Quantifying Loss in Automated Market Makers

We consider the market microstructure of automated market making and, specifically, constant function market makers (CFMMs), from the economic perspective of passive liquidity providers (LPs). In a frictionless, continuous-time Black-Scholes setting and in the absence of trading fees, we decompose the return of an LP into a instantaneous market risk component and a non-negative, non-decreasing, and predictable component which we call "loss-versus-rebalancing'' (ŁVR, pronounced "lever''). Market risk can be fully hedged, but once eliminated, ŁVR remains as a running cost that must be offset by trading fee income in order for liquidity provision to be profitable. ŁVR is distinct from the more commonly known metric of "impermanent loss'' or "divergence loss''; this latter metric is more fundamentally described as "loss-versus-holding'' and is not a true running cost. We express ŁVR simply and in closed-form: instantaneously, it is the scaled product of the variance of prices and the marginal liquidity available in the pool. As such, ŁVR is easily calibrated to market data and specific CFMM structure. ŁVR provides tradeable insight in both the ex ante and ex post assessment of CFMM LP investment decisions, and can also inform the design of CFMM protocols. For a more complete version of this paper, please refer to https://arxiv.org/pdf/2208.06046.pdf.