Constant Product Automated Market Maker

A Constant Product Automated Market Maker (AMM) is a type of decentralized exchange mechanism used in decentralized finance (DeFi) platforms. It allows users to trade cryptocurrencies directly from their wallets without the need for a traditional order book or centralized intermediary.

Constant Product Formula

A constant product AAM is an AMM where the price of the tokens are determined by the equation

$$XY=k$$

Where:

• $X$ = amount of token A
• $Y$ = amount of taken B
• $k$ a constant product

Mechanisms

Swap Function: Calculating $dY$ for $dX$?

For example, if we are going to sell Token A and buy Token B, we can write it as:

$$\begin{array}{rl}dX& =\text{Amount of Token A in}\\ dy& =\text{Amount of Token B out}\end{array}$$

Now we have the two equations

$$\begin{array}{rl}XY& =k\\ (X+dX)(Y-dY)& =k\end{array}$$

Solving for $dY$ gives us:

$$dY=\frac{Y\cdot dX}{X+dX}$$

Thus, the amount of Token B you receive ($dY$) when you sell an amount of Token A ($dX$) is determined by the current reserves of both tokens in the liquidity pool.

Add Liquidity Function: Balancing $dX$ and $dY$?

Adding liquidity to the pool requires maintaining the price ratio, which means the amounts added must satisfy:

$$\begin{array}{rl}\frac{X+dX}{Y+dY}& =\frac{X}{Y}\\ X\cdot dY& =Y\cdot dX\\ \frac{X}{Y}& =\frac{dX}{dY}\\ dY& =\frac{Y}{X\cdot dX}\end{array}$$

Share Calculation in Add Liquidity: Minting New Shares

The share calculation is pivotal for fair distribution:

$$\text{Shares Minted}=\frac{dX}{X}\times \text{Total Shares}$$