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A Derivative Pricing Perspective on Liquidity Tokens in Constant Product Market Makers

Maxim Bichuch, Zachary Feinstein

TL;DR

The paper addresses mispricing and hedging of liquidity tokens in constant product market makers (CPMMs) by reframing LP positions as derivative claims on underlying pool assets and developing a risk-neutral valuation framework. It derives a Bermudan-style pricing problem with a critical fee-threshold $\hat{\gamma}^*$, yielding a closed-form price $V_0(P_0)$ when $\hat{\gamma} \ge \hat{\gamma}^*$ and a corresponding set of Greeks for hedging. The authors further define implied volatility notions $\sigma^I$ and propose data-driven calibration $\sigma^M$ to obtain arbitrage-free prices, demonstrated on Uniswap-like data and showing reduced hedging error under re-pricing. Finally, they propose two novel AMM designs with liquidity-dependent minting/burning costs and fee rates to allow fully dynamic, arbitrage-free pricing of liquidity tokens, with implications for DeFi risk management and AMM design. Overall, the work provides a rigorous financial-derivatives framing for CPMMs, offers practical calibration methods, and suggests design directions to stabilize liquidity markets in DeFi.

Abstract

In decentralized finance, any individual can pool their assets into an automated market maker (AMM) -- herein we focus on the constant product market maker (CPMM) -- in exchange for a claim on a fraction of future pool assets and fees earned from the market making operations. This position is represented by a liquidity token, whose prevailing on-chain price is effectively the initial deposited assets. Though this price is well-defined, we treat the liquidity token as a derivative position in the prices of the underlying assets for the CPMM in order to deduce risk-neutral pricing and hedging formulas, not dissimilar to the Black-Scholes result. Adopting this perspective, in a frictionless environment, hedging the CPMM liquidity token under fair valuation should produce a riskless process, which therefore grows at the risk-free rate, something that is not seen in empirical case studies under the prevailing price. With our novel pricing formula, we construct a method to calibrate a volatility to data which provides an updated (non-market) valuation which is consistent with the (near-continuous) replication strategy out-of-sample. We conclude with a discussion of novel AMM design considerations motivated by this derivative-pricing perspective.

A Derivative Pricing Perspective on Liquidity Tokens in Constant Product Market Makers

TL;DR

The paper addresses mispricing and hedging of liquidity tokens in constant product market makers (CPMMs) by reframing LP positions as derivative claims on underlying pool assets and developing a risk-neutral valuation framework. It derives a Bermudan-style pricing problem with a critical fee-threshold , yielding a closed-form price when and a corresponding set of Greeks for hedging. The authors further define implied volatility notions and propose data-driven calibration to obtain arbitrage-free prices, demonstrated on Uniswap-like data and showing reduced hedging error under re-pricing. Finally, they propose two novel AMM designs with liquidity-dependent minting/burning costs and fee rates to allow fully dynamic, arbitrage-free pricing of liquidity tokens, with implications for DeFi risk management and AMM design. Overall, the work provides a rigorous financial-derivatives framing for CPMMs, offers practical calibration methods, and suggests design directions to stabilize liquidity markets in DeFi.

Abstract

In decentralized finance, any individual can pool their assets into an automated market maker (AMM) -- herein we focus on the constant product market maker (CPMM) -- in exchange for a claim on a fraction of future pool assets and fees earned from the market making operations. This position is represented by a liquidity token, whose prevailing on-chain price is effectively the initial deposited assets. Though this price is well-defined, we treat the liquidity token as a derivative position in the prices of the underlying assets for the CPMM in order to deduce risk-neutral pricing and hedging formulas, not dissimilar to the Black-Scholes result. Adopting this perspective, in a frictionless environment, hedging the CPMM liquidity token under fair valuation should produce a riskless process, which therefore grows at the risk-free rate, something that is not seen in empirical case studies under the prevailing price. With our novel pricing formula, we construct a method to calibrate a volatility to data which provides an updated (non-market) valuation which is consistent with the (near-continuous) replication strategy out-of-sample. We conclude with a discussion of novel AMM design considerations motivated by this derivative-pricing perspective.
Paper Structure (18 sections, 4 theorems, 16 equations, 4 figures, 1 table)

This paper contains 18 sections, 4 theorems, 16 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivating Example
  3. Background on Automated Market Maker
  4. Hedging a CPMM
  5. Constant Product Market Making
  6. Market Model
  7. Risk-Neutral Valuation
  8. Greeks
  9. Delta:
  10. Gamma:
  11. Vega:
  12. The Implied Volatility and Estimating the Arbitrage-Free Price of a Liquidity Token
  13. Implied Volatility
  14. Estimating the Arbitrage-Free Price of a Liquidity Token
  15. Discussion
  16. ...and 3 more sections

Key Result

Theorem 3.2

Fix the risk-free rate $r \geq 0$ and let the price process follow the geometric Brownian motion as in Assumption ass:gbm. Assume the current time ($t = 0$) is a block time. A risk-neutral investor will deposit liquidity in the constant product market maker if, and only if, where $\Phi$ is the CDF of the standard normal distribution. Provided $\hat{\gamma} \geq\hat{\gamma}^{*}$, the value of the

Figures (4)

  • Figure 1: Example \ref{['ex:motivating']}: Comparison of the discounted values of a 1 USDC investment in liquidity tokens and its delta hedged position from market creation (December 20, 2021) until June 21, 2025.
  • Figure 2: Valuation and vega with $P_0 = 1$, $\gamma = 5bps$ (i.e., $\hat{\gamma} = 5.0025bps$), $r = 5\%$ (annualized) and $\Delta t = 2$ seconds.
  • Figure 3: Example \ref{['ex:impliedvol']}: The shaded region indicates fee-volatility $(\gamma,\sigma)$ pairs that provide arbitrage opportunities.
  • Figure 4: Example \ref{['ex:repricing']}: Comparison of the delta hedged position of 1 USDC investment in a liquidity token under market pricing (blue dashed line) and with the calibrated arbitrage-free price (black solid line). Volatility is calibrated on January 2023 and applied throughout calendar year 2023.

Theorems & Definitions (20)

  • Remark 1
  • Example 2.3
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 10 more