Uniswap AMM Core Mechanics

How the AMM Actually Works

Uniswap's AMM is built on a single invariant: x * y = k. When you deposit liquidity into a pool, you're committing tokens to a shared reserve. The AMM uses these reserves to facilitate trades, charging a fee on every swap and distributing that fee proportionally to all LP token holders.

The beauty of the constant-product formula is that it creates a self-regulating market without any external control. If someone buys a lot of ETH, the ETH reserve decreases and the USDC reserve increases. The new price (USDC/ETH) is automatically higher — because the pool now has less ETH relative to USDC, each unit of ETH costs more USDC. This is the market's way of discouraging further buys and encouraging sells.

The 0.30% fee (in V2 and standard V3 pools) is taken at the input side — before the swap math is applied. So when you swap 10 ETH, only 9.97 ETH goes into the constant-product calculation, and the remaining 0.03 ETH becomes the first piece of LP fee revenue. This fee is taken upfront so that the pool's k is computed correctly with the net input amount.

Liquidity Provision Math

When you provide liquidity, you deposit both tokens in the proportion that matches the current pool ratio. If the pool has 1,000 ETH and 2,000,000 USDC, the ratio is 2,000 USDC per ETH. If you want to add $5,000 of liquidity, you'd deposit 2.5 ETH and 5,000 USDC — maintaining the ratio.

You receive LP tokens proportional to your share. If the total pool is worth $2,000,000 and you deposit $10,000, you own 0.5% of the pool and receive LP tokens representing that 0.5% share. Every time someone trades, 0.30% of the trade value goes to the pool, increasing the value of all LP tokens proportionally.

To withdraw, you burn your LP tokens and receive your proportional share of the current pool — which now includes accumulated fees. If the pool grew from $2,000,000 to $2,100,000 due to trading fees, your 0.5% share is worth $10,500, and you receive that plus your initial principal back in the form of the token pair.

Impermanent Loss: The Hidden Cost

Impermanent loss (IL) is the difference between holding tokens in the AMM vs holding them in a wallet. It occurs because the AMM automatically sells your appreciating asset as the price rises — leaving you with less of it than if you'd just held. IL is "impermanent" in the sense that it becomes real only when you withdraw; if prices return to the original ratio, the loss disappears.

The IL formula for a price ratio change of r (where r = new_price / old_price) is: IL = 2*?r / (1+r) - 1. For a 2x price increase (r=2), IL = 2*?2/3 - 1 = -5.72%. For a 5x increase (r=5), IL = -25.5%. This means even if the pool earns 20% in fees, a 5x price move would leave the LP underwater compared to just holding.

V3 concentrated positions amplify IL geometrically. A 2% width position experiences ~50x the IL of a full-range V2 position for the same price move — because the same capital is concentrated into a tighter price window where each price step has a proportionally larger effect.

Multi-Hop Swaps

Uniswap's router can find paths through multiple pools. If you want to swap USDC for WBTC but there's no direct WBTC/USDC pool, the router might find: USDC → ETH (ETH/USDC pool) → WBTC (WBTC/ETH pool). Each hop adds a fee, so multi-hop swaps are more expensive than direct swaps.

The router uses a "split search" algorithm — it checks all possible paths up to a maximum number of hops (typically 3), computes the output for each path, and returns the best one. For same-chain swaps, paths with more hops are generally worse due to compounding fees, but they may be necessary for illiquid pairs.

Key Concepts

Constant product invariant (x*y=k)

The mathematical foundation of Uniswap's pricing. The product of the two token reserves must remain constant after every swap. This creates a hyperbolic price curve where buying one token increases its price exponentially as you consume more liquidity.

Trading fee (0.30% standard)

Charged on the input amount of every swap. For a 10 ETH swap at 0.30% tier, 0.03 ETH goes to LPs and 9.97 ETH goes into the constant-product calculation. The fee is built into the swap math, not added afterward.

LP tokens

ERC-20 tokens issued when you provide liquidity. They represent your proportional share of a pool's reserves plus accumulated fees. Each pool has its own LP token (e.g., ETH/USDC LP token is different from ETH/WBTC LP token).

Price impact

The difference between the mid price before your swap and your actual average fill price. It's caused by walking up the AMM curve — each unit of the asset you buy costs more than the previous unit. Price impact grows with swap size relative to pool liquidity.

Virtual reserves

In V3 concentrated positions, the token amounts in a hypothetical full-range V2 pool that would produce the same swap behavior. The constant product formula applies to virtual reserves, enabling the same depth with a fraction of the capital.

Slippage tolerance

Your specified maximum acceptable price deviation from the quoted price. If the execution price moves beyond your slippage tolerance, the transaction reverts. Setting a higher tolerance protects against failed transactions but increases MEV exposure.

Minimum output

The minimum amount of output token you'll accept from a swap. Calculated by the router based on the quoted price minus your slippage tolerance. If the swap would produce less than this, it reverts and you pay no gas.