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Pricing and hedging for liquidity provision in Constant Function Market Making

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.

Pricing and hedging for liquidity provision in Constant Function Market Making

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.
Paper Structure (47 sections, 6 theorems, 91 equations, 15 figures, 1 table)

This paper contains 47 sections, 6 theorems, 91 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Intrinsic Liquidity and Liquidity Profile
  3. Dimensionally Consistent Liquidity
  4. Liquidity Profile and Reserves
  5. Liquidity Profile Value and Option Replication
  6. Replicating Market Makers
  7. Pricing and Hedging for General Liquidity Profiles
  8. Static Replication of Impermanent Loss by Vanilla Options
  9. Dynamic Decomposition: Loss-Versus-Rebalancing
  10. IL as Hedging Cost plus LVR
  11. Implications for LVR Design and Pricing
  12. Risk Neutral Pricing and Hedging for Impermanent Loss
  13. Pricing IL as a European-Style Contingent Claim
  14. Delta of IL
  15. Gamma of IL
  16. ...and 32 more sections

Key Result

Theorem 2.2

Let the bonding function $f(x,y)$ be smooth, strictly increasing, and convex in $(x,y)$. For the CFMM $f(x,y)=K$, the reserve levels $(x,y)$ are uniquely determined by the spot price $p$ and the local intrinsic liquidity $\ell$ via the following integral representations: where $\ell(q)$ is the intrinsic liquidity at price $q$ defined in eqn:ell.

Figures (15)

  • Figure 1: The expected P&L $v(\epsilon)$ defined in \ref{['eqn:optimal_epsilon']} as a function of the log-exit price $\epsilon$. Left (Case 1): A unique maximizer $\epsilon^* > 0$ exists because $\mu > r$. Middle (Case 2): A unique interior maximizer exists despite $\mu < r$, due to the interplay of fee income and last-passage statistics. Right (Case 3):$v(\epsilon)$ increases monotonically toward the supremum $\frac{\varphi}{r} = 0.5$ as $\epsilon \to \infty$, indicating an optimal strategy of never withdrawing.
  • Figure 2: IL contribution for the 30bp ETH/USDC pool (Nov 17, 2025 snapshot; representative expiry). The left column reports IL contribution; the remaining columns show the corresponding single-resolution IV outputs from the shared plotting routine.
  • Figure 3: IL contribution for the 5bp ETH/USDC pool (Nov 17, 2025 snapshot; representative expiry). The left column reports IL contribution; the remaining columns show the corresponding single-resolution IV outputs from the shared plotting routine.
  • Figure 4: Multi-resolution implied-volatility overlay for the 30bp ETH/USDC pool (Nov 17, 2025 snapshot). Curves correspond to $n\in\{1,3,6,12,N\}$, with $N$ the finest initialized-tick partition. Columns are quarterly expiries; rows are Black--Scholes IV (top) and normalized Bachelier IV $\bar{\sigma}_{\rm B}=\sigma_{\rm B}/P_0$ (bottom).
  • Figure 5: Multi-resolution implied-volatility overlay for the 5bp ETH/USDC pool (Nov 17, 2025 snapshot). Fine-scale oscillations are most visible at $n=N$, while coarser aggregations (e.g., $n=6$ and $n=12$) recover a stable macro smile/skew structure.
  • ...and 10 more figures

Theorems & Definitions (16)

  • Example 2.1: G3M Liquidity
  • Theorem 2.2: Canonical Parametrization
  • proof
  • Example 2.3: Payoff of a Covered Call at Expiry
  • Example 2.4: Black--Scholes Covered Call
  • Example 3.1
  • Proposition 3.2
  • proof
  • Example 3.3: LVR-neutral profile in the CEV model
  • Example 3.4: Delta of Concentrated Liquidity
  • ...and 6 more