Automated Market Makers Designs Beyond Constant Functions

Popular automated market makers (AMMs) use constant function markets (CFMs) to clear the demand and supply in the pool of liquidity. A key drawback in the implementation of CFMs is that liquidity providers (LPs) are currently providing liquidity at a loss, on average. In this paper, we propose new designs for AMMs where the price of liquidity is determined by LPs who post quotes around the marginal exchange rate of the pool. We develop two new models, the constant quote volume model (CQV) and the fractional quote volume model (FQV). In our models, the price of liquidity depends on the LP's (i) choice of impact functions that determine how liquidity taking orders impact the marginal rate, (ii) tolerance to inventory risk, and (iii) views on the demand for liquidity. We use stochastic control to obtain quotes that maximise the LP's expected profit, while controlling exposure to inventory risk. Our models admit closed-form solutions and are computationally efficient. We show that given a trading function of a CFM, there are impact functions and quotes in a CQV that replicate the marginal rate dynamics and the execution costs in the CFM; furthermore, we demonstrate that these quotes are suboptimal in the CQV pool. Finally, we use transaction data from Binance and Uniswap v3 to showcase optimal strategies for LPs and the profitability of liquidity provision in a CQV model.