Authors:

Alvaro Cartea ´
Fayc¸al Drissia
Marcello Monga Authors: Authors: Alvaro Cartea ´
Fayc¸al Drissia
Marcello Monga Alvaro Cartea ´ Fayc¸al Drissia Marcello Monga Table Of Links Table Of Links Abstract Abstract 1. Introduction 1. Introduction 1. Introduction 2. Constant function markets and concentrated liquidity 2. Constant function markets and concentrated liquidity 2. Constant function markets and concentrated liquidity Constant function markets
Concentrated liquidity market Constant function markets Concentrated liquidity market 3. The wealth of liquidity providers in CL pools 3. The wealth of liquidity providers in CL pools 3. The wealth of liquidity providers in CL pools Position value
Fee income

Fee income: pool fee rate
Fee income: spread and concentration risk
Fee income: drift and asymmetry


Rebalancing costs and gas fees Position value Fee income

Fee income: pool fee rate
Fee income: spread and concentration risk
Fee income: drift and asymmetry Fee income: pool fee rate
Fee income: spread and concentration risk
Fee income: drift and asymmetry Fee income: pool fee rate Fee income: spread and concentration risk Fee income: drift and asymmetry Rebalancing costs and gas fees 4. Optimal liquidity provision in CL pools 4. Optimal liquidity provision in CL pools 4. Optimal liquidity provision in CL pools The problem
The optimal strategy
Discussion: profitability, PL, and concentration risk
Discussion: drift and position skew The problem The optimal strategy Discussion: profitability, PL, and concentration risk Discussion: drift and position skew 5. Performance of strategy 5. Performance of strategy 5. Performance of strategy Methodology
Benchmark
Performance results Methodology Benchmark Performance results 6. Discussion: modelling assumptions 6. Discussion: modelling assumptions 6. Discussion: modelling assumptions Discussion: related work Discussion: related work 7. Conclusions And References 7. Conclusions And References 7. Conclusions And References Abstract Abstract Constant product markets with concentrated liquidity (CL) are the most popular type of automated market makers. In this paper, we characterise the continuous-time wealth dynamics of strategic LPs who dynamically adjust their range of liquidity provision in CL pools. Their wealth results from fee income, the value of their holdings in the pool, and rebalancing costs. Next, we derive a self-financing and closed-form optimal liquidity provision strategy where the width of the LP’s liquidity range is determined by the profitability of the pool (provision fees minus gas fees), the predictable losses (PL) of the LP’s position, and concentration risk. Concentration risk refers to the decrease in fee revenue if the marginal exchange rate (akin to the midprice in a limit order book) in the pool exits the LP’s range of liquidity. When the drift in the marginal rate is stochastic, we show how to optimally skew the range of liquidity to increase 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 a significant loss, and to show that the out-of-sample performance of our strategy is superior to the historical performance of LPs in the pool we consider. Introduction Traditional electronic exchanges are organised around limit order books to clear demand and supply of liquidity. In contrast, the takers and providers of liquidity in constant function markets (CFMs) interact in liquidity pools; liquidity providers (LPs) deposit their assets in the liquidity pool and liquidity takers (LTs) exchange assets directly with the pool. At present, constant product markets (CPMs) with concentrated liquidity (CL) are the most popular type of CFM, with Uniswap v3 as a prime example; see Adams et al. (2021). In CPMs with CL, LPs specify the rate intervals (i.e., tick ranges) over which they deposit their assets, and this liquidity is counterparty to trades of LTs when the marginal exchange rate of the pool is within the liquidity range of the LPs. When LPs deposit liquidity, fees paid by LTs accrue and are paid to LPs when they withdraw their assets from the pool. The amount of fees accrued to LPs is proportional to the share of liquidity they hold in each liquidity range of the pool. Existing research characterises the losses of LPs, but does not offer tools for strategic liquidity provision. In this paper, we study strategic liquidity provision in CPMs with CL. We derive the continuous-time dynamics of the wealth of strategic LPs which consists of the position they hold in the pool (position value), fee income, and costs from repositioning their liquidity position. The width of the range where the assets are deposited affects the value of the LP’s position in the pool; specifically, we show that the predictable loss (PL) incurred by LPs increases as the width of the liquidity range decreases. PL measures the unhedgeable losses of LPs stemming from the depreciation of their holdings in the pool and from the opportunity costs from locking their assets in the pool; see Cartea et al. (2023b). PL is similar to loss-versus-rebalancing (LVR) in Milionis et al. (2022) which describes the losses in traditional CFMs when LPs hedge their price exposure in an exogenous market.1 Also, we show that fee income is subject to a tradeoff between the width of the LP’s liquidity range and the volatility of the marginal rate in the pool. More precisely, CL increases fee revenue when the rate is in the range of the LP, but also increases concentration risk. Concentration risk refers to the risk that the LP faces when her position is concentrated in narrow ranges; the LP stops collecting fees when the rate exits the range of her position. We derive an optimal dynamic strategy to provide liquidity in a CPM with CL. In our model, the LP maximises the expected utility of her terminal wealth, which consists of the accumulated trading fees and the gains and losses from the market making strategy. The dynamic strategy controls the width and the skew of liquidity that targets the marginal exchange rate. For the particular case of log-utility, we obtain the strategy in closed-form and show how the solution balances the opposing effects between PL and fee collection. When volatility increases, PL increases, so there is an incentive for the LP to widen the range of liquidity provision to reduce the strategy’s exposure to PL. In particular, in the extreme case of very high volatility, the LP must withdraw from the pool because the exposure to PL is too high. Also, when there is an increase in the potential provision fees that the LP may collect because of higher liquidity taking activity, the strategy balances two opposing forces. One, there is an incentive to increase fee collection by concentrating the liquidity of the LP in a tight range around the exchange rate of the pool. Two, there is an incentive to limit the losses due to concentration risk by widening the range of liquidity provision. Finally, when the drift of the marginal exchange rate is stochastic (e.g., a predictive signal), the strategy skews the range of liquidity to increase fee revenue, by capturing the LT trading flow, and to increase the position value, by profiting from the expected changes in the marginal rate. Finally, we use Uniswap v3 data to motivate our model and to test the performance of the strategy we derive. The LP and LT data are from the pool ETH/USDC (Ethereum and USD coin) between the inception of the pool on 5 May 2021 and 18 August 2022. To illustrate the performance of the strategy we use in-sample data to estimate model parameters and out-of-sample data to test the strategy. Our analysis of the historical transactions in Uniswap v3 shows that LPs have traded at a significant loss, on average, in the ETH/USDC pool. We show that the out-ofsample performance of our strategy is considerably superior to the average LP performance we observe in the ETH/USDC pool. Early works on AMMs are in Chiu and Koeppl (2019), Angeris et al. (2021), Lipton and Treccani (2021), Capponi and Jia (2021), Engel and Herlihy (2021a,b), Angeris et al. (2022); see also the recent review Biais et al. (2023). Capponi et al. (2023a) study price formation in AMMs. Some works in the literature study strategic liquidity provision in CFMs and CPMs with CL. Heimbach et al. (2022) discuss the tradeoff between risks and returns that LPs face in Uniswap v3, Cartea et al. (2023b) study the predictable losses of LPs in a continuous-time setup, Milionis et al. (2023) study the impact of fees on the profits of arbitrageurs in CFMs, Fukasawa et al. (2023) study the hedging of the impermanent losses of LPs, and Lı et al. (2023) study the economics of liquidity provision. Closest to our work are the models in Fan et al. (2021) and Fan et al. (2022) which focus on fee revenue and use approximation techniques to obtain dynamic strategies. Finally, there is a growing literature on AMM design for fair competition between LPs and LTs. Goyal et al. (2023) study an AMM with dynamic trading functions that incorporate beliefs of LPs, Lommers et al. (2023) study AMMs where the LP’s strategy adjusts dynamically to market information, and Cartea et al. (2023c) generalise CFMs and propose AMM designs where LPs express their beliefs and risk preferences; see also Bergault et al. (2024) and He et al. (2024). Our work is related to the algorithmic trading and optimal market making literature. Early works on liquidity provision in traditional markets are Ho and Stoll (1983), Biais (1993), and Avellaneda and Stoikov (2008) with extensions in many directions; see Cartea et al. (2014, 2017), Gueant ´ (2017), Bergault et al. (2021), Drissi (2022). We refer the reader to Cartea et al. (2015), Gueant ´ (2016), and Donnelly (2022) for a comprehensive review of algorithmic trading models for takers and makers of liquidity in traditional markets. Also, our work is related to those in Cartea et al. (2018), Barger and Lorig (2019), Cartea and Wang (2020), Donnelly and Lorig (2020), Forde et al. (2022), Bergault et al. (2022) which implement market signals in algorithmic trading strategies. The remainder of the paper proceeds as follows. Section 2 describes CL pools. Section 3 studies the continuous-time dynamics of the wealth of LPs as a result of the position value, the fee revenue, and rebalancing costs. In particular, we use Uniswap v3 data to study the fee revenue component of the LP’s wealth and our results motivate the assumptions in our model. Section 4 introduces our liquidity provision model and uses stochastic control to derive a closed-form optimal strategy. Next, we study how the strategy controls the width and the skew of the liquidity range as a function of the pool’s profitability, PL, concentration risk, and the drift in the marginal rate. Finally, Section 5 uses Uniswap v3 data to test the performance of the strategy and showcases its superior performance. This paper is available on arxiv under CC0 1.0 Universal license. This paper is available on arxiv under CC0 1.0 Universal license. available on arxiv rxiv

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