Automated Market Makers: Mean-Variance Analysis of LPs Payoffs and Design of Pricing Functions

Paper: arXiv 2212.00336

Authors: Philippe Bergault, Louis Bertucci, David Bouba, Olivier Guéant

Abstract

With the emergence of decentralized finance, new trading mechanisms called Automated Market Makers have appeared. The most popular Automated Market Makers are Constant Function Market Makers. They have been studied both theoretically and empirically. In particular, the concept of impermanent loss has emerged and explains part of the profit and loss of liquidity providers in Constant Function Market Makers. In this paper, we propose another mechanism in which price discovery does not solely rely on liquidity takers but also on an external exchange rate or price oracle. We also propose to compare the different mechanisms from the point of view of liquidity providers by using a mean / variance analysis of their profit and loss compared to that of agents holding assets outside of Automated Market Makers. In particular, inspired by Markowitz’ modern portfolio theory, we manage to obtain an efficient frontier for the performance of liquidity providers in the idealized case of a perfect oracle. Beyond that idealized case, we show that even when the oracle is lagged and in the presence of adverse selection by liquidity takers and systematic arbitrageurs, optimized oracle-based mechanisms perform better than popular Constant Function Market Makers.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a novel mean-variance framework grounded in Markowitz portfolio theory to evaluate liquidity provider payoffs in AMMs. It successfully introduces and analyzes oracle-guided pricing functions, demonstrating their superiority over traditional CFMMs even under adverse selection and oracle lag.

Research Flowchart

  flowchart TD
    A[Goal: Compare LP Returns in CFMMs vs Oracle-Based AMMs] --> B[Input Data: External Oracle Price, Lag & Arbitrage Dynamics]
    B --> C[Methodology: Apply Markowitz Mean-Variance Analysis]
    C --> D[Model Idealized Perfect Oracle AMM]
    C --> E[Model Real-World Lagged Oracle with Adverse Selection]
    D & E --> F[Outcomes: Perfect Oracle Yields Markowitz Efficient Frontier]
    D & E --> G[Outcomes: Optimized Oracle AMMs Outperform Traditional CFMMs]