. Given a target maximum acceptable IL probability ( $\xi$ ), the model solves for the minimum fee parameters required to keep the system within the IG zone.

C. Experimental Validation

To validate the theory, the authors conducted a controlled on-chain experiment on the Polygon network using two private USDT/WMATIC pools:

Pool A: 0.03% swap fee.
Pool B: 0% swap fee.
Control Mechanism: Both pools utilized a whitelisted arbitrage bot to eliminate external MEV competition and isolate the LP-arbitrageur interaction. This allowed for a clean test of the theoretical "symbiotic" model.

3. Key Contributions

Formal Model of Joint Profitability: The paper formulates and solves the arbitrageur's optimization problem for CFMMs, explicitly deriving the LP's resulting profit function and identifying the exact conditions for symbiotic equilibrium.
Analytical Characterization of IG Zones: It provides the first closed-form expressions for the boundaries of the price ratio region where LPs experience net impermanent gain in Uniswap V2.
Probabilistic Risk Framework: It establishes a direct link between market volatility, protocol fees, and LP risk by modeling the probability of IL occurrence within a single block.
Fee Optimization Insight: The paper demonstrates how to set protocol fees as a constraint to meet specific LP risk tolerance levels, moving fee selection from heuristic to mathematical optimization.
Empirical Validation: Through on-chain experiments, the authors proved that 76% of arbitrage transactions in a fee-based pool occurred within the predicted win-win zone.

4. Key Results

Existence of Win-Win Zones: The theoretical bounds were validated. In the 0.03% fee pool (Pool A), 76% of arbitrage transactions occurred in the IG zone, while 24% were in the IL zone.
Fee Regularization Effect:
- Pool A (0.03% fee): LPs experienced minimal IL (ending at -1 MATIC). The fee acted as a "regularizer," filtering out low-profit, high-volatility arbitrage trades that would otherwise harm LPs.
- Pool B (0% fee): Generated 50% more arbitrage transactions and higher arbitrageur profits (45 MATIC vs. 36 MATIC). However, LPs suffered significant IL (down to -19 MATIC).
Total System Efficiency: Despite Pool B having more activity, Pool A generated higher total combined wealth (Arbitrageur Profit + LP Gain/Loss).
- Pool A Total: ~35 MATIC.
- Pool B Total: ~26 MATIC.
- This indicates that fees do not just redistribute value but optimize collective economic efficiency by preventing toxic, volatile arbitrage.
Risk Mitigation: The 0.03% fee reduced the LP's impermanent loss exposure significantly compared to the zero-fee pool, demonstrating that fees serve as a coordination mechanism to stabilize returns.

5. Significance and Implications

Reframing IL: The paper shifts the narrative of Impermanent Loss from an unavoidable hazard to a manageable parameter. It proves that IL is not inherent to arbitrage but is a function of price deviation magnitude relative to fees.
Protocol Design: Provides a mathematical tool for protocol designers to optimize fee structures based on measurable market risk and volatility, rather than competitive guessing.
LP Strategy: Offers LPs a principled basis for selecting pools and understanding the trade-offs between fee levels and risk exposure.
Market Stability: Suggests that "toxic" order flow (high-frequency, low-margin arbitrage that harms LPs) can be mitigated through appropriate fee structures, enhancing overall market stability.
Future Directions: The authors highlight the need for dynamic fee mechanisms that adjust to real-time volatility and the exploration of decentralized profit-sharing models for zero-fee pools to capture the efficiency of high-volume trading without sacrificing LP safety.

In conclusion, this work bridges rigorous theoretical modeling with practical on-chain experimentation, demonstrating that symbiotic profitability between LPs and arbitrageurs is not only theoretically possible but practically achievable through carefully calibrated fee structures.