Constant Product AMM

This is a constant rpoduct AMM which is similar to Uniswap V2

What is a constant product AMM ?

a constant product AMM mean that X*Y=K with : X the amount of token X Y the amount of token Y K remain the same with a swap but change during adding or removing liquidities.

How to calculate the amount out of a swap ?

previously:

XY = K now :

(X+dx)(Y-dy)=K dx the amount of x in dy the amount of y out so: (x+dx)(Y-dy)=XY XY-Xdy+Ydx-dxdy=XY -dxdy-Xdy=-Ydx dy(X+dx)=Ydx dy=ydx/(X+dx)

How many token to add ?

The only constrain is tha the price much remain the same:

P= X/Y X is still the amount of token X Y is still the amount of token Y (x+dx)/(Y+dy)=X/Y (x+dx)Y = X(Y+dy) YX+ Ydx= XY+ Xdy Ydx= Xdy dx/dy= X/Y It's totally logical it means that the increase of liquidity must be propotional

How do we measure liquidity ?

liquidity = sqrt(X*Y)

How many shares to mint ?

the increase of shares will be proportional with increase of liquidity We have: T the total shares S the shares minted L1 the liquidity after L0 the liquidity before

L1/L0 = (T+S)/T TL1/L0 = (T+S) (TL1-TL0)/L0 = S

S = T(L1-L0)/L0 S = T(sqrt((X+dx)(Y+dy))-sqrt(XY))/sqrt(XY)

after some algebra

S=(dx/X)T=(dy/Y)T