Liquidity provision of utility indifference type in decentralized exchanges

ArXiv ID: 2502.01931 “View on arXiv”

Authors: Unknown

Abstract

We present a mathematical formulation of liquidity provision in decentralized exchanges. We focus on constant function market makers of utility indifference type, which include constant product market makers with concentrated liquidity as a special case. First, we examine no-arbitrage conditions for a liquidity pool and compute an optimal arbitrage strategy when there is an external liquid market. Second, we show that liquidity provision suffers from impermanent loss unless a transaction fee is levied under the general framework with concentrated liquidity. Third, we establish the well-definedness of arbitrage-free reserve processes of a liquidity pool in continuous-time and show that there is no loss-versus-rebalancing under a nonzero fee if the external market price is continuous. We then argue that liquidity provision by multiple liquidity providers can be understood as liquidity provision by a representative liquidity provider, meaning that the analysis boils down to that for a single liquidity provider. Last, but not least, we give an answer to the fundamental question in which sense the very construction of constant function market makers with concentrated liquidity in the popular platform Uniswap v3 is optimal.

Keywords: Constant Function Market Maker, Liquidity Provision, Impermanent Loss, Arbitrage Strategy, Uniswap v3, Cryptocurrency

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats
• Why: The paper employs advanced continuous-time stochastic calculus, Legendre transforms, and rigorous theoretical proofs for arbitrage and impermanent loss, indicating high mathematical complexity. However, it lacks backtesting, code, or empirical data, focusing purely on theoretical derivations rather than implementation-heavy analysis.

  flowchart TD
    A["Research Goal: Analyze Liquidity Provision in DEXs<br/>CFMMs of Utility Indifference Type"] --> B["Methodology: Mathematical Formulation<br/>& Theoretical Analysis"]
    B --> C["Input: External Market Prices &<br/>Liquidity Pool Parameters"]
    C --> D["Computational Processes"]
    D --> D1["1. No-Arbitrage Conditions<br/>& Optimal Arbitrage Strategy"]
    D --> D2["2. Impermanent Loss Analysis<br/>w/ Concentrated Liquidity"]
    D --> D3["3. Continuous-Time Well-Definedness<br/>& Loss-Versus-Rebalancing"]
    D1 & D2 & D3 --> E["Key Findings: Uniswap v3 is optimal<br/>LPs act as representative agent"]