UniswapV3 Notes

uniswapv3_notes.md

Uniswap v3 Notes

N. P. O'Donnell, 2022

Links

• Core contracts
• Whitepaper

Constant Product Formula (CPF)

Uniswap v1 and v2 are based on constant product formula:

x * y = k

Where:

• x = Number of units of asset x (a.k.a. base currency)
• y = Number of units of asset y (a.k.a. quote currency)
• k = A constant

Uniswap v3 Improvements over v2

Concentrated Liquidity

• In v1 and v2, liquidity was spread across the entire price range from 0 to Inf. In v3, liquidity can be concentrated in ranges. e.g. (10.0, 1000.0). This allows buyers and sellers to get a better price (less slippage) and LPs to earn more fees.

Concentrated Liquidity

Ticks

Each possible price corresponds to a tick (implemented as 24-bit signed integer).

• The tick index at point a is denoted i_a - lower bound of position.
• The tick index at point b is denoted i_b - upper bound of position.
• The tick index at point c is denoted i_c

Glossary

Terms

• Position: 2 points a and b on curve, between two ticks i_l and i_h inclusive.

Acronyms

• CPF = Constant product formula
• LP = Liquidity pool

Variables

• x: Number of units of asset x (a.k.a. base currency) in LP
• y: Number of units of asset y (a.k.a. quote currency) in LP
• k: Constant in CPF
• L: Global liquidity
• l: Per-position liquidity
• a: Lower bound of a position (Asset x has lowest reserves thus highest price, asset y has highest reserves)
• b: Upper bound of a position (Asset x has highest reserves thus lowest price, asset y has lowest reserves)
• c: Some arbitrary point on the curve in the range (a, b)

Formulae

• xy=k: CPF
• L=sqrt(k): Liquidity as a function of k
• L=sqrt(xy): Liquidity as a function of x and y
• sqrt(P)=sqrt(y/x): SqrtPrice as a function of x and y (obviously this implies that P=y/x)
• x=L/sqrt(P): x (base currency virtual reserves) as a function of liquidity and SqrtPrice
• y=L*sqrt(P): y (quote currency virtual reserves) as a function of liquidity and SqrtPrice