Reference
Math
Weighted Math

Weighted Math

Overview

Weighted Math is designed to allow for swaps between any assets whether or not they have any price correlation. Prices are determined by the pool balances, pool weights, and amounts of the tokens that are being swapped.

Balancer's Weighted Math equation is a generalization of the xy=kx*y=k constant product formula, accounting for cases with n2n \geq2 tokens as well as weightings that are not an even 50/50 split.

For more formulas and derivations of the below formulas, please refer to the Balancer Whitepaper (opens in a new tab).

Implementations

TypeScript

Developers can use the TypeScript math implementations used by the Smart Order router

Python

There are also Python implementations in progress

Invariant

The value function VVis defined as:

V=tBtWtV= \prod_t B_t^{W_t}

Where

  • tt ranges over the tokens in the pool
  • BtB_t is the balance of the token in the pool
  • WtW_t​is the normalized weight of the tokens, such that the sum of all normalized weights is 1.

Spot Price

Each pair of tokens in a pool has a spot price defined entirely by the weights and balances of just that pair of tokens. The spot price between any two tokens,SpotPriceioSpotPrice^o_i, or in short SPioSP^o_i, is the the ratio of the token balances normalized by their weights:

SPio=BiWiBoWoSP^o_i = \frac{\frac{B_i}{W_i}}{\frac{B_o}{W_o}}
  • BiB_i is the balance of token ii, the token being sold by the swapper which is going into the pool
  • BoB_o is the balance of token oo, the token being bought by the swapper which is going out of the pool
  • WiW_i is the weight of token ii
  • WoW_o is the weight of token oo

Spot Price with Swap Fees

When we consider swap fees, we do exactly the same calculations as without fees, but using Ai(1swapFee)A_i \cdot (1-swapFee) instead of AiA_i since fees are taken out of the input amount. The equation then becomes:

SPio=BiWiBoWo11swapFeeSP^o_i = \frac{\frac{B_i}{W_i}}{\frac{B_o}{W_o}} \cdot \frac{1}{1-swapFee}

Swap Equations

outGivenIn

When a user sends tokens ii to get tokens oo, all other token balances remain the same. Therefore, if we define AiA_i and AoA_o as the amount of tokens ii and oo exchanged, and since the value function VV must be constant before and after the swap, we can calculate the amount AoA_o a users gets when sending AiA_i.

Ao=Bo(1(BiBi+Ai)WiWo)A_o = B_o \cdot \left(1-\left(\frac{B_i}{B_i + A_i}\right)^{\frac{W_i}{W_o}}\right)

::: info If you're computing this value yourself, remember that the pool collects swap fees as a percentage of the input token. In the equation above,AiA_i is the amount that the pool actually swaps into the output token, not the amount sent by a swapper, AsentA_{sent}. To calculate through, we must compute:Ai=Asent(1swapFee)A_i = A_{sent} * (1-swapFee) :::

inGivenOut

It is also very useful for swappers to know how much they need to send of the input token AiA_i to get a desired amount of output token AoA_o:

Ai=Bi((BoBoAo)WoWi1)A_i = B_i \cdot \left(\left(\frac{B_o}{B_o - A_o}\right)^{\frac{W_o}{W_i}}-1\right)

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