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The Geometry of Constant Function Market Makers

Author

Listed:
  • Guillermo Angeris
  • Tarun Chitra
  • Theo Diamandis
  • Alex Evans
  • Kshitij Kulkarni

Abstract

Constant function market makers (CFMMs) are the most popular type of decentralized trading venue for cryptocurrency tokens. In this paper, we give a very general geometric framework (or 'axioms') which encompass and generalize many of the known results for CFMMs in the literature, without requiring strong conditions such as differentiability or homogeneity. One particular consequence of this framework is that every CFMM has a (unique) canonical trading function that is nondecreasing, concave, and homogeneous, showing that many results known only for homogeneous trading functions are actually fully general. We also show that CFMMs satisfy a number of intuitive and geometric composition rules, and give a new proof, via conic duality, of the equivalence of the portfolio value function and the trading function. Many results are extended to the general setting where the CFMM is not assumed to be path-independent, but only one trade is allowed. Finally, we show that all 'path-independent' CFMMs have a simple geometric description that does not depend on any notion of a 'trading history'.

Suggested Citation

  • Guillermo Angeris & Tarun Chitra & Theo Diamandis & Alex Evans & Kshitij Kulkarni, 2023. "The Geometry of Constant Function Market Makers," Papers 2308.08066, arXiv.org.
    • Handle: RePEc:arx:papers:2308.08066
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    References listed on IDEAS

    as
    1. Guillermo Angeris & Tarun Chitra, 2020. "Improved Price Oracles: Constant Function Market Makers," Papers 2003.10001, arXiv.org, revised Jun 2020.
    2. 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.
    3. Guillermo Angeris & Tarun Chitra & Alex Evans & Stephen Boyd, 2022. "Optimal Routing for Constant Function Market Makers," Papers 2204.05238, arXiv.org.
    4. Masaaki Fukasawa & Basile Maire & Marcus Wunsch, 2023. "Weighted variance swaps hedge against impermanent loss," Quantitative Finance, Taylor & Francis Journals, vol. 23(6), pages 901-911, June.
    5. Jason Milionis & Ciamac C. Moallemi & Tim Roughgarden, 2023. "A Myersonian Framework for Optimal Liquidity Provision in Automated Market Makers," Papers 2303.00208, arXiv.org, revised Nov 2023.
    Full references (including those not matched with items on IDEAS)

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