Lp Math

x * y = k

Where:

• x = reserve amount of token X (e.g., SOL)
• y = reserve amount of token Y (e.g., USDC)
• k = constant product (increases over time from fees)

Spot Price

P = x / y    (price of Y in terms of X)
P = y / x    (price of X in terms of Y)

For a pool with 100 SOL and 10,000 USDC: price of SOL = 10,000 / 100 = 100 USDC.

Trade Execution

When a trader swaps Δx of token X into the pool:

# Output amount (before fees)
delta_y = y * delta_x / (x + delta_x)

# With fee (e.g., 0.3%)
delta_y_after_fee = delta_y * (1 - fee_rate)

# New reserves
x_new = x + delta_x
y_new = y - delta_y_after_fee

The key insight: larger trades get worse prices because each unit moves the ratio further.

Inverse Calculation

To get a specific output amount Δy, the required input is:

delta_x = x * delta_y / (y - delta_y)

Price After Trade

price_new = y_new / x_new

Worked Example

Pool: 100 SOL / 10,000 USDC (k = 1,000,000), fee = 0.3%

Buy 5 SOL worth of USDC:

1. Gross output: 10,000 * 5 / (100 + 5) = 476.19 USDC
2. Fee: 476.19 * 0.003 = 1.43 USDC
3. Net output: 474.76 USDC
4. Effective price: 474.76 / 5 = 94.95 USDC/SOL (vs spot 100)
5. Price impact: (100 - 94.95) / 100 = 5.05%
6. New reserves: 105 SOL / 9,525.24 USDC
7. New k: 105 * 9,525.24 = 1,000,150.2 (k increased from fees)

See references/amm_formulas.md for complete derivations.

2. Concentrated Liquidity (CLMM)

Used by Orca Whirlpool, Raydium CLMM, and Meteora DLMM. Liquidity is only active within a chosen price range [P_lower, P_upper].

Key Concepts

L = sqrt(x * y)           # Liquidity within the active range
price_at_tick = 1.0001^tick  # Tick-to-price conversion

Capital Efficiency

Concentrating liquidity in a narrow range provides more depth per dollar:

# Capital efficiency ratio
efficiency = sqrt(P_upper / P_lower) / (sqrt(P_upper / P_lower) - 1)

# Example: ±5% range around $100 SOL
P_lower, P_upper = 95, 105
efficiency = sqrt(105/95) / (sqrt(105/95) - 1)  # ≈ 20.5x

A ±5% range is ~20x more capital-efficient than full-range, but the position goes 100% into one asset if price moves outside the range.

Position Value

For a CLMM position with liquidity L in range [P_lower, P_upper] at current price P:

if P <= P_lower:
    # All in token X (below range)
    value_x = L * (1/sqrt(P_lower) - 1/sqrt(P_upper))
    value_y = 0
elif P >= P_upper:
    # All in token Y (above range)
    value_x = 0
    value_y = L * (sqrt(P_upper) - sqrt(P_lower))