Why Constant-Product AMM Liquidity Provision Has Negative Gamma
Summary
The document asks why providing liquidity to Uniswap is described as a negative-gamma strategy. It frames liquidity provision as a dynamic exposure: traders exchange the pooled assets, so the provider’s holdings shift as the relative price changes. This inventory change is commonly called impermanent loss, and the question relates it to the curvature of the provider’s profit-and-loss profile and the effect of volatility.
Trading fees can offset some of that loss, with fee income depending on pool volume and the provider’s share of total liquidity. The document gives a fee-rate example and describes fee accrual as lifting the price-based return curve with each trade. The answer points to research on a replicating portfolio for a constant-product AMM, but does not derive the replication or quantify the balance between fees and price movement. It also does not establish when fees compensate for adverse price exposure, so the discussion is conceptual rather than a complete profitability analysis.
Key ideas
- A constant-product AMM changes a liquidity provider’s asset inventory as traders exchange tokens.
- The resulting exposure is described as negative gamma because the profit-and-loss curve is concave with respect to price changes.
- Impermanent loss names the shortfall associated with holding the changing AMM portfolio relative to a benchmark.
- Trading fees add income that depends on pool volume and the provider’s share of liquidity.
- The discussion references a replicating portfolio but does not quantify net returns or fee sufficiency.
Tags
Full text
# Why is providing liquidity on Uniswap a "negative gamma" trading strategy? # Why is providing liquidity on Uniswap a "negative gamma" trading strategy? I know the basics about options greeks, but I heard traders extrapolating the concept to portfolios composed not just of options. Providing liquidity on Uniswap, an automated market maker (AMM) built on top of the Ethereum blockchain, is said to have negative gamma. The capital deployed would be "impermanently lost" as per the following function: Uniswap charges a trading fee of 30 bips, so a liquidity provider would make up the loss when there's a lot of volume on the exchange. Why can you refer to this investment strategy as being "negative gamma"? Is it because volatility harms your portfolio - i.e. concavity? Update As per Argyll's comment, I'm adding more details about providing liquidity on Uniswap: Understanding Uniswap Returns The strategy is dynamic, in that you provide liquidity once and then people buy and sell assets through the blockchain, using your capital. The assets that you own may change during the liquidity provision - this is what is dubbed as "impermanent loss" and is explained by the chart above. The instruments that can be provided can be anything that is minted on Ethereum. (e.g. two USD-backed stablecoins). The P&L varies according to how much volume there is on Uniswap. The curve from the image above is shifted up by `0.30% * your liquidity / total liquidity in the pool` on every trade. ## Answer by Joseph Clark (score 2) https://quant.stackexchange.com/a/61892 Exact replicating portfolio for constant product AMM here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3550601 It's irritating that people use a new term 'impermanent loss' for something that happens with any option and has been well understood for decades!
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