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Constant product markets with concentrated liquidity (CL) are the mostpopular type of automated market makers. In this paper, we characterise thecontinuous-time wealth dynamics of strategic LPs who dynamically adjust theirrange of liquidity provision in CL pools. Their wealth results from fee incomeand the value of their holdings in the pool. Next, we derive a self-financingand closed-form optimal liquidity provision strategy where the width of theLP's liquidity range is determined by the profitability of the pool (provisionfees minus gas fees), the predictable losses (PL) of the LP's position, andconcentration risk. Concentration risk refers to the decrease in fee revenue ifthe marginal exchange rate (akin to the midprice in a limit order book) in thepool exits the LP's range of liquidity. When the marginal rate is driven by astochastic drift, we show how to optimally skew the range of liquidity toincrease fee revenue and profit from the expected changes in the marginal rate.Finally, we use Uniswap v3 data to show that, on average, LPs have traded at asignificant loss, and to show that the out-of-sample performance of ourstrategy is superior to the historical performance of LPs in the pool weconsider.

Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision