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Pricing and hedging for liquidity provision in Constant Function Market Making

Author

Listed:
  • Jimmy Risk
  • Shen-Ning Tung
  • Tai-Ho Wang

Abstract

This paper develops a robust mathematical framework for Constant Function Market Makers (CFMMs) by transitioning from traditional token reserve analyses to a coordinate system defined by price and intrinsic liquidity. We establish a canonical parametrization of the bonding curve that ensures dimensional consistency across diverse trading functions, such as those employed by Uniswap and Balancer, and demonstrate that asset reserves and value functions exhibit a linear dependence on this intrinsic liquidity. This linear structure facilitates a streamlined approach to arbitrage-free pricing, delta hedging, and systematic risk management. By leveraging the Carr-Madan spanning formula, we characterize Impermanent Loss (IL) as a weighted strip of vanilla options, thereby defining a fine-grained implied volatility structure for liquidity profiles. Furthermore, we provide a path-dependent analysis of IL using the last-passage time. Empirical results from Uniswap v3 ETH/USDC pools and Deribit option markets confirm a volatility smile consistent with crypto-asset dynamics, validating the framework's utility in characterizing the risk-neutral fair value of liquidity provision.

Suggested Citation

  • Jimmy Risk & Shen-Ning Tung & Tai-Ho Wang, 2026. "Pricing and hedging for liquidity provision in Constant Function Market Making," Papers 2603.01344, arXiv.org.
    • Handle: RePEc:arx:papers:2603.01344
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    References listed on IDEAS

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    1. Alex Lipton & Vladimir Lucic & Artur Sepp, 2025. "Unified approach for hedging impermanent loss of liquidity provision," Digital Finance, Springer, vol. 7(3), pages 429-477, September.
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    3. Shen-Ning Tung & Tai-Ho Wang, 2024. "A mathematical framework for modelling CLMM dynamics in continuous time," Papers 2412.18580, arXiv.org.
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    5. Jimmy Risk & Shen-Ning Tung & Tai-Ho Wang, 2025. "Dynamics of Liquidity Surfaces in Uniswap v3," Papers 2509.05013, arXiv.org.
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