[Paper Review] Automated Market Making and Arbitrage Profits in the Presence of Fees

Jason Milionis, Ciamac C. Moallemi+1 more|arXiv (Cornell University)|May 24, 2023

Financial Markets and Investment Strategies16 citations

TL;DR

This paper develops a tractable model for arbitrage profits against an AMM when trading fees and discrete block arrivals are present, yielding closed-form steady-state mispricing and a semi-closed form rate for arbitrage profits.

ABSTRACT

We consider the impact of trading fees on the profits of arbitrageurs trading against an automated market maker (AMM) or, equivalently, on the adverse selection incurred by liquidity providers (LPs) due to arbitrage. We extend the model of Milionis et al. [2022] for a general class of two asset AMMs to introduce both 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 blocks which present profitable trading opportunities to arbitrageurs. This fraction decreases with an increasing block rate, hence our model yields an important practical insight: faster blockchains will result in reduced LP losses. Further introducing gas fees (fixed costs) in our model, we show that, in the fast block asymptotic regime, lower gas fees lead to smaller losses for LPs.

Motivation & Objective

• Quantify how trading fees affect arbitrage profits against AMMs (and adverse selection costs) in a two-asset CFMM setting.
• Incorporate discrete block arrivals via a Poisson process to reflect real-world blockchain trading constraints.
• Derive the stationary distribution of mispricing and the instantaneous rate of arbitrage profits under fees.
• Characterize how fees scale arbitrage profits in fast-block regimes and identify a phase transition induced by fees.

Proposed method

• Extend the Milionis et al. (2022) framework to include trading fees and Poisson-distributed arbitrageur arrivals.
• Model the mispricing process z_t as a Markovian jump-diffusion with a bounded jump at arrival times.
• Derive the stationary distribution pi(z) of mispricing and the trade-entrance probability P_trade from pi.
• Provide a semi-closed form expression for the instantaneous arbitrage rate ARB_bar in terms of an integral over A_+ and A_- functions.
• Specialize the results to constant product market makers to obtain explicit rates and compare to the frictionless LVR benchmark.

Experimental results

Research questions

• RQ1 How do trading fees modify arbitrage profits when arbitrageurs arrive at discrete times?
• RQ2 What is the stationary distribution of the mispricing process under fees and discrete arrivals?
• RQ3 What is the instantaneous rate of arbitrage profit in this fee-and-discrete-arrival setting, and how does it relate to the frictionless benchmark?
• RQ4 How do asymptotics behave in the fast-block regime, and is there a phase transition due to fees?

Key findings

• The mispricing process z_t is ergodic with a unique invariant distribution pi(z) that has three regions separated by the fee bounds ±γ.
• The steady-state probability of hitting a profitable trade is P_trade = 1 / (1 + sqrt(2λ) γ / σ).
• The instantaneous rate of arbitrage profit ARB_bar equals λ E_pi[A(P,z)], yielding ARB_bar = λ P_trade (√(2λ)/σ) ∫_0^∞ (A_+(P,x+γ) + A_-(P, -x-γ))/2 · e^{-√(2λ)x/σ} dx.
• In the fast-block regime (λ → ∞) and small γ, ARB_bar approximates the frictionless LVR scaled by P_trade, ARB_bar ≈ LVR × P_trade.
• For constant product AMMs, ARB_bar/V(P) is given explicitly and exhibits a phase transition: ARB_bar is finite when σ^2/8 < λ and infinite otherwise (with practical thresholds noted).
• Overall, fees effectively scale arbitrage profits by the fraction of time trades are profitable, acting like a time rescaling in fast-block settings.