Closed-form solutions for generic N-token AMM arbitrage

ArXiv ID: 2402.06731 “View on arXiv”

Authors: Unknown

Abstract

Convex optimisation has provided a mechanism to determine arbitrage trades on automated market markets (AMMs) since almost their inception. Here we outline generic closed-form solutions for $N$-token geometric mean market maker pool arbitrage, that in simulation (with synthetic and historic data) provide better arbitrage opportunities than convex optimisers and is able to capitalise on those opportunities sooner. Furthermore, the intrinsic parallelism of the proposed approach (unlike convex optimisation) offers the ability to scale on GPUs, opening up a new approach to AMM modelling by offering an alternative to numerical-solver-based methods. The lower computational cost of running this new mechanism can also enable on-chain arbitrage bots for multi-asset pools.

Keywords: Automated Market Maker (AMM), Arbitrage, Convex Optimization, Geometric Mean Market Maker, GPU Computing, Cryptocurrency (DeFi)

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats
  • Why: The paper presents advanced mathematical derivations for closed-form solutions using Lagrange multipliers and introduces a trade signature concept for N-token pools, indicating high mathematical complexity. However, the empirical validation relies on synthetic simulations and lacks backtest-ready code, datasets, or real-world implementation details, resulting in low empirical rigor.
  flowchart TD
    A["Research Goal: Find closed-form solutions<br>for N-token AMM arbitrage"] --> B["Methodology: Derive generic equations<br>for Geometric Mean Market Maker"]
    
    B --> C{"Data / Inputs"}
    C --> D1["Synthetic Data"]
    C --> D2["Historic Market Data"]
    C --> D3["AMM Pool Parameters<br>e.g., constant product k"]
    
    D1 & D2 & D3 --> E["Computational Process:<br>Parallel GPU Execution"]
    E --> F["Comparison: Closed-form vs.<br>Convex Optimization Solvers"]
    
    F --> G{"Key Findings / Outcomes"}
    G --> H1["Better arbitrage opportunities<br>identified"]
    G --> H2["Capitalises on opportunities<br>sooner (faster execution)"]
    G --> H3["Lower computational cost<br>enables on-chain bots"]