От Impermanent Loss к Impermanent Gain | Making Money Onchain

Янович Юрий, Руководитель группы и старший преподаватель, Сколтех

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1. Yury Yanovich Skoltech Impermanent Loss Gain 2026
2. Before We Start [1] R. Vlasov, V. Gorgadze, and A. Barger, “The Impact of the Exchange Fees on Impermanent Loss of Liquidity Providers for Conservative Automated Market Makers,” The Journal of The British Blockchain Association, 2025. [2] I. Melnikov, R. Vlasov, V. Gorgadze, A. Seoev, Y. Yanovich, “From Impermanent Loss to Sustainable Gain: Quantifying Profitability Zones for Liquidity Providers on DEX,” (under review), 2026. My research interest are not limited to the current topic! 2/19
3. Background: Impermanent Gain 01
4. Automated Market Maker (UniSwap, 2018) ETH/ERC20 Exchange: constant product formula X ⋅ Y = K= const X ⋅ Y = (X+ΔX)(Y-ΔY) ΔY = Y ⋅ ΔX/(X+ΔX) Token type Amount (old) Amount (new) A X X+ΔX B Y Y-ΔY 4/19
5. Impermanent Loss Impermanent loss = “HOLD strategy” – “On DEX strategy” Impermanent Loss as a Function of Price Change Token type Amount (old) Amount (new) Price (old) Price (new) A X X+ΔX Y/X (Y-ΔY)/(X+ΔX) B Y Y-ΔY 1 1 = 5/19
6. Impermanent Loss: Numerical Example New reserves Price slippage 6/19
7. Impermanent Gain Fee before swap: 𝜙1 = 1 − 𝛾1 Fee after swap: 𝜙2 = 1 − 𝛾2 7/19
8. Impermanent Gain: Assumptions The reserve ratio (x, y) satisfies the conservative function 𝐹 𝑥, 𝑦, 𝐼 𝑥, 𝑦 = 𝐹 𝑥, 𝑦, 𝐾 = 0, where 𝐼: 𝑅+ 2 → 𝑅+ is a homogeneous, concave, piecewise differentiable function that is increasing in each argument. Examples: * Uniswap V2 (CPMM) 𝐹𝑢 𝑥, 𝑦, 𝐾 = 𝑥𝑦 − 𝐾2 * Balancer (CGMM) 𝐹𝑏 𝑥, 𝑦, 𝐾 = 𝑥𝑤𝑥𝑦𝑤𝑦 − 𝐾𝑤𝑥+𝑤𝑦 * Curve (CHMM) 𝐹𝑐 𝑥, 𝑦, 𝐾 = 4𝐴 𝑥 + 𝑦 − 4𝐴 − 1 𝐾 − 𝐾3 4𝑥𝑦 8/19
9. Impermanent Gain: Theory Theorem [1]. Estimation of the size of the impermanent loss region up to first-order accuracy Δ𝑥𝑚𝑎𝑥 ≈ 2 ϕ1 + ϕ2 𝑝 𝑥0 , 𝐾0 𝜕𝑝 𝑥0 , 𝐾0 𝜕𝑥 . Impermanent Gain area 9/19
10. Impermanent Gain + Arbitrage 02
11. Rational User: Arbitrageur Change in user’s capital: 𝛿𝑃 Δx; Δy = Δx ∙ 1 − ϕ1 ∙ pA + Δy ∙ pB − α, α − network fee Arbitrageur’s goal: 𝛿𝑃 Δx, Δy → 𝑚𝑎𝑥 For constant fee optimal Δx, Δy: 𝑑(𝛿𝑃 Δx, Δy ) 𝑑(Δx) = −𝑝𝐴 + 𝑥 ∙ 𝑦 ∙ 1 − ϕ1 ∙ 𝑝𝐵 𝑥 + 1 − ϕ1 ∙ Δx = 0 Δx∗ = ൗ 𝑥 ∙ 𝑦 (1 − ϕ1 ) ∙ 𝑝𝐴 /𝑝𝐵 − 𝑥 (1 − ϕ1 ) ; Δy∗ = 𝑥 ∙ 𝑦 𝑥 + Δx∗ ∙ (1 − ϕ1 ) − 𝑦 11/19
12. Time to Mix: The Big Picture Market model: • LP and Arbitrageur • DEX pool and External pool (aka CEX) • Block-wise decisions LP and arbitrageur profit functions for Uniswap (solid) and Balancer with wx = 0.2, wy = 0.8 (dashed). The blue shaded area indicates the no- arbitrage zone, while green areas represent the joint profitability (IG) zone. 12/19
13. The Win-Win Zone: Theory Win-win zone (Uniswap): 𝑝 𝑐𝑒𝑥 /𝑝 𝑑𝑒𝑥 ∈ [1/𝜏, 𝜏], where 𝜏 = (𝛾₂(2 − 𝛾₁)²) / (𝛾₁(2𝛾₂ − 1)²). Probability of Impermanent Loss (Upper Bound) where Φ is the standard normal CDF. 13/19
14. The Win-Win Zone: Experiment CDF of the block number until first IL occurrence. The geometric distribution (GD) model with experimental PIL closely matches simulations. Minimum fee required to achieve target IL probability ξ at different volatility levels. Theoretical bounds (black lines) provide close approximations to simulation results (points with error bars). 14/19
15. Private Pool: Setup Private pool Public pool arbitrage Whitelist: our bot only Δx: p 0 Merged pool Δx: p 1 15/19
16. Private Pool: Experiment 16/19
17. From Impermanent Loss to Sustainable Design Key Results • Reframing IL: It's not just a loss; it's one side of a value transfer. A "Win-Win" zone exists. • Validation: Controlled experiments prove this zone is real and accessible with even tiny fees. • New Paradigm: Fees are coordination mechanisms, not just revenue. They align incentives, stabilize returns, and enable sustainable LP-arbitrageur symbiosis. Future Work • When is a private pool better in terms of APR? • IG by design via dynamic fees • DEX profit decomposition. 17/19
18. Social media accounts: Contacts: y.yanovich@skoltech.ru Tg: @YuryYa Blockchain seminar Yury Yanovich Research profile: