Optimal Rebalancing in Dynamic AMMs

ArXiv ID: 2403.18737 “View on arXiv”

Authors: Unknown

Abstract

Dynamic AMM pools, as found in Temporal Function Market Making, rebalance their holdings to a new desired ratio (e.g. moving from being 50-50 between two assets to being 90-10 in favour of one of them) by introducing an arbitrage opportunity that disappears when their holdings are in line with their target. Structuring this arbitrage opportunity reduces to the problem of choosing the sequence of portfolio weights the pool exposes to the market via its trading function. Linear interpolation from start weights to end weights has been used to reduce the cost paid by pools to arbitrageurs to rebalance. Here we obtain the $\textit{“optimal”}$ interpolation in the limit of small weight changes (which has the downside of requiring a call to a transcendental function) and then obtain a cheap-to-compute approximation to that optimal approach that gives almost the same performance improvement. We then demonstrate this method on a range of market backtests, including simulating pool performance when trading fees are present, finding that the new approximately-optimal method of changing weights gives robust increases in pool performance. For a BTC-ETH-DAI pool from July 2022 to June 2023, the increases of pool P&L from approximately-optimal weight changes is $\sim25%$ for a range of different strategies and trading fees.

Keywords: Dynamic AMM, Portfolio Rebalancing, Optimal Interpolation, Arbitrage Opportunity, Trading Function, Cryptocurrency/DeFi

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematical concepts including optimization with KKT conditions, transcendental equations (Lambert W function), and stochastic calculus for AMM rebalancing, driving a high math score. It also demonstrates empirical rigor through backtests on real market data (BTC-ETH-DAI from 2022-2023) with specific P&L improvements (~25%) and simulations including trading fees, making it highly data- and implementation-focused.

  flowchart TD
    A["Research Goal: Find the Optimal Weight Transition<br/>for Dynamic AMMs to Minimize Rebalancing Costs"] --> B["Analyze Rebalancing Mechanism<br/>via Arbitrage Opportunities"]

    B --> C{"Methodology"}
    C --> D["Formulate Optimal<br/>Interpolation Problem"]
    C --> E["Derive Analytical Solution<br/>for Small Weight Changes"]

    D --> F["Compute Approximation<br/>to Optimal Strategy"]
    E --> F

    F --> G{"Validation"}
    G --> H["Data: BTC-ETH-DAI<br/>Jul 2022 - Jun 2023"]
    G --> I["Simulation: Market Backtests<br/>w/ & w/o Trading Fees"]

    H --> J
    I --> J["Computational Process<br/>Pool Performance Simulation"]

    J --> K["Key Findings"]
    K --> L["~25% Increase in Pool P&L<br/>vs. Linear Interpolation"]
    K --> M["Robust Performance Gains<br/>Across Fee Structures & Strategies"]