Learning Objectives:

  • Master how DEXs work and their key mechanisms

  • Understand AMM mathematics and formulas

  • Learn about different AMM models (Uniswap, Curve, Balancer)

  • Analyze DEX risks and optimization strategies

5.2.1: What are DEXs?

Definition and Purpose

Decentralized Exchanges (DEXs) are peer-to-peer marketplaces where users can trade cryptocurrencies directly with each other without intermediaries. Unlike centralized exchanges, DEXs are non-custodial and operate through smart contracts.

 
Order Book DEX vs AMM:

Order Book DEX:
┌─────────────────────────────────────────────────────────────────────┐
│  Buy Orders      Sell Orders                                      │
│  ┌─────────────┐  ┌─────────────┐                               │
│  │ Price $100  │  │ Price $105  │                               │
│  │ 100 tokens  │  │ 200 tokens  │                               │
│  └─────────────┘  └─────────────┘                               │
│  ┌─────────────┐  ┌─────────────┐                               │
│  │ Price $99   │  │ Price $106  │                               │
│  │ 150 tokens  │  │ 100 tokens  │                               │
│  └─────────────┘  └─────────────┘                               │
│                                                                   │
│  Problems:                                                        │
│  • Low liquidity for low-volume tokens                           │
│  • Order book must be filled                                     │
│  • Slippage issues                                              │
└─────────────────────────────────────────────────────────────────────┘

AMM (Automated Market Maker):
┌─────────────────────────────────────────────────────────────────────┐
│  Liquidity Pool:                                                  │
│  ┌─────────────────────────────────────────────────────────────┐   │
│  │  Token A: 100 ETH                                         │   │
│  │  Token B: 200,000 USDC                                   │   │
│  │  Price determined by formula: x × y = k                  │   │
│  └─────────────────────────────────────────────────────────────┘   │
│                                                                   │
│  Benefits:                                                        │
│  • Always available liquidity                                    │
│  • No order book required                                       │
│  • Anyone can provide liquidity                                  │
│  • Price discovery through math                                  │
└─────────────────────────────────────────────────────────────────────┘

5.2.2: AMM Mathematics – The Constant Product Formula

The Core AMM Formula (Uniswap):

The constant product formula is the foundation of most AMMs:

text
x × y = k

Where:
- x = Reserve of token A
- y = Reserve of token B
- k = Constant product (invariant)

Price Calculation:
Price_A_in_B = y / x
Price_B_in_A = x / y

Swap Calculation:
Given: Swap dx amount of token A for token B
Δy = (y × dx) / (x + dx)

After Swap:
x' = x + dx
y' = y - Δy
k' = x' × y' = k (remains constant)

Example:
Pool: 100 ETH and 200,000 USDC
k = 100 × 200,000 = 20,000,000

Price: 1 ETH = 2,000 USDC

Swap 1 ETH for USDC:
dx = 1 ETH
Δy = (200,000 × 1) / (100 + 1) = 1,980.2 USDC

After Swap:
x' = 101 ETH
y' = 198,019.8 USDC
k' = 101 × 198,019.8 = 20,000,000

New Price: 1 ETH = 1,960.6 USDC

Slippage Calculation:

text
Slippage = (Expected_Price - Actual_Price) / Expected_Price × 100%

Example (Large Swap):
Expected Price: 2,000 USDC/ETH
Actual Price: 1,960.6 USDC/ETH
Slippage = (2,000 - 1,960.6) / 2,000 × 100 = 1.97%

Slippage Factors:
- Trade size relative to pool size
- Pool depth (liquidity)
- Trading volume

Mitigation:
- Split large trades
- Use DEX aggregators
- Set slippage tolerance

5.2.3: The Constant Sum Formula (Curve)

Curve’s Stable Swap Formula:

Curve is optimized for stablecoins (USDC, USDT, DAI) where prices are expected to remain stable.

text
Constant Sum Formula:
x + y = k

Benefits:
- Minimal slippage for stable assets
- Efficient trading
- Low fees

Problems:
- No price discovery
- Can be manipulated

Solution: Hybrid Formula
Curve uses a combination of constant product and constant sum:

x + y + (x × y / (x + y)²) × k = k

This provides:
- Low slippage near equilibrium
- Price discovery when imbalanced
- Protection against manipulation

5.2.4: Balancer’s Weighted Pools

Balancer’s Formula:

Balancer extends the AMM concept to multiple tokens with custom weights.

text
Balancer Formula:
Σ (x_i^w_i) = k

Where:
- x_i = Reserve of token i
- w_i = Weight of token i
- k = Invariant

Example (80/20 Pool):
- Token A: 80% weight
- Token B: 20% weight

Exit fee: 0.1% - 0.5%
Trading fee: 0.1% - 1%

Benefits:
- Custom portfolio weights
- Single-sided liquidity
- Flexible fee structures
- Multi-token pools

5.2.5: Liquidity Provision and LP Tokens

Providing Liquidity:

text
Liquidity Provision Process:

1. Deposit:
   - User deposits equal value of both tokens
   - Example: 1 ETH + 2,000 USDC

2. LP Tokens Minted:
   - Represent share of pool
   - Redeemable for underlying tokens
   - Earn trading fees

3. Share Calculation:
   Shares = (Deposited_Value / Total_Pool_Value) × Total_Shares

Example:
Pool: 100 ETH + 200,000 USDC (Total Value: $400,000)
Total Shares: 100,000

Deposit: 1 ETH + 2,000 USDC ($4,000)
Shares = (4,000 / 400,000) × 100,000 = 1,000 shares

5.2.6: Impermanent Loss

Understanding Impermanent Loss:

Impermanent loss is the temporary loss experienced by liquidity providers when the price ratio of deposited tokens changes.

text
Impermanent Loss Example:

Initial Deposit:
- 1 ETH + 2,000 USDC
- ETH Price: $2,000
- Pool Share: 1%

After Price Change:
- ETH Price: $4,000 (doubles)
- Pool arbitrageurs rebalance
- Pool now has: 0.707 ETH + 2,828 USDC

IF liquidated now:
- Value of LP position: 0.707 × 4,000 + 2,828 = $5,656
- Value of HODL: 1 × 4,000 + 2,000 = $6,000
- Impermanent Loss: $344 (5.7%)

Why "Impermanent":
- Loss only realized when withdrawing
- If prices return, loss disappears
- Can be offset by trading fees

IL by Price Change:
┌─────────────────────────────────────────────────────────────────────┐
│  Price Change   |   Impermanent Loss                              │
│  1.25x         |   0.6%                                          │
│  1.5x          |   2.0%                                          │
│  2x            |   5.7%                                          │
│  3x            |   13.4%                                         │
│  5x            |   25.5%                                         │
│  10x           |   42.5%                                         │
└─────────────────────────────────────────────────────────────────────┘

5.2.7: DEX Aggregators

How Aggregators Work:

DEX aggregators find the best prices across multiple DEXs.

text
Aggregation Process:

1. Split Order:
   ┌─────────────────────────────────────────────────────────────┐
   │  User wants to swap 100 ETH                              │   │
   └─────────────────────────────────────────────────────────────┘
                            │
                            ▼
   ┌─────────────────────────────────────────────────────────────┐
   │  Query Multiple DEXs:                                    │   │
   │  • Uniswap: 200,000 USDC                                 │   │
   │  • Curve: 201,000 USDC                                   │   │
   │  • Balancer: 199,000 USDC                               │   │
   └─────────────────────────────────────────────────────────────┘
                            │
                            ▼
   ┌─────────────────────────────────────────────────────────────┐
   │  Optimal Routing:                                         │   │
   │  • 30 ETH → Balancer: 59,700 USDC                        │   │
   │  • 70 ETH → Curve: 140,700 USDC                          │   │
   │  • Total: 200,400 USDC                                   │   │
   └─────────────────────────────────────────────────────────────┘

Popular Aggregators:

 
 
Aggregator Features Supported DEXs
1inch Best price, limit orders 50+ DEXs
ParaSwap Privacy, low slippage 30+ DEXs
CowSwap Intent-based, MEV protection 20+ DEXs
0x API Open-source, customizable 30+ DEXs

5.2.8: DEX Security and Risks

Common DEX Risks:

 
 
Risk Description Mitigation
Impermanent Loss Temporary loss from price divergence Choose stable pairs, monitor
Smart Contract Risk Bugs or exploits Audits, use established DEXs
Front-Running MEV exploitation Use slippage protection
Rug Pulls Malicious liquidity removal Verify contracts, use audited DEXs
Slippage Price change during trade Set slippage limits
Oracle Manipulation Price manipulation attacks Use multiple oracles
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