Optimal Dynamic Fees for Automated Market Makers: A Stochastic Control Approach to Loss-Versus-Rebalancing

Battle Count: 81

Highly relevant to quantitative trading in DeFi and crypto markets. The paper provides a rigorous, implementable framework for dynamic fee setting in AMMs, directly applicable to Uniswap v4 hook-based fee policies. The pro-cyclical volatility feedback rule, the ergodic HJB characterization, and the calibration to liquid large-cap conditions offer actionable guidance for LPs and protocol designers. The connection to classical market-making literature (bid-ask spread analogy) and the treatment of gas costs via dead-bands make it practically deployable. However, the modest improvement over well-chosen static fees (5.7 bps) tempers the practical urgency, and the single-agent, non-strategic-arbitrageur assumptions limit direct applicability to competitive multi-venue environments.

The theoretical framework is mathematically sophisticated (ergodic HJB, Poisson equations, CIR process, impulse control), but the implementation recipe is parsimonious: a lookup table from filtered variance to fee, precomputed offline from the first-order condition. The numerical solution requires a 1D finite-difference scheme for the Poisson equation (tridiagonal system), which is straightforward. The main practical challenges are: (1) obtaining a manipulation-resistant external volatility signal, (2) periodic re-estimation of uninformed demand parameters, (3) implementing the dead-band for gas cost management, and (4) on-chain deployment via Uniswap v4 hooks. The single-agent formulation simplifies implementation relative to full equilibrium models.

The paper provides detailed calibration parameters (block rate, CIR parameters, uninformed flow parameters, fee cap), describes the finite-difference scheme for the Poisson equation, and specifies the simulation design (6000 common variance paths, 1.5-year horizon, full-truncation Euler scheme). However, code is only 'available from the author on request' rather than publicly hosted. The mathematical derivations are complete with proofs in appendices. The numerical scheme (tridiagonal system with upwind correction, reflecting/linear-growth boundary conditions) is well-described but not fully specified in code.