DeFi Core Mathematics: AMM / Interest Rate Models / Liquidation Mechanisms

AMM: Constant Product Formula (Uniswap V2)

Core Formula: x * y = k (x, y are the quantities of two tokens, k is a constant)

Price Calculation

Current Price = y / x (how much y you get for 1 x)

Example: Pool has 10 ETH + 20000 USDC
ETH Price = 20000 / 10 = 2000 USDC

Swap Calculation (with 0.3% fee)

Input Δx, output Δy:
Δy = y * Δx * 997 / (x * 1000 + Δx * 997)

Example: Swap 1 ETH for USDC, pool has 10ETH + 20000USDC
Δy = 20000 * 1 * 997 / (10 * 1000 + 1 * 997)
   = 19940000 / 10997
   ≈ 1813 USDC (price slippage ~9.35%)

Price Impact

The larger the trade size / pool depth, the greater the price impact
Price Impact = 1 - x / (x + Δx)

Example: 1 ETH / 10 ETH pool = 10% impact (very high)
         1 ETH / 1000 ETH pool = 0.1% impact (normal)

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Uniswap V3: Concentrated Liquidity

V3 introduces the price range concept, where LPs provide liquidity only within specified ranges:

tick: Price is represented by tick, price(tick) = 1.0001^tick
tickLower/tickUpper: Price range where liquidity is active

Virtual liquidity formulas:
L = Δx / (1/√Pa - 1/√Pb)  ← Calculate required token X
L = Δy / (√Pb - √Pa)       ← Calculate required token Y

Impact of Concentrated Liquidity:

• With the same capital, narrower price ranges yield higher capital efficiency (but greater IL risk)
• When price moves out of range, LP holds 100% of the depreciated asset

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Interest Rate Model (Compound / Aave)

Utilization Rate

U = Total Borrowed / Total Deposits

Example: 10000 USDC deposits, 7000 USDC borrowed
U = 7000 / 10000 = 70%

Kinked Rate Model

Slope1 (normal range, U < optimal point):
Borrow Rate = BaseRate + U * Slope1

Slope2 (high-risk range, U > optimal point):
Borrow Rate = BaseRate + Optimal * Slope1 + (U - Optimal) * Slope2

Example (Aave USDC parameters):
BaseRate = 0%, Optimal = 90%, Slope1 = 4%, Slope2 = 60%

U=50%: Borrow Rate = 0 + 50%*4% = 2%
U=90%: Borrow Rate = 0 + 90%*4% = 3.6% (optimal point)
U=95%: Borrow Rate = 3.6% + 5%*60% = 6.6% (steep penalty)
U=99%: Borrow Rate = 3.6% + 9%*60% = 9%

Deposit Rate = Borrow Rate * U * (1 - Protocol Fee)

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Liquidation Mechanism

Health Factor (Aave)

HF = Σ(Collateral Value × Liquidation Threshold) / Total Borrowed Value

HF > 1: Safe
HF = 1: Liquidation threshold
HF < 1: Subject to liquidation

Example:
Deposit $10000 ETH (liquidation threshold 82.5%)
Borrow $6000 USDC
HF = 10000 * 0.825 / 6000 = 1.375 (safe)

ETH drops 30% → $7000
HF = 7000 * 0.825 / 6000 = 0.9625 → Subject to liquidation

Liquidation Bonus

Liquidator repays 50% of debt (close factor = 50%)
Receives collateral + 5-15% bonus (liquidation bonus)

Example:
User owes $6000 USDC, collateral $7000 ETH (bonus 5%)
Liquidator repays $3000 USDC
Receives ETH = $3000 * 1.05 = $3150 worth of ETH
Arbitrage profit = $150

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Impermanent Loss (IL)

IL = 2√r/(1+r) - 1, where r = price change ratio

Price Change → IL
+25%        → -0.6%
+50%        → -2.0%
+100%       → -5.7% (2x price = 5.7% loss)
+200%       → -13.4%
-50%        → -5.7%

Price returns to original → IL = 0 (no permanent loss)

Solidity Implementation Reference:

// Calculate sqrt for V3 liquidity calculations
function sqrt(uint256 x) internal pure returns (uint256 y) {
  if (x == 0) return 0;
  uint256 z = (x + 1) / 2;
  y = x;
  while (z < y) { y = z; z = (x / z + z) / 2; }
}

// Q96 fixed-point number (Uniswap V3 internal format)
// sqrtPriceX96 = sqrt(price) * 2^96
uint160 sqrtPriceX96 = ...;
uint256 price = uint256(sqrtPriceX96) * uint256(sqrtPriceX96) >> 192;