Automated Market Maker (AMM)

Optimal Exit Time for Liquidity Providers in Automated Market Makers ArXiv ID: 2509.06510 “View on arXiv” Authors: Philippe Bergault, Sébastien Bieber, Leandro Sánchez-Betancourt Abstract We study the problem of optimal liquidity withdrawal for a representative liquidity provider (LP) in an automated market maker (AMM). LPs earn fees from trading activity but are exposed to impermanent loss (IL) due to price fluctuations. While existing work has focused on static provision and exogenous exit strategies, we characterise the optimal exit time as the solution to a stochastic control problem with an endogenous stopping time. Mathematically, the LP’s value function is shown to satisfy a Hamilton-Jacobi-Bellman quasi-variational inequality, for which we establish uniqueness in the viscosity sense. To solve the problem numerically, we develop two complementary approaches: a Euler scheme based on operator splitting and a Longstaff-Schwartz regression method. Calibrated simulations highlight how the LP’s optimal exit strategy depends on the oracle price volatility, fee levels, and the behaviour of arbitrageurs and noise traders. Our results show that while arbitrage generates both fees and IL, the LP’s optimal decision balances these opposing effects based on the pool state variables and price misalignments. Lastly, we find the optimal fee level for the representative LP when they play the exit strategy we derived. This work contributes to a deeper understanding of dynamic liquidity provision in AMMs and provides insights into the sustainability of passive LP strategies under different market regimes. ...

Dynamics of Liquidity Surfaces in Uniswap v3 ArXiv ID: 2509.05013 “View on arXiv” Authors: Jimmy Risk, Shen-Ning Tung, Tai-Ho Wang Abstract This paper presents a comprehensive study on the empirical dynamics of Uniswap v3 liquidity, which we model as a time-tick surface, $L_t(x)$. Using a combination of functional principal component analysis (FPCA) and dynamic factor methods, we analyze three distinct pools over multiple sample periods. Our findings offer three main contributions: a statistical characterization of automated market maker liquidity, an interpretable and portable basis for dimension reduction, and a robust analysis of liquidity dynamics using rolling window metrics. For the 5 bps pools, the leading empirical eigenfunctions explain the majority of cross-tick variation and remain stable, aligning closely with a low-order Legendre polynomial basis. This alignment provides a parsimonious and interpretable structure, similar to the dynamic Nelson-Siegel method for yield curves. The factor coefficients exhibit a time series structure well-captured by AR(1) models with clear GARCH-type heteroskedasticity and heavy-tailed innovations. ...

Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options ArXiv ID: 2508.02971 “View on arXiv” Authors: Srisht Fateh Singh, Reina Ke Xin Li, Samuel Gaskin, Yuntao Wu, Jeffrey Klinck, Panagiotis Michalopoulos, Zissis Poulos, Andreas Veneris Abstract This paper mathematically models a constant-function automated market maker (CFAMM) position as a portfolio of exotic options, known as perpetual American continuous-installment (CI) options. This model replicates an AMM position’s delta at each point in time over an infinite time horizon, thus taking into account the perpetual nature and optionality to withdraw of liquidity provision. This framework yields two key theoretical results: (a) It proves that the AMM’s adverse-selection cost, loss-versus-rebalancing (LVR), is analytically identical to the continuous funding fees (the time value decay or theta) earned by the at-the-money CI option embedded in the replicating portfolio. (b) A special case of this model derives an AMM liquidity position’s delta profile and boundaries that suffer approximately constant LVR, up to a bounded residual error, over an arbitrarily long forward window. Finally, the paper describes how the constant volatility parameter required by the perpetual option can be calibrated from the term structure of implied volatilities and estimates the errors for both implied volatility calibration and LVR residual error. Thus, this work provides a practical framework enabling liquidity providers to choose an AMM liquidity profile and price boundaries for an arbitrarily long, forward-looking time window where they can expect an approximately constant, price-independent LVR. The results establish a rigorous option-theoretic interpretation of AMMs and their LVR, and provide actionable guidance for liquidity providers in estimating future adverse-selection costs and optimizing position parameters. ...

Optimal Dynamic Fees in Automated Market Makers ArXiv ID: 2506.02869 “View on arXiv” Authors: Unknown Abstract Automated Market Makers (AMMs) are emerging as a popular decentralised trading platform. In this work, we determine the optimal dynamic fees in a constant function market maker. We find approximate closed-form solutions to the control problem and study the optimal fee structure. We find that there are two distinct fee regimes: one in which the AMM imposes higher fees to deter arbitrageurs, and another where fees are lowered to increase volatility and attract noise traders. Our results also show that dynamic fees that are linear in inventory and are sensitive to changes in the external price are a good approximation of the optimal fee structure and thus constitute suitable candidates when designing fees for AMMs. ...

FLUXLAYER: High-Performance Design for Cross-chain Fragmented Liquidity ArXiv ID: 2505.09423 “View on arXiv” Authors: Xin Lao, Shiping Chen, Qin Wang Abstract Autonomous Market Makers (AMMs) rely on arbitrage to facilitate passive price updates. Liquidity fragmentation poses a complex challenge across different blockchain networks. This paper proposes FluxLayer, a solution to mitigate fragmented liquidity and capture the maximum extractable value (MEV) in a cross-chain environment. FluxLayer is a three-layer framework that integrates a settlement layer, an intent layer, and an under-collateralised leverage lending vault mechanism. Our evaluation demonstrates that FluxLayer can effectively enhance cross-chain MEV by capturing more arbitrage opportunities, reducing costs, and improving overall liquidity. ...

Automated Market Makers: A Stochastic Optimization Approach for Profitable Liquidity Concentration ArXiv ID: 2504.16542 “View on arXiv” Authors: Simon Caspar Zeller, Paul-Niklas Ken Kandora, Daniel Kirste, Niclas Kannengießer, Steffen Rebennack, Ali Sunyaev Abstract Concentrated liquidity automated market makers (AMMs), such as Uniswap v3, enable liquidity providers (LPs) to earn liquidity rewards by depositing tokens into liquidity pools. However, LPs often face significant financial losses driven by poorly selected liquidity provision intervals and high costs associated with frequent liquidity reallocation. To support LPs in achieving more profitable liquidity concentration, we developed a tractable stochastic optimization problem that can be used to compute optimal liquidity provision intervals for profitable liquidity provision. The developed problem accounts for the relationships between liquidity rewards, divergence loss, and reallocation costs. By formalizing optimal liquidity provision as a tractable stochastic optimization problem, we support a better understanding of the relationship between liquidity rewards, divergence loss, and reallocation costs. Moreover, the stochastic optimization problem offers a foundation for more profitable liquidity concentration. ...

Equilibrium Reward for Liquidity Providers in Automated Market Makers ArXiv ID: 2503.22502 “View on arXiv” Authors: Unknown Abstract We find the equilibrium contract that an automated market maker (AMM) offers to their strategic liquidity providers (LPs) in order to maximize the order flow that gets processed by the venue. Our model is formulated as a leader-follower stochastic game, where the venue is the leader and a representative LP is the follower. We derive approximate closed-form equilibrium solutions to the stochastic game and analyze the reward structure. Our findings suggest that under the equilibrium contract, LPs have incentives to add liquidity to the pool only when higher liquidity on average attracts more noise trading. The equilibrium contract depends on the external price, the pool reference price, and the pool reserves. Our framework offers insights into AMM design for maximizing order flow while ensuring LP profitability. ...

QubitSwap: The Informational Edge in Decentralised Exchanges ArXiv ID: 2504.06281 “View on arXiv” Authors: Unknown Abstract Decentralised exchanges (DEXs) have transformed trading by enabling trustless, permissionless transactions, yet they face significant challenges such as impermanent loss and slippage, which undermine profitability for liquidity providers and traders. In this paper, we introduce QubitSwap, an innovative DEX model designed to tackle these issues through a hybrid approach that integrates an external oracle price with internal pool dynamics. This is achieved via a parameter $z$, which governs the balance between these price sources, creating a flexible and adaptive pricing mechanism. Through rigorous mathematical analysis, we derive a novel reserve function and pricing model that substantially reduces impermanent loss and slippage compared to traditional DEX frameworks. Notably, our results show that as $z$ approaches 1, slippage approaches zero, enhancing trading stability. QubitSwap marks a novel approach in DEX design, delivering a more efficient and resilient platform. This work not only advances the theoretical foundations of decentralised finance but also provides actionable solutions for the broader DeFi ecosystem. ...

Better market Maker Algorithm to Save Impermanent Loss with High Liquidity Retention ArXiv ID: 2502.20001 “View on arXiv” Authors: Unknown Abstract Decentralized exchanges (DEXs) face persistent challenges in liquidity retention and user engagement due to inefficiencies in conventional automated market maker (AMM) designs. This work proposes a dual-mechanism framework to address these limitations: a ``Better Market Maker (BMM)’’, which is a liquidity-optimized AMM based on a power-law invariant ($X^nY = K$, $n = 4$), and a dynamic rebate system (DRS) for redistributing transaction fees. The segment-specific BMM reduces impermanent loss by 36% compared to traditional constant-product ($XY = K$) models, while retaining 3.98x more liquidity during price volatility. The DRS allocates fees ($γV$, $γ\in {“0.003, 0.005, 0.01"}$) with a rebate ratio $ρ\in [“0.3, 0.4”]$ to incentivize trader participation and maintain continuous capital injection. Simulations under high-volatility conditions demonstrate impermanent loss reductions of 36.0% and 40% higher user engagement compared to static fee models. By segmenting markets into high-, mid-, and low-volatility regimes, the framework achieves liquidity depth comparable to centralized exchanges (CEXs) while maintaining decentralized governance and retaining value within the cryptocurrency ecosystem. ...

Automated Market Making: the case of Pegged Assets ArXiv ID: 2411.08145 “View on arXiv” Authors: Unknown Abstract In this paper, we introduce a novel framework to model the exchange rate dynamics between two intrinsically linked cryptoassets, such as stablecoins pegged to the same fiat currency or a liquid staking token and its associated native token. Our approach employs multi-level nested Ornstein-Uhlenbeck (OU) processes, for which we derive key properties and develop calibration and filtering techniques. Then, we design an automated market maker (AMM) model specifically tailored for the swapping of closely related cryptoassets. Distinct from existing models, our AMM leverages the unique exchange rate dynamics provided by the multi-level nested OU processes, enabling more precise risk management and enhanced liquidity provision. We validate the model through numerical simulations using real-world data for the USDC/USDT and wstETH/WETH pairs, demonstrating that it consistently yields efficient quotes. This approach offers significant potential to improve liquidity in markets for pegged assets. ...