We study the adverse selection cost incurred by liquidity providers (LPs) due to arbitrage in an automated market maker (AMM). We consider a model with trading fees (which introduce friction to arbitrage) and discrete Poisson block generation times. The "mispricing process" --- the difference between the AMM and market log-prices --- is a Markovian jump-diffusion process. We establish that this process is ergodic and identify its steady state distribution in closed form. We are then able to compute the expected instantaneous rate of arbitrage profit in closed form.

When trading fees are low, in the fast block asymptotic regime, the impact of fees takes a particularly simple form: fees simply scale down arbitrage profits by the fraction of blocks which present profitable trading opportunities to arbitrageurs. This fraction decreases with an increasing block rate, and hence our model yields an important practical insight: faster blockchains have smaller LP losses. Further introducing gas fees (fixed costs) in our model, we show that, in the fast block asymptotic regime, lower gas fees lead to smaller losses for LPs.

Joint work with Jason Milionis (Columbia) and Ciamac Moallemi (Columbia).