The math behind DEXs

Hello anon,

Everybody used Uniswap / Sushiswap in its crypto experience, but how Decentralized Exchanges actually work technically? What’s the math behind them?

• Decentralized exchanges (DEXs) enable the exchange between a pair of tokens through a liquidity pool, and the exchange rates between two tokens are determined through a predefined function of the supply of tokens in the pool

• This is fundamentally different from centralized exchanges (e.g., Binance and Coinbase) which match between buy orders and sell orders.

  • The price on centralized exchanges is the price of the most recent trade.

• The pricing function implemented by DEXs like Uniswap and Sushiswap is the “constant product function”

Consider a liquidity pool associated with token A and token B.

• The constant product function means the amount of token A times the amount of token B is a constant K

• When a liquidity transaction happens, the liquidity supply in the pool is changed, so K will be changed, but if it is a trading transaction, the liquidity supply does not change and K keeps the same

We use two numeric examples to demonstrate how liquidity and trading transactions work in Uniswap

1) How Liquidity transactions work:

Assume the initial liquidity supply is 100 token A and 10,000 token B in the liquidity pool.

• Since the value of token A is always equal to the value of token B, the current exchange rate between token A and token B is 100 : 1.

• Whenever liquidity is deposited into a pool, unique tokens known as LP tokens are minted and sent to the liquidity providers as a receipt of depositing liquidity.

• The amount of LP tokens a liquidity provider receives is proportional to the amount of liquidity the liquidity provider deposits into the liquidity pool

• If it is a new liquidity pool, the amount of LP tokens the liquidity provider will receive is equal to sqrt(x × y), where x and y is the amount of token A and token B deposited

• Suppose a liquidity provider deposits an equal value of both tokens, 10 token A and 1000 token B. It is 10% share of the liquidity and the amount of LP tokens the liquidity provider receives is 100

• The liquidity supply becomes 110 token A and 11,000 token B in the liquidity pool.

2) How Trading transactions work:

Assume currently there are 100 token A and 10,000 token B in the liquidity pool.

• The question is how many token B should be put into the pool if a trader would like to get 20 token A

• Since the pricing curve is defined by x × y = 1,000,000 and the trading transaction does not change the constant, we will get (100 - 20) × (10,000 + y ) = 1,000,000

• y = 2,500, so the trader needs to deposit 2,500 token B to get 20 token A. To facilitate the trading transaction, the trader needs to pay 0.03% of the amount of input token, 0.75 token B as trading fees

• The 0.03% fee is added to the pool as new liquidity

• The fee is distributed proportionally to all liquidity providers in the pool upon completion of the trading transaction.

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