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

We study the fee policy of a liquidity provider (LP) in a constant-product automated market maker (AMM) whose fee can be adjusted continuously, as enabled by programmable hooks. Building on the loss-versus-rebalancing (LVR) framework of Milionis et al. (2022) and its extension to nonzero fees by Milionis et al. (2024), we model the LP's wealth relative to the continuously rebalanced benchmark as a controlled process in which the fee governs two opposing forces: it raises revenue per uninformed trade while discouraging uninformed volume, and it widens the no-arbitrage band, which lowers the rate at which arbitrageurs extract value. Because the fee enters only the drift of relative wealth and never its diffusion, the LP's expected-utility problem reduces to an ergodic control problem whose solution is a pointwise volatility feedback. We prove that the growth-optimal fee is independent of the LP's wealth and of its constant relative risk aversion, that it collapses to a static constant when volatility is constant, and that it is strictly increasing in instantaneous variance, so that the optimal schedule is pro-cyclical. When volatility is stochastic, we characterise the optimal fee through a scalar ergodic Hamilton-Jacobi-Bellman equation and a linear Poisson equation, solved by a finite-difference scheme. We further show that the optimal fee is invariant to price jumps under logarithmic preferences, relate the optimal fee to a stylised model of competition among venues, and treat gas costs through an impulse-control dead-band. In a calibration to liquid large-capitalisation conditions, the optimal dynamic fee weakly dominates every static and volatility-linked heuristic fee on each simulated path, improving the LP's growth rate over the best static fee by a modest but uniformly positive margin, with a dead-band rendering gas costs negligible.