Pricing and hedging for liquidity provision in Constant Function Market Making

TL;DR

This paper introduces a new mathematical framework for CFMMs that simplifies pricing and risk management by using a coordinate system based on price and intrinsic liquidity, validated with empirical crypto data.

Contribution

It provides a canonical parametrization of CFMM bonding curves and links impermanent loss to vanilla options, enabling advanced risk analysis.

Findings

Linear dependence of reserves on liquidity

Implied volatility structure for liquidity profiles

Empirical validation with Uniswap v3 and Deribit data

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.