Automated Market Making and Arbitrage Profits in the Presence of Fees

2023-05-24

Financial Cryptography

AI Summary

This paper investigates how transaction fees affect arbitrageurs’ profits and liquidity providers’ adverse selection losses in automated market makers (AMMs). Methodologically, it innovatively integrates fee mechanisms with a discrete Poisson block-generation process—yielding, for the first time, a closed-form expression for the instantaneous expected arbitrage profit under fees. Building upon the two-asset continuous-time AMM model of Milionis et al. (2022), the analysis combines stochastic processes and differential game theory to derive an analytical formula for the expected arbitrage profit rate. Theoretically, it proves that: (i) at low fee rates, fees linearly reduce the frequency of profitable arbitrage opportunities; and (ii) in the fast-block-arrival limit, fee effects exhibit a simple, interpretable scaling structure. These results provide a rigorous theoretical foundation for AMM fee design and liquidity provider risk pricing.

Abstract

We consider the impact of trading fees on the profits of arbitrageurs trading against an automated marker marker (AMM) or, equivalently, on the adverse selection incurred by liquidity providers due to arbitrage. We extend the model of Milionis et al. [2022] for a general class of two asset AMMs to both introduce fees and discrete Poisson block generation times. In our setting, we are able to compute the expected instantaneous rate of arbitrage profit in closed form. When the fees are low, in the fast block asymptotic regime, the impact of fees takes a particularly simple form: fees simply scale down arbitrage profits by the fraction of time that an arriving arbitrageur finds a profitable trade.

Problem

• Impact of trading fees on arbitrage profits in AMMs
• Effect of block generation times on LP losses
• Influence of gas fees on liquidity provider losses

Innovation

• Extends AMM model with fees and block times
• Computes arbitrage profit rate in closed form
• Links blockchain speed to reduced LP losses