Optimal Dynamic Fees in Automated Market Makers

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.