Alexangelj / concentrated

Simple concentrated liquidity AMM with limited price grid.

Concentrated AMM

What's a concentrated AMM? Well, it means liquidity at prices. Those prices are derived from the tokens in the reserves.

Run the swap test with: yarn test:forge

Spec

Allocate

Add tokens to reserves at some price and give me liquidity tokens.

Swap

Input some tokens and get some tokens out at a specific price.

Remove

Remove tokens from the reserves, burn my liquidity at some price.

Uniswap V3 reference

• Tracks sqrt(Price) instead of reserves.

• Ticks are mapped to prices with the formula 1.0001^i -> 0.01% -> 1 bps difference in prices

• Ticks are ordered by price, which lets us loop through them when we swap.

• Something like Price = Y / X

• L = sqrt(XY) = (L^1)^2 = ( XY^(1/2) )^2 = L^2 = XY

• sqrt(Price) = sqrt(Y/X)

• L^2 = XY

• L^2 = k

• XY=k

• L^2 / Y = X

• L^2 / X = Y

• X = L / sqrt(Price)

• X = L / sqrt(Y / X)

• X^2 = L^2 / (Y / X)

• X^2 = L^2 * X / Y

• X = L^2 / Y

• (X + deltaX) = L^2 / (Y - deltaY)

• L = sqrt(Y / X) * X

• Y = L * sqrt(Price)

• Y = L * sqrt(XY)

• L = Y / sqrt(XY)

Swap X -> Y

• (X + deltaX) = L / sqrt((Y - deltaY) / X)

liquidityToMint = 100
reserve0 = 50
reserve1 = 50
totalLiquidity = L = sqrt(XY) = 50
amount0 = liquidityToMint * reserve0 / totalLiquidity = 100 * 50 / 50 = 100
amount1 = liquidityToMint * reserve1 / totalLiquidity = 100 * 50 / 50 = 100

liquidityToMint = amount0 * totalLiquidity / reserve0 = 100 * 50 / 50 = 100
liquidityToMint = amount1 * totalLiquidity / reserve1 = 100 * 50 / 50 = 100

Price & Reserves

We have reserves X and Y. We have a Price.

Price = Y / X

X = Y / Price

Y = X * Price

Swap

deltaX + X = (Y - delatY) / Price deltaY + Y = (X - delatX) * Price

Problem

• No way to loop over the prices and see how much liquidity there is.