Chasing price drains liquidity
Yizhou Cao, Yepeng Ding, Ruichao Jiang, Long Wen
TL;DR
The paper analyzes liquidity dynamics in Uniswap v3–style AMMs when liquidity chases the current price. Using two market models—a GBM exogenous price and a mean-reverting price tied to a CEX—it derives the liquidity SDE under a price-tracking strategy and shows deterministic exponential decay $L_t=L_0\exp\left(-\frac{\sigma^2 t}{8(\sqrt{\alpha}-1)}\right)$ in the GBM case. It then demonstrates that mean reversion can still lead to decay, but proposes an arbitrage-enabled liquidity-increasing strategy that can raise liquidity without compounding fees. Empirical estimation on crypto data supports the theoretical findings and highlights the practical challenges of maintaining liquidity when prices move, especially under continuous-price assumptions and during arbitrage events.
Abstract
Assuming that the price in a Uniswap v3 style Automated Market Maker (AMM) follows a Geometric Brownian Motion (GBM), we prove that the strategy that adjusts the position of liquidity to track the current price leads to a deterministic and exponentially fast decay of liquidity. Next, assuming that there is a Centralized Exchange (CEX), in which the price follows a GBM and the AMM price mean reverts to the CEX price, we show numerically that the same strategy still leads to decay. Last, we propose a strategy that increases the liquidity even without compounding fees earned through liquidity provision.