Optimal Fees for Geometric Mean Market Makers (2104.00446v1)

Abstract: Constant Function Market Makers (CFMMs) are a family of automated market makers that enable censorship-resistant decentralized exchange on public blockchains. Arbitrage trades have been shown to align the prices reported by CFMMs with those of external markets. These trades impose costs on Liquidity Providers (LPs) who supply reserves to CFMMs. Trading fees have been proposed as a mechanism for compensating LPs for arbitrage losses. However, large fees reduce the accuracy of the prices reported by CFMMs and can cause reserves to deviate from desirable asset compositions. CFMM designers are therefore faced with the problem of how to optimally select fees to attract liquidity. We develop a framework for determining the value to LPs of supplying liquidity to a CFMM with fees when the underlying process follows a general diffusion. Focusing on a popular class of CFMMs which we call Geometric Mean Market Makers (G3Ms), our approach also allows one to select optimal fees for maximizing LP value. We illustrate our methodology by showing that an LP with mean-variance utility will prefer a G3M over all alternative trading strategies as fees approach zero.

• The paper demonstrates that optimal LP utility is achieved when trading fees approach zero without reaching it, effectively maintaining no-arbitrage boundaries.
• The study utilizes stochastic control theory to model dynamic interactions between fees and portfolio adjustments, revealing nuanced market behaviors.
• The findings suggest current DeFi fee structures need reevaluation to enhance LP compensation and reduce arbitrage costs, influencing future protocol designs.

Optimal Fees in Geometric Mean Market Makers: An Analytical Exploration

This paper addresses the complexities involved in determining optimal fees for Geometric Mean Market Makers (G3Ms), a prevalent subclass within Constant Function Market Makers (CFMMs), which includes platforms like Uniswap, Sushiswap, and Balancer. The research aims to achieve a balance between compensating Liquidity Providers (LPs) for arbitrage-induced losses and maintaining price accuracy within these decentralized financial protocols.

Analytical Framework and Methodology

The authors introduce a modeling framework centered around the stochastic control theory to analyze LP returns in a G3M with non-zero trading fees. This framework extends prior studies that generally assumed zero fees, thus disregarding the nuanced effects of transaction costs. The core of the analysis involves understanding the dynamic interplay between trading fees, portfolio weight adjustments, and the incentives for arbitrageurs.

To achieve this, the authors model the portfolio weights' dynamics as a stochastic process dependent on trading fees and market conditions, particularly the properties of geometric diffusion processes. A significant insight provided is that as fees approach zero, LPs achieve higher utility, drawing on established mean-variance utility frameworks for randomness in the asset price processes.

Key Results and Contributions

The paper presents several important findings about the impact of fees on LP value:

1. Boundary Conditions and No-Arbitrage Intervals: The paper calculates specific conditions under which fees become economically equivalent to zero when viewed in the boundary limit. It demonstrates that optimal LP strategies arise from choosing fees that allow them to maintain a desired no-arbitrage interval effectively.
2. Optimal Fee Structure: The theoretical model demonstrates that the optimal LP value is achieved as fees approach, but do not reach, zero. This theoretically contradicts assumptions that increasing fees always compensate LPs better. Importantly, this optimal fee structure mitigates the cost incurred from arbitrage activities.
3. Utility Maximization: Adapting the mean-variance utility of LPs, the analysis concludes that minimizing fees maximizes LP utility. This shapes a broader discourse on strategic fee setting in DeFi platforms where liquidity provision and trading activity must be harmonized.

Implications for Future Research and Practice

The conclusions drawn have significant implications for both theoretical research and practical application in the field of decentralized finance (DeFi). From a strategic perspective, the findings encourage a reevaluation of current fee structures in automated market makers to enhance LPs' returns. Practically, these insights might drive new design paradigms in CFMMs, influencing protocol governance and incentivization structures.

Future work could explore broader applications of these findings, including empirical validations in existing DeFi platforms and extending the analysis to other CFMM varieties with different invariants. Additionally, assessing the impact on market liquidity and potential systemic risks in extreme market conditions could offer more comprehensive insights into optimizing decentralized exchange infrastructures.

Conclusion

This paper significantly contributes to the understanding of how trading fees in G3Ms affect LP profitability under the influence of arbitrage. By employing a rigorous analytical approach, the authors provide a framework that intricately ties together financial optimization principles with the strategic design of DeFi protocols, yielding valuable guidelines for enhancing the efficiency and utility of decentralized exchanges.