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

TL;DR

This paper models liquidity tokens in constant product market makers as derivatives, deriving risk-neutral pricing and hedging formulas, and introduces a calibration method for volatility to improve valuation accuracy in decentralized finance.

Contribution

It presents a novel derivative pricing framework for liquidity tokens in AMMs, enabling better risk management and valuation calibration in DeFi markets.

Findings

Hedging the liquidity token should produce a riskless process in theory.

The new pricing formula allows calibration of volatility from data.

Updated valuations are consistent with replication strategies.

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.