Paper: arXiv 2506.02869

Abstract

Automated Market Makers (AMMs) are emerging as a popular decentralised trading platform. In this work, we determine the optimal dynamic fees in a constant function market maker. We find approximate closed-form solutions to the control problem and study the optimal fee structure. We find that there are two distinct fee regimes: one in which the AMM imposes higher fees to deter arbitrageurs, and another where fees are lowered to increase volatility and attract noise traders. Our results also show that dynamic fees that are linear in inventory and are sensitive to changes in the external price are a good approximation of the optimal fee structure and thus constitute suitable candidates when designing fees for AMMs.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper formulates and solves complex stochastic control problems with HJB equations and approximate closed-form solutions (high math). It also includes simulations comparing optimal policies to alternatives, validating the theoretical findings against data (moderate-to-high empirical rigor).

Research Flowchart

  flowchart TD
  A["Research Goal<br>Determine Optimal Dynamic Fees in AMMs"] --> B["Methodology<br>Stochastic Control Problem Formulation"]
  B --> C{"Data/Inputs<br>Market Price Dynamics & Trading Flow"}
  C --> D["Computational Process<br>Approximate Closed-Form Solutions"]
  D --> E{"Findings/Outcomes"}
  
  E --> F["Regime 1: High Fees<br>Deter Arbitrageurs"]
  E --> G["Regime 2: Low Fees<br>Attract Noise Traders"]
  E --> H["Optimal Structure<br>Linear in Inventory & Sensitive to External Price"]

Cited References (45)

  1. Equilibrium Reward for Liquidity Providers in Automated Market Makers — → 2503.22502
  2. A unifying framework for generalised Bayesian online learning in non-stationary environments (2025)
  3. Efficient online bayesian inference for neural bandits (2022)
  4. am-AMM: An Auction-Managed Automated Market Maker (2025) — link
  5. Automated Market Making and Arbitrage Profits in the Presence of Fees — → 2305.14604
  6. Automated Market Making and Loss-versus-Rebalancing (2022) — link
  7. Decentralized Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision (2024)
  8. Predictable Losses of Liquidity Provision in Constant Function Markets and Concentrated Liquidity Markets (2023)
and 37 more
  1. Execution and Statistical Arbitrage with Signals in Multiple Automated Market Makers (2023)
  2. Weighted Variance Swaps Hedge Against Impermanent Loss (2023)
  3. Optimal Fees for Geometric Mean Market Makers (2021)
  4. A Note on Optimal Fees for Constant Function Market Makers (2021)
  5. QLAMMP: A Q-Learning Agent for Optimizing Fees on Automated Market Making Protocols (2023)
  6. The Adoption of Blockchain-based Decentralized Exchanges (2021) — link
  7. Finding the Right Curve: Optimal Design of Constant Function Market Makers (2023)
  8. A Theory of Automated Market Makers in DeFi (2021)
  9. Improved Price Oracles: Constant Function Market Makers (2020)
  10. Market Making via Reinforcement Learning (2018)
  11. Macroscopic Market Making Games — → 2406.05662
  12. Market Making with Exogenous Competition — → 2407.17393
  13. Price-aware automated market makers: Models beyond B rownian prices and static liquidity — → 2405.03496
  14. An Analysis of Uniswap markets (2021)
  15. Automated Market Makers: Mean-Variance Analysis of LPs Payoffs and Design of Pricing Functions (2024)
  16. Replicating Market Makers (2023)
  17. Strategic Bonding Curves in Automated Market Makers (2024)
  18. The Case for Variable Fees in Constant Product Markets: An Agent-Based Simulation (2023)
  19. Uniswap v2 Core (2020)
  20. Uniswap v3 Core (2021)
  21. Uniswap v4 Core (2024)
  22. The Need for Fees at a DEX: How Increases in Fees Can Increase DEX Trading Volume (2022)
  23. An Overview of Uniswap v4 for Researchers (2025)
  24. High-Frequency Trading in a Limit Order Book (2008)
  25. Algorithmic and High-Frequency Trading (2015)
  26. The financial mathematics of market liquidity. From optimal execution to market making (2016)
  27. Market Making with Fads, Informed, and Uninformed Traders — → 2501.03658
  28. Optimal Exiting for Liquidity Provision in Constant Function Market Makers (2025)
  29. A Structural Model of Automated Market Making (2023)
  30. Optimal Design of Automated Market Makers on Decentralized Exchanges — → 2404.13291
  31. Liquidity provision of utility indifference type in decentralized exchanges — → 2502.01931
  32. Model-Free Hedging of Impermanent Loss in Geometric Mean Market Makers with Proportional Transaction Fees (2024)
  33. Unified Approach for Hedging Impermanent Loss of Liquidity Provision — → 2407.05146
  34. ZeroSwap: Data-driven Optimal Market Making in DeFi (2023) — link
  35. Adaptive curves for optimally efficient market making — → 2406.13794
  36. Decentralised Finance and Automated Market Making: Execution and Speculation (2022)
  37. Option market-making and vol arbitrage (2025)

10 of 45 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.