AMM Math Deep Dive

Every Uniswap pool runs the same mathematical invariant: x · y = k. Understanding this formula unlocks how price impact, slippage, fee revenue, and impermanent loss all flow from one simple idea.

🧮 Constant Product Explorer

Adjust the two token reserves and the trade amount to see how x·y=k determines the output. Notice how large trades push the pool far from its pre-trade price.

Token X Reserve 1,000 ETH Token Y Reserve (USDC) 2,000,000 USDC Swap Amount (ETH bought) 10 ETH Fee Tier 0.30%

Spot Price

$2,000

Execution Price

$2,020

Price Impact

1.00%

USDC Out

$20,202

Fee Revenue (LP)

$60.60

📉 Slippage Calculator

For a given pool depth and trade size, compute the expected slippage across fee tiers. Deep pools with high liquidity have lower slippage; thin pools can move 5–10% on a single large trade.

Pool Depth (USDC) $1,000,000

Trade Size (USDC) $100,000

Fee Tier	Effective Depth	Output Amount	Slippage	Fee Revenue

💡 Key insight: In a 0.05% stablecoin pool, $1M depth and $100k trade → only 0.05% slippage. The same $100k in a 1.00% exotic pool with $500k depth might slip 0.8%. Higher fee tiers compensate LPs for higher price impact risk, but they also drive traders to lower-fee alternatives.

📐 Constant Product vs StableSwap

The same x·y=k formula behaves very differently for pegged vs. volatile assets. Curve's StableSwap flattens the curve near the peg so that USDC/USDT trades feel almost frictionless. See the comparison:

🟣 Constant Product 🔵 StableSwap (A=100)

📊 The Math: Swaps, k, and Fee Math

Swap output formula (after fee):
Δy = (Δx × fee) → input with fee
x' = x + Δx
y' = k / x' // invariant stays k
Δy_out = y − y' = y − k / (x + Δx)
Price impact:
PI = |price_after − price_before| / price_before
 = |y'/x' − y/x| / (y/x)
Fee revenue to LPs:
Fee = Δx_in × fee_tier × spot_price
Daily ≈ Volume_24h × fee_tier

$1B daily volume, 0.30% fee

→ $3M daily LP revenue

→ ~$1.1B annual fee revenue

USDC/USDT 0.05% tier

→ $500M daily vol → $250k/day

→ Negligible IL for stablecoin pair

How x·y=k produces a market

The constant product invariant is deceptively simple. A pool starting with 1,000 ETH and 2,000,000 USDC has k = 2,000,000,000. The pre-trade spot price is simply y/x = 2,000 USDC per ETH. When a trader submits a swap for Δx ETH, the protocol first deducts the fee (e.g. 0.30% = 0.003 × Δx), then computes the new reserves as x' = x + (Δx − fee) and y' = k / x'. The trader receives y − y' USDC.

The critical insight is that every unit of Δx pushed into the pool moves the price — and larger pushes move it more. For the first 10 ETH swapped into the 1,000 ETH / 2M USDC pool, the output is approximately 19,802 USDC (about 1% below the spot price). For the next 10 ETH, the output drops to about 19,604 USDC — the execution price for the second tranche is worse than the first. This is slippage: the average price paid per unit is between the spot price and the execution price.

The fee is what makes LP economically viable. Every swap pays a fee that is distributed pro-rata to all existing LPs, so even though the AMM's constant rebalancing causes impermanent loss, the fee revenue from sufficient trading volume more than compensates. For a pool doing $10M daily volume at 0.30% fee, LPs earn $30,000 per day — enough to offset IL from moderate price moves within days.

Key concepts

Invariant k: The product of the two token reserves. In V2, k is constant for the life of the pool unless an LP adds or removes liquidity. In V3, the concept of "liquidity" (√Δ) is separated from k, but the swap formula still reads as if the pool has full-range reserves when active — this is what makes V3 swaps equivalent to constant-product within the active range.
Spot price: The ratio of reserves: y/x. This is also the price at which an infinitesimally small trade would execute. Any real trade, no matter how small, executes at a worse price due to the discrete nature of the EVM math and the rounding that happens at the bytecode level.
Execution price: The average price paid: (USDC paid) / (ETH received). For a trade split into multiple tranches (if routed through multiple pools), the execution price is the volume-weighted average of each tranche's execution price.
Price impact: The percentage difference between the execution price and the pre-trade spot price. A 1% price impact means the trade moves the market 1%. Price impact is non-linear: it grows faster as the trade size approaches a meaningful fraction of the pool depth.
Slippage tolerance: The maximum acceptable price impact set by the trader as a guard against market movement between transaction signing and settlement. A 0.5% slippage tolerance means the swap reverts if the execution price is more than 0.5% worse than the quoted spot price — protecting against MEV sandwiching where an attacker floods the mempool with a large trade to move the price before the victim's trade executes.
Fee revenue model: At 0.30% fee, each swap splits: 0.06% goes to protocol (if fee switch is on), 0.24% goes to LPs. On a $1M swap, LPs earn $2,400. The APY for LPs in a pool depends on daily volume / TVL ratio: $10M daily volume in a $100M pool implies 36.5% annualized fee return (before IL).