Paper: arXiv 2409.11339

Authors: Maxim Bichuch, Zachary Feinstein

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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a novel Black-Scholes-style derivative pricing framework for AMM liquidity tokens, resolving empirical arbitrage anomalies in Uniswap data. It features strong continuous-time stochastic calculus math alongside empirical validation on blockchain data, though full reproducible code is not explicitly linked in the text excerpt.

Research Flowchart

  flowchart TD
    A["Research Goal: Reconcile CPMM Liquidity Token Pricing with Continuous Delta Hedging"] --> B["Conceptualize Liquidity Token as a Financial Derivative"]
    B --> C["Inputs: Historical Asset Prices & On-Chain Pool Data"]
    C --> D["Calibrate Implied Volatility & Derive Black-Scholes-like Risk-Neutral Pricing Model"]
    D --> E["Execute Out-of-Sample Dynamic Delta Hedging Strategy"]
    E --> F["Outcomes: Fair Non-Market Valuation, Riskless Replicated Portfolio & AMM Design Insights"]