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. Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options Document https://doi.org/ 10.4230/LIPIcs.AFT.2025.6 Export XML Export ACM-XML Export DOAJ-XML Export Schema.org Export BibTeX Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options Authors Srisht Fateh Singh , Reina Ke Xin Li , Samuel Gaskin , Yuntao Wu , Jeffrey Klinck , Panagiotis Michalopoulos , Zissis Poulos , Andreas Veneris Part of: Volume: 7th Conference on Advances in Financial Technologies (AFT 2025) Part of: Series: Leibniz International Proceedings in Informatics (LIPIcs) Part of: Conference: Advances in Financial Technologies (AFT) License: Creative Commons Attribution 4.0 International license Publication Date: 2025-10-06 PDF Files PDF LIPIcs.AFT.2025.6.pdf Filesize: 0.98 MB 23 pages HTML (experimental) LIPIcs.AFT.2025.6.html Document Identifiers DOI: 10.4230/LIPIcs.AFT.2025.6 URN: urn:nbn:de:0030-drops-247256 Related Versions Full Version https://arxiv.org/abs/2508.02971 Subject Classification ACM Subject Classification Applied computing → Economics Keywords blockchain decentralized finance automated market makers mathematical finance perpetual options continuous installments Metrics Access Statistics Total Accesses (updated on a weekly basis) 0 Document 0 Metadata 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. Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options. In 7th Conference on Advances in Financial Technologies (AFT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 354, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.AFT.2025.6 BibTex @InProceedings{singh_et_al:LIPIcs.AFT.2025.6, author = {Singh, Srisht Fateh and Li, Reina Ke Xin and Gaskin, Samuel and Wu, Yuntao and Klinck, Jeffrey and Michalopoulos, Panagiotis and Poulos, Zissis and Veneris, Andreas}, title = {{Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options}}, booktitle = {7th Conference on Advances in Financial Technologies (AFT 2025)}, pages = {6:1--6:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-400-0}, ISSN = {1868-8969}, year = {2025}, volume = {354}, editor = {Avarikioti, Zeta and Christin, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AFT.2025.6}, URN = {urn:nbn:de:0030-drops-247256}, doi = {10.4230/LIPIcs.AFT.2025.6}, annote = {Keywords: blockchain, decentralized finance, automated market makers, mathematical finance, perpetual options, continuous installments} } @InProceedings{singh_et_al:LIPIcs.AFT.2025.6, author = {Singh, Srisht Fateh and Li, Reina Ke Xin and Gaskin, Samuel and Wu, Yuntao and Klinck, Jeffrey and Michalopoulos, Panagiotis and Poulos, Zissis and Veneris, Andreas}, title = {{Modeling Loss-Versus-Rebalancing in Automated Market Makers via Continuous-Installment Options}}, booktitle = {7th Conference on Advances in Financial Technologies (AFT 2025)}, pages = {6:1--6:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-400-0}, ISSN = {1868-8969}, year = {2025}, volume = {354}, editor = {Avarikioti, Zeta and Christin, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, Basile Maire and Marcus Wunsch. Market neutral liquidity provision. Ledger, 9:73-88, November 2024. URL: https://doi.org/10.5195/ledger.2024.389 . Jason Milionis, Ciamac C Moallemi, Tim Roughgarden, and Anthony Lee Zhang. Automated market making and loss-versus-rebalancing. CoRR, 2024. URL: https://arxiv.org/abs/2208.06046 . Srisht Fateh Singh, Reina Ke Xin Li, Samuel Gaskin, Yuntao Wu, Jeffrey Klinck, Panagiotis Michalopoulos, Zissis Poulos, and Andreas Veneris. Modeling loss-versus-rebalancing in automated market makers via continuous-installment options, 2025. URL: https://arxiv.org/abs/2508.02971 . Anatoly Yakovenko. Solana: A new architecture for a high performance blockchain v0.

Automated market makers (AMMs) allocate fee revenue \textit{proportional} to the amount of liquidity investors deposit. In this paper, we study the economic consequences of the competition between passive liquidity providers (LPs) caused by this allocation rule. We employ a game-theoretic model in which $N$ strategic agents optimally provide liquidity and two types of liquidity traders trade. In this setting, we find that competition drives LPs to provide excess liquidity. Excess liquidity is costly as more capital is exposed to adverse selection costs. One of our main results is that the price of anarchy, defined over the liquidity provider performance, is $O(N)$, implying that the welfare loss scales linearly with the number of liquidity providers. This inefficient capital allocation is masked when considering the welfare of elastic liquidity traders as the total price of anarchy is $O(1)$. Since this result is driven by elastic liquidity traders benefiting from the liquidity provided because of inelastic liquidity traders, we show that different types of liquidity traders complement each other. Finally, we show that AMM designs that reduce the arbitrage intensity per unit of liquidity do increase utility for liquidity traders but importantly not for LPs nor do they necessarily decrease total arbitrage volume. [2402.18256] The Cost of Permissionless Liquidity Provision in Automated Market Makers The Cost of Permissionless Liquidity Provision in Automated Market Makers Julian Ma Robust 1 Introduction Automated Market Makers (AMMs) are one of the most used applications on blockchains. AMMs allow users to exchange tokens by trading with a smart contract on a blockchain. Liquidity Providers (LPs) deposit tokens into the liquidity pool of the AMM. LPs are passive and do not actively quote prices; instead, prices are determined by a predefined formula based on the current liquidity in the pool, referred to as the bonding curve. A distinct advantage of AMMs over, for example, electronic limit order books, which are the dominant trading mechanism in traditional finance, is that AMMs minimize the computational cost of token exchange. This is important because computation and storage in blockchains are costly. Moreover, AMMs are well-suited for illiquid assets since they do not require a dedicated, active market maker. A downside of AMMs is that passive LPs cannot update their quotes like in traditional finance. New information flows into the market via informed traders trading against the liquidity that the AMM holds in its liquidity pool. These adverse selection costs that passive LPs face in AMMs, known as Loss-Versus-Rebalancing (LVR) [ 12 ] , are a big problem for decentralized finance and recent AMM design focuses on mitigating LVR. Each player can deposit a fraction of their endowment into the exogenous investment opportunity, and the other part of their endowment is provided as liquidity to the AMM. Capital provided as liquidity is subject to adverse selection costs but also attracts fee revenues. Automated Market Maker. The automated market maker employs a bonding curve, f ​ ( x , y ) = k 𝑓 𝑥 𝑦 𝑘 f(x,y)=k where f : ℝ 2 → ℝ : 𝑓 → superscript ℝ 2 ℝ f:\mathbb{R}^{2}\rightarrow\mathbb{R} that determines the exchange rate between the two assets it holds, one risky asset X 𝑋 X and a numéraire Y 𝑌 Y . The reserves of the liquidity pool of the AMM are given by ( x , y ) 𝑥 𝑦 (x,y) . We define the amount of liquidity in the AMM as the pool value, following [ 12 ] V ​ ( P ) = P ​ x + y 𝑉 𝑃 𝑃 𝑥 𝑦 V(P)=Px+y (1) where P 𝑃 P is the external market price of the risky asset in terms of the numéraire. Fee Policy. Let f 𝑓 f denote the trading fee in the AMM. The level of fees has two consequences: it determines the revenues from trading for liquidity providers, and it determines the height of the adverse selection costs [ 11 ] . We use the arbitrage intensity per unit of liquidity per unit of time as defined in [ 11 ] as the expected adverse selection costs that liquidity providers face in this study. From the objective function of Problem 2 , we can see that the investor obtains a portion of the aggregate profits from the AMM proportional to the fraction of the total liquidity provided by the investor. We refer to this as the pro-rata allocation rule . This rule is a crucial driver of our results as it encourages competition between the agents in the game. Finally, we assume that an investor provides the smallest amount of liquidity if an investor is indifferent between multiple levels. Liquidity Provider Performance. We measure the liquidity provider performance in an automated market maker by the aggregate profits of all liquidity providers. We assume the liquidity providers hedge their liquidity provision positions, for example, by trading the rebalancing portfolio [ 12 ] , such that the payoffs of the liquidity providers are not dependent on price movements. Furthermore, we assume that there is a set of arbitrageurs that monitors the liquidity pool and executes on arbitrage opportunities, both at the beginning of a block as an arbitrage between the external market and the automated market maker and if liquidity traders move the price in the block away from the external market price, as a reverse trade arbitrage opportunity [ 3 ] . The potential of self-regulation for front-running prevention on dexes, 2023. Available at: https://arxiv.org/abs/2306.05756 . [9] Josojo. Mev capturing amm (mcamm), August 2022. URL: https://ethresear.ch/t/mev-capturing-amm-mcamm/13336/4 . [10] Conor McMenamin, Vanesa Daza, and Bruno Mazorra. Diamonds are forever, loss-versus-rebalancing is not. Cryptology ePrint Archive, Paper 2022/1420, 2022. https://eprint.iacr.org/2022/1420 . [11] Jason Milionis, Ciamac C. Moallemi, and Tim Roughgarden. Automated market making and arbitrage profits in the presence of fees. In Financial Cryptography and Data Security (FC 2024) , 2024. Forthcoming. [12] Jason Milionis, Ciamac C.