IDEAS home Printed from
   My bibliography  Save this paper

Automated Market Making and Loss-Versus-Rebalancing


  • Jason Milionis
  • Ciamac C. Moallemi
  • Tim Roughgarden
  • Anthony Lee Zhang


We consider the market microstructure of automated market making and, specifically, constant function market makers (CFMMs), from the economic perspective of passive liquidity providers (LPs). In a frictionless, continuous-time Black-Scholes setting and in the absence of trading fees, we decompose the return of an LP into a instantaneous market risk component and a non-negative, non-decreasing, and predictable component which we call "loss-versus-rebalancing" (LVR, pronounced "lever"). Market risk can be fully hedged, but once eliminated, LVR remains as a running cost that must be offset by trading fee income in order for liquidity provision to be profitable. We show how LVR can be interpreted in many ways: as the cost of pre-commitment, as the time value for giving up future optionality, as the compensator in a Doob-Meyer decomposition, as an adverse selection cost in the form of the profits of arbitrageurs trading against the pool, and as an information cost because the pool does not have access to accurate market prices. LVR is distinct from the more commonly known metric of "impermanent loss" or "divergence loss"; this latter metric is more fundamentally described as "loss-versus-holding" and is not a true running cost. We express LVR simply and in closed-form: instantaneously, it is the scaled product of the variance of prices and the marginal liquidity available in the pool, i.e., LVR is the floating leg of a generalized variance swap. As such, LVR is easily calibrated to market data and specific CFMM structure. LVR provides tradeable insight in both the ex ante and ex post assessment of CFMM LP investment decisions, and can also inform the design of CFMM protocols.

Suggested Citation

  • Jason Milionis & Ciamac C. Moallemi & Tim Roughgarden & Anthony Lee Zhang, 2022. "Automated Market Making and Loss-Versus-Rebalancing," Papers 2208.06046,, revised Sep 2022.
  • Handle: RePEc:arx:papers:2208.06046

    Download full text from publisher

    File URL:
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    1. Guillermo Angeris & Tarun Chitra, 2020. "Improved Price Oracles: Constant Function Market Makers," Papers 2003.10001,, revised Jun 2020.
    2. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
    3. Peter P. Carr & Robert A. Jarrow, 2008. "The Stop-Loss Start-Gain Paradox and Option Valuation: A new Decomposition into Intrinsic and Time Value," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 4, pages 61-84, World Scientific Publishing Co. Pte. Ltd..
    4. Alex Evans, 2020. "Liquidity Provider Returns in Geometric Mean Markets," Papers 2006.08806,, revised Jul 2020.
    5. Agostino Capponi & Ruizhe Jia, 2021. "The Adoption of Blockchain-based Decentralized Exchanges," Papers 2103.08842,, 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. Guillermo Angeris & Alex Evans & Tarun Chitra, 2021. "Replicating Market Makers," Papers 2103.14769,
    2. Neelesh Tiruviluamala & Alexander Port & Erik Lewis, 2022. "A General Framework for Impermanent Loss in Automated Market Makers," Papers 2203.11352,
    3. Sam M. Werner & Daniel Perez & Lewis Gudgeon & Ariah Klages-Mundt & Dominik Harz & William J. Knottenbelt, 2021. "SoK: Decentralized Finance (DeFi)," Papers 2101.08778,, revised Sep 2022.
    4. Guillermo Angeris & Alex Evans & Tarun Chitra, 2021. "Replicating Monotonic Payoffs Without Oracles," Papers 2111.13740,
    5. Philippe Bergault & Louis Bertucci & David Bouba & Olivier Gu'eant, 2022. "Automated Market Makers: Mean-Variance Analysis of LPs Payoffs and Design of Pricing Functions," Papers 2212.00336,, revised Dec 2022.
    6. Andrea Barbon & Angelo Ranaldo, 2021. "On The Quality Of Cryptocurrency Markets: Centralized Versus Decentralized Exchanges," Papers 2112.07386,, revised Oct 2022.
    7. Johannes Rude Jensen & Mohsen Pourpouneh & Kurt Nielsen & Omri Ross, 2021. "The Homogenous Properties of Automated Market Makers," Papers 2105.02782,
    8. Guillermo Angeris & Akshay Agrawal & Alex Evans & Tarun Chitra & Stephen Boyd, 2021. "Constant Function Market Makers: Multi-Asset Trades via Convex Optimization," Papers 2107.12484,
    9. Guillermo Angeris & Alex Evans & Tarun Chitra, 2020. "When does the tail wag the dog? Curvature and market making," Papers 2012.08040,
    10. Lioba Heimbach & Eric Schertenleib & Roger Wattenhofer, 2022. "Risks and Returns of Uniswap V3 Liquidity Providers," Papers 2205.08904,, revised Sep 2022.
    11. Bujar Huskaj & Marcus Nossman, 2013. "A Term Structure Model for VIX Futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(5), pages 421-442, May.
    12. Alexander, Carol & Rauch, Johannes, 2021. "A general property for time aggregation," European Journal of Operational Research, Elsevier, vol. 291(2), pages 536-548.
    13. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    14. Liang Wang & Weixuan Xia, 2022. "Power‐type derivatives for rough volatility with jumps," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(7), pages 1369-1406, July.
    15. Jiahua Xu & Krzysztof Paruch & Simon Cousaert & Yebo Feng, 2021. "SoK: Decentralized Exchanges (DEX) with Automated Market Maker (AMM) Protocols," Papers 2103.12732,, revised Dec 2022.
    16. Maxim Bichuch & Zachary Feinstein, 2022. "Axioms for Automated Market Makers: A Mathematical Framework in FinTech and Decentralized Finance," Papers 2210.01227,
    17. Kraft, Holger, 2007. "Pitfalls in static superhedging of barrier options," Finance Research Letters, Elsevier, vol. 4(1), pages 2-9, March.
    18. López, Raquel & Esparcia, Carlos, 2021. "Analysis of the performance of volatility-based trading strategies on scheduled news announcement days: An international equity market perspective," International Review of Economics & Finance, Elsevier, vol. 71(C), pages 32-54.
    19. Yang, Ben-Zhang & Yue, Jia & Wang, Ming-Hui & Huang, Nan-Jing, 2019. "Volatility swaps valuation under stochastic volatility with jumps and stochastic intensity," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 73-84.
    20. Chen Mao & Guanqi Liu & Yuwen Wang, 2021. "A Closed-Form Pricing Formula for Log-Return Variance Swaps under Stochastic Volatility and Stochastic Interest Rate," Mathematics, MDPI, vol. 10(1), pages 1-17, December.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:2208.06046. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: .

    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: .

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.