Complex DeFi network illustrating closed-form solutions for crypto arbitrage in automated market makers.

Unlock the Secrets of Crypto Arbitrage: How Closed-Form Solutions Are Changing the Game

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Lena Voss in Tech & Innovation December 2025 • 4 min read.

"Discover how new analytical expressions are revolutionizing arbitrage trading in Automated Market Makers (AMMs), offering faster and more efficient opportunities in the crypto market."


Automated market makers (AMMs) have become a cornerstone of decentralized finance (DeFi), offering users a way to trade cryptocurrencies without traditional intermediaries. These systems rely on liquidity providers who deposit assets into pools, enabling traders to exchange tokens against the deposited capital. The prices within these pools are maintained close to market value through arbitrage, where traders capitalize on price differences to make a profit, simultaneously correcting those differences.

A novel approach has emerged that leverages closed-form analytical expressions to identify and execute arbitrage trades in multi-token AMMs. Unlike traditional methods that rely on numerical solutions and convex optimization, this new approach offers a direct, calculable solution, promising faster execution and potentially higher returns. This is particularly relevant in multi-token scenarios where the arbitrage opportunity involves trading multiple assets in and out of the pool, a process more complex than simple token swaps.

The implications of closed-form solutions extend beyond mere efficiency. They open new avenues for mathematical analysis of arbitrage trades, enabling a more profound understanding of market dynamics. Furthermore, the intrinsic parallelism offered by this approach allows for scaling on GPUs, presenting a shift in AMM modeling and potentially enabling on-chain arbitrage bots capable of handling multi-asset pools.

The Power of Closed-Form Solutions in AMM Arbitrage

Complex DeFi network illustrating closed-form solutions for crypto arbitrage in automated market makers.

Traditionally, arbitrage in AMMs has been addressed using numerical methods, such as linear programming or convex optimization. While effective, these methods can be computationally intensive, especially as the number of tokens in the pool increases. This is where closed-form solutions offer a distinct advantage. By providing an analytical expression for the optimal arbitrage trade, these solutions eliminate the need for iterative calculations, leading to significantly faster execution times.

The benefits of this speed are twofold. First, it allows arbitrageurs to capitalize on fleeting opportunities that might be missed by slower numerical methods. Second, the reduced computational cost makes it feasible to run arbitrage strategies on-chain, directly within the blockchain environment. This opens up new possibilities for automated arbitrage bots that can respond in real-time to market fluctuations.

  • Faster Execution: Closed-form solutions provide immediate results, essential in fast-moving markets.
  • Reduced Computational Cost: Enables on-chain arbitrage and lowers the barrier to entry for smaller players.
  • Improved Analysis: Offers deeper insights into market dynamics and arbitrage opportunities.
  • Parallel Processing: Easily scalable using GPUs, further accelerating computations.
Beyond speed and efficiency, closed-form solutions offer a more nuanced understanding of arbitrage trades. By expressing the optimal trade as a direct function of pool parameters and market prices, these solutions facilitate sensitivity analysis and risk management. Traders can use these expressions to assess the impact of fees, slippage, and other factors on their profitability, allowing for more informed decision-making.

The Future of Arbitrage is Here

The introduction of closed-form solutions marks a significant step forward in the evolution of AMM arbitrage. These solutions empower traders with the speed, efficiency, and analytical capabilities needed to thrive in increasingly competitive markets. As DeFi continues to mature and AMMs become more complex, the ability to quickly and accurately identify arbitrage opportunities will be crucial for maintaining market efficiency and profitability. Whether you're a seasoned arbitrageur or a newcomer to the world of DeFi, understanding and leveraging closed-form solutions is essential for navigating the future of crypto trading.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2402.06731,

Title: Closed-Form Solutions For Generic N-Token Amm Arbitrage

Subject: q-fin.tr q-fin.mf

Authors: Matthew Willetts, Christian Harrington

Published: 09-02-2024

Everything You Need To Know

1

What are automated market makers (AMMs), and what role do they play in decentralized finance (DeFi)?

Automated market makers (AMMs) are a key component of decentralized finance (DeFi). They allow users to trade cryptocurrencies without traditional intermediaries. Liquidity providers deposit assets into pools, enabling traders to exchange tokens against that deposited capital. Arbitrage then helps to maintain prices within these pools close to market value.

2

How do closed-form solutions differ from traditional methods used for arbitrage in AMMs, such as numerical methods and convex optimization?

Traditional arbitrage methods in AMMs rely on numerical solutions and convex optimization, which can be computationally intensive, especially in multi-token pools. Closed-form solutions, however, offer a direct, calculable analytical expression for the optimal arbitrage trade. This eliminates the need for iterative calculations, leading to faster execution times and opening possibilities for on-chain arbitrage bots.

3

What advantages do closed-form solutions provide in terms of speed and computational cost when applied to AMM arbitrage?

Closed-form solutions provide immediate results, which is essential in fast-moving markets. Because these solutions eliminate iterative calculations, the reduced computational cost makes it feasible to run arbitrage strategies on-chain, directly within the blockchain environment. This lowers the barrier to entry for smaller players and enables automated arbitrage bots to respond in real-time to market fluctuations. Additionally, their easily scalable nature using GPUs further accelerates computations.

4

Beyond speed and efficiency, what other benefits do closed-form solutions offer for understanding and managing arbitrage trades in AMMs?

Closed-form solutions facilitate sensitivity analysis and risk management by expressing the optimal trade as a direct function of pool parameters and market prices. Traders can use these expressions to assess the impact of fees, slippage, and other factors on their profitability, allowing for more informed decision-making. Numerical methods do not expose this information directly.

5

What are the implications of using closed-form solutions for the future of AMM arbitrage, and how might they impact market efficiency and profitability?

Closed-form solutions represent a significant advancement in AMM arbitrage, empowering traders with the speed, efficiency, and analytical capabilities needed to thrive in competitive markets. As DeFi matures and AMMs become more complex, the ability to quickly and accurately identify arbitrage opportunities will be crucial for maintaining market efficiency and profitability. The intrinsic parallelism offered by this approach allows for scaling on GPUs, enabling on-chain arbitrage bots capable of handling multi-asset pools, which marks a shift in AMM modeling.

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