Predictable Losses of Liquidity Provision in Constant Function Markets and Concentrated Liquidity Markets

We introduce a new comprehensive and model-free measure for the unhedgeable and predictable loss (PL) incurred by liquidity providers in constant function markets (CFMs) and in concentrated liquidity markets. PL compares the value of the LP's holdings in the CFM liquidity pool (assuming no fee revenue) with that of a self-financing portfolio that (i) continuously replicates the dynamic holdings of the LP in the pool to offset the market risk of the LP's position, and (ii) invests in a risk-free account. We provide closed-form formulae for PL in CFMs with and without concentrated liquidity, and show that the losses stem from two sources: \textit{convexity cost}, which depends on liquidity taking activity and the convexity of the pool's trading function; and \textit{opportunity cost}, which is due to locking the LP's assets in the pool. For liquidity providers, PL is the appropriate measure to assess the cost of liquidity provision in CFMs, so that fees and compensation to LPs provide the right incentives for a well-functioning market. When prices form outside of the pool, we show that PL is reduced when liquidity taking is costly, i.e., when the convexity of the pool's trading function is high. On the other hand, when prices form in the pool, PL is reduced when liquidity taking is cheap, i.e., when the convexity of the trading function is low. Finally, we use Uniswap v3 and Binance transaction data to compute PL and fees collected by LPs and show that, at present, liquidity provision in CFMs is a loss-leading activity