IT

A Deep Dive into the Fundamental Principles of Uniswap’s XYK Model

XYK means: put a pair of assets into a “pool”, and require that the product of the two asset amounts always stays equal: x * y = k. All swaps must follow this rule, and therefore price, slippage, yield and risk all naturally come from this formula.

Let’s walk through the math step by step.

1. Traditional Trading: Why Do We Need an Order Book?

In traditional exchanges, the core mechanism is the order book:

  • A places a buy order at 100 for 1 ETH
  • B places a sell order at 101 for 2 ETH
  • The system matches orders and completes the trade

This design has several characteristics:

1. Requires professional market makers

  • Without someone providing orders, ordinary traders cannot trade
  • Market makers must watch the market and continuously update orders

2. Requires a centralized matching engine

  • Run by the exchange
  • You have to trust the intermediary

3. Ordinary users cannot participate in market-making easily

  • Hard to provide liquidity
  • Most profits go to institutions

For blockchain, which emphasizes open participation, this is too centralized.

2. AMM: Turning Market Making into a Liquidity Pool

The innovation of AMM (Automated Market Maker) is transforming trading from order matching into a liquidity pool that anyone can join.

A liquidity pool holds two assets (e.g., ETH and USDT) and follows:

x * y = k

This is the XYK constant product market maker model.

When someone buys ETH with USDT, ETH decreases and USDT increases, while keeping the product constant.

AMM replaces the order book:

  • no buy/sell orders
  • no professional market makers
  • no centralized matching engine
  • anyone can join

So essentially:

AMM = liquidity pool + mathematical rule

Price, slippage and liquidity depth all follow from this.

3. How Is Price Determined?

Example:

Pool state:

  • ETH x = 100
  • USDT y = 10000

Then the price is naturally:

price = y / x = 10000 / 100 = 100 USDT

Key points:

  1. Price is not quoted manually
  2. Price changes automatically with pool state
  3. External arbitrage aligns the pool price to market price

So in AMM:

Price = asset ratio in pool

4. How Do Trades Change Price?

Initial State 

x = 100 ETH
y = 10000 USDT
k = 1,000,000
P₀ = 100

A user wants to spend 1000 USDT to buy ETH.

Transaction rule

Must satisfy x * y = k before and after the swap

After the swap

User adds:

Δy = +1000
y₂ = 11000
x₂ = 1,000,000 / 11000 ≈ 90.909
Δx ≈ 9.091

Average price ≈ 110 Final price ≈ 121

You observe:

  1. Average paid ≈ 110
  2. Final price ≈ 121
  3. Initial price = 100

That feeling of:

“I buy → the price jumps!”

That is slippage.

5. Slippage: Not a Fee, But Price You Push Higher

Slippage is not a fee:

  • You moved the price by trading

You are literally buying along the curve from cheap → expensive. Bigger trades cause more slippage.

6. Why Bigger Pools Have Lower Slippage?

Compare:

Small pool: 100 / 10000

Price moves 100 → 121

Big pool: 1000 / 100000

Price moves 100 → 102

Conclusion:

More liquidity → milder price curve → lower slippage

It’s like:

  • a basin vs. a lake
  • adding a bucket changes the level much more in a basin

7. What Exactly Is Liquidity?

Liquidity = actual asset amounts in the pool

  • more liquidity → flatter curve → stable prices
  • less liquidity → steep curve → price impact increases

Anyone can provide liquidity, therefore:

Market-making becomes open

8. How Do You Join as a Liquidity Provider?

Pool state:

x = 100 ETH
y = 10000 USDT

Price = 100 USDT/ETH

To add liquidity, you must deposit proportionally:

  • add 1 ETH
  • plus 100 USDT

Otherwise you change the price.

You receive LP tokens representing your share.

When you withdraw, you take a share of the pool, not exactly what you put in.

This is important for understanding impermanent loss.

9. How Do LPs Earn Fees?

Each trade pays a small fee (e.g., 0.3%), and this fee:

  • does NOT go to “the platform”
  • goes into the pool
  • distributed to LPs

Think of it as:

Traders use your assets, and you charge toll fees

LP earnings:

  • trading fees
  • token incentives

10. Impermanent Loss

LPs earn fees but face impermanent loss.

Example pool:

1 ETH + 100 USDT

You add:

1 ETH + 100 USDT

Pool:

2 ETH + 200 USDT
(You own 50%)

If ETH price falls, the pool becomes:

4 ETH + 100 USDT

You withdraw:

2 ETH + 50 USDT = 100 USDT value

You deposited value: 200 You withdraw value: 100

You lose value because:

  • pool moved toward the depreciating asset
  • arbitrage takes the increasing asset out

The pool automatically rebalances toward cheaper assets.

11. What Problems Does XYK Solve?

  1. No order book
  2. Price determined algorithmically
  3. Anyone can become a market maker
  4. Infrastructure becomes public and open

12. Limitations & Evolution

XYK drawbacks

  1. Large slippage → specialized stable-swap curves (Curve)
  2. Poor capital efficiency → concentrated liquidity (Uniswap v3)
  3. Impermanent loss → LP bears volatility risk

Still:

XYK is the entrance door to AMM design

Understanding XYK makes everything else easier.

13. The One Simple Formula

Back to:

x * y = k

From this follows:

  • price: P = y / x
  • slippage
  • liquidity depth
  • LP revenue
  • impermanent loss

XYK is the F = ma of DeFi:

deceptively simple, yet the foundation of decentralized trading.