IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2310.09782.html

All AMMs are CFMMs. All DeFi markets have invariants. A DeFi market is arbitrage-free if and only if it has an increasing invariant

Author

Listed:
  • Roger Lee

Abstract

In a universal framework that expresses any market system in terms of state transition rules, we prove that every DeFi market system has an invariant function and is thus by definition a CFMM; indeed, all automated market makers (AMMs) are CFMMs. Invariants connect directly to arbitrage and to completeness, according to two fundamental equivalences. First, a DeFi market system is, we prove, arbitrage-free if and only if it has a strictly increasing invariant, where arbitrage-free means that no state can be transformed into a dominated state by any sequence of transactions. Second, the invariant is, we prove, unique if and only if the market system is complete, meaning that it allows transitions between all pairs of states in the state space, in at least one direction. Thus a necessary and sufficient condition for no-arbitrage (respectively, for completeness) is the existence of the increasing (respectively, the uniqueness of the) invariant, which, therefore, fulfills in nonlinear DeFi theory the foundational role parallel to the existence (respectively, uniqueness) of the pricing measure in the Fundamental Theorem of Asset Pricing for linear markets. Moreover, a market system is recoverable by its invariant if and only if it is complete; and in all cases, complete or incomplete, every market system has, and is recoverable by, a multi-invariant. A market system is arbitrage-free if and only if its multi-invariant is increasing. Our examples illustrate (non)existence of various specific types of arbitrage in the context of various specific types of market systems -- with or without fees, with or without liquidity operations, and with or without coordination among multiple pools -- but the fundamental theorems have full generality, applicable to any DeFi market system and to any notion of arbitrage expressible as a strict partial order.

Suggested Citation

  • Roger Lee, 2023. "All AMMs are CFMMs. All DeFi markets have invariants. A DeFi market is arbitrage-free if and only if it has an increasing invariant," Papers 2310.09782, arXiv.org, revised Dec 2023.
    • Handle: RePEc:arx:papers:2310.09782
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2310.09782
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    2. 'Alvaro Cartea & Fayc{c}al Drissi & Marcello Monga, 2023. "Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision," Papers 2309.08431, arXiv.org, revised Jun 2024.
    3. Sondermann, Dieter, 1980. "Utility representations for partial orders," Journal of Economic Theory, Elsevier, vol. 23(2), pages 183-188, October.
    4. 'Alvaro Cartea & Fayc{c}al Drissi & Marcello Monga, 2023. "Decentralised Finance and Automated Market Making: Execution and Speculation," Papers 2307.03499, arXiv.org, revised Jun 2025.
    5. Maxim Bichuch & Zachary Feinstein, 2022. "Axioms for Automated Market Makers: A Mathematical Framework in FinTech and Decentralized Finance," Papers 2210.01227, arXiv.org, revised Feb 2025.
    6. Agostino Capponi & Ruizhe Jia, 2021. "The Adoption of Blockchain-based Decentralized Exchanges," Papers 2103.08842, arXiv.org, revised Jul 2021.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wen-Ting Wang, 2026. "Reinforcement Learning for Execution under Dynamic Fees in a Closed-Loop DEX Simulator," Papers 2607.10960, arXiv.org.
    2. Alif Aqsha & Philippe Bergault & Leandro S'anchez-Betancourt, 2025. "Equilibrium Reward for Liquidity Providers in Automated Market Makers," Papers 2503.22502, arXiv.org.
    3. Farbod Ghasemlu, 2026. "Optimal Dynamic Fees for Automated Market Makers: A Stochastic Control Approach to Loss-Versus-Rebalancing," Papers 2606.21769, arXiv.org.
    4. Philippe Bergault & S'ebastien Bieber & Leandro S'anchez-Betancourt, 2025. "Optimal Exit Time for Liquidity Providers in Automated Market Makers," Papers 2509.06510, arXiv.org, revised Oct 2025.
    5. Chu, Gang & Dowling, Michael & Li, Xiao, 2026. "Impermanent loss in cryptocurrency," Journal of International Money and Finance, Elsevier, vol. 160(C).
    6. Wenpin Tang, 2026. "Proof of Stake economy under centralized exchanges--a mean field model," Papers 2606.09003, arXiv.org.
    7. Hamed Amini & Maxim Bichuch & Zachary Feinstein, 2023. "Decentralized Prediction Markets and Sports Books," Papers 2307.08768, arXiv.org, revised Jan 2025.
    8. Jason Milionis & Ciamac C. Moallemi & Tim Roughgarden, 2023. "Complexity-Approximation Trade-offs in Exchange Mechanisms: AMMs vs. LOBs," Papers 2302.11652, arXiv.org, revised Apr 2023.
    9. Jimmy Risk & Shen-Ning Tung & Tai-Ho Wang, 2025. "Dynamics of Liquidity Surfaces in Uniswap v3," Papers 2509.05013, arXiv.org.
    10. Alan Beggs, 2021. "Afriat and arbitrage," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(2), pages 167-176, October.
    11. Nicola Jean & Giacomo Le Pera & Lorenzo Giada & Claudio Nordio, 2025. "A Framework for Waterfall Pricing Using Simulation-Based Uncertainty Modeling," Papers 2507.13324, arXiv.org.
    12. Alessandro Doldi & Marco Frittelli & Marco Maggis, 2025. "Collective completeness and pricing hedging duality," Mathematics and Financial Economics, Springer, volume 19, number 6, March.
    13. Jean-Yves Datey & Genevieve Gauthier & Jean-Guy Simonato, 2003. "The Performance of Analytical Approximations for the Computation of Asian Quanto-Basket Option Prices," Multinational Finance Journal, Multinational Finance Journal, vol. 7(1-2), pages 55-82, March-Jun.
    14. Marcelo F. Perillo, 2021. "Valuación de Títulos de Deuda Indexados al Comportamiento de un Índice Accionario: Un Modelo sin Riesgo de Crédito," CEMA Working Papers: Serie Documentos de Trabajo. 784, Universidad del CEMA.
    15. Chris Kenyon & Andrew Green, 2015. "Self-Financing Trading and the Ito-Doeblin Lemma," Papers 1501.02750, arXiv.org.
    16. Damir Filipovi'c & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Papers 1711.08043, arXiv.org, revised Jul 2019.
    17. Daniele Maria Di Nosse & Federico Gatta & Fabrizio Lillo & Sebastian Jaimungal, 2025. "Deviations from Tradition: Stylized Facts in the Era of DeFi," Papers 2510.22834, arXiv.org.
    18. Kimmel, Robert L., 2004. "Modeling the term structure of interest rates: A new approach," Journal of Financial Economics, Elsevier, vol. 72(1), pages 143-183, April.
    19. Enrico Biffis & Beniamin Goldys & Cecilia Prosdocimi & Margherita Zanella, 2015. "A pricing formula for delayed claims: Appreciating the past to value the future," Papers 1505.04914, arXiv.org, revised Jul 2022.
    20. Yeap, Claudia & Kwok, Simon S. & Choy, S. T. Boris, 2016. "A Flexible Generalised Hyperbolic Option Pricing Model and its Special Cases," Working Papers 2016-14, University of Sydney, School of Economics.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2310.09782. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.