LP Math — AMM Liquidity Provision Mathematics

Automated Market Makers (AMMs) replace traditional orderbooks with liquidity pools. Instead of matching buyers and sellers, a mathematical formula determines prices based on reserve ratios. Liquidity providers (LPs) deposit both assets into a pool and earn fees from every trade.

Understanding the math behind AMMs is essential for:

Evaluating whether providing liquidity is profitable after impermanent loss
Estimating price impact before executing large trades
Comparing capital efficiency across pool types (constant product vs concentrated)
Calculating expected fee revenue for a given pool position

Related skills: See

impermanent-loss

for IL calculations,

yield-analysis

for LP yield modeling,

liquidity-analysis

for pool depth assessment.

1. Constant Product AMM (xy = k)

The foundational AMM model used by Raydium V4 and most Solana DEXes.

Core Invariant

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:

python

# 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:

python

delta_x = x * delta_y / (y - delta_y)

Price After Trade

python

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:

Gross output:
```
10,000 * 5 / (100 + 5) = 476.19 USDC
```
Fee:
```
476.19 * 0.003 = 1.43 USDC
```
Net output:
```
474.76 USDC
```
Effective price:
```
474.76 / 5 = 94.95 USDC/SOL
```
(vs spot 100)
Price impact:
```
(100 - 94.95) / 100 = 5.05%
```
New reserves: 105 SOL / 9,525.24 USDC
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:

python

# 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:

python

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))
else:
    # In range — holds both tokens
    value_x = L * (1/sqrt(P) - 1/sqrt(P_upper))
    value_y = L * (sqrt(P) - sqrt(P_lower))

Range Strategy Comparison

Range	Efficiency	IL Risk	Fee Capture	Best For
±2%	~50x	Very high	High if in range	Stablecoins, tight pegs
±5%	~20x	High	Good for trending	Active management
±25%	~4x	Moderate	Consistent	Semi-passive
±100%	~2x	Low	Lower per $	Passive, volatile pairs
Full range	1x	Baseline	Always earning	Set and forget

See

references/amm_formulas.md

for full CLMM derivations.

3. Price Impact

Constant Product Impact

python

# Price impact as a fraction
price_impact = delta_x / (x + delta_x)

# As percentage of pool
pool_fraction = trade_value / pool_tvl

# Rule of thumb: impact ≈ 2 * pool_fraction for constant product

Multi-Hop Impact

For a route through multiple pools, compound the impacts:

python

def multi_hop_impact(hops: list[dict]) -> float:
    """Calculate total price impact across route legs.

    Args:
        hops: List of {reserve_in, trade_amount} for each leg.

    Returns:
        Total price impact as a fraction.
    """
    remaining = 1.0
    for hop in hops:
        leg_impact = hop["trade_amount"] / (hop["reserve_in"] + hop["trade_amount"])
        remaining *= (1 - leg_impact)
    return 1 - remaining

Impact Thresholds

Impact	Assessment	Action
< 0.1%	Negligible	Proceed normally
0.1–0.5%	Low	Acceptable for most trades
0.5–2%	Moderate	Consider splitting across pools
2–5%	High	Split trade, use TWAP
> 5%	Severe	Reduce size or find deeper pools

4. LP Share Calculations

Initial Deposit (Empty Pool)

python

shares = sqrt(x_deposited * y_deposited)

The first depositor sets the ratio and receives shares equal to the geometric mean.

Subsequent Deposits

python

shares_minted = min(
    x_added / x_reserve,
    y_added / y_reserve
) * total_shares

Deposits must be proportional to the current reserve ratio. Any excess of one token is not used (or returned, depending on implementation).

Withdrawal

python

x_out = (shares_burned / total_shares) * x_reserve
y_out = (shares_burned / total_shares) * y_reserve

You always receive both tokens in the current ratio.

Share Value

python

share_value = pool_tvl / total_shares
your_value = your_shares * share_value

5. Fee Accrual

Fees accumulate inside the pool, increasing k:

python

# Before trade: k = x * y
# After trade with fee:
# k_new = (x + delta_x) * (y - delta_y_net) > k
# The difference is the fee retained in the pool

# Fee APR estimation
daily_volume = 500_000  # USD
fee_rate = 0.003        # 0.3%
daily_fees = daily_volume * fee_rate  # $1,500
tvl = 2_000_000         # $2M pool
fee_apr = (daily_fees * 365) / tvl    # 27.4%

For CLMM positions, fee earnings depend on:

Whether price stays within your range (out-of-range = no fees)
Your share of active liquidity in that range
Total volume routed through the pool

python

# CLMM fee estimation
your_liquidity = 50_000     # Your L
total_liquidity = 1_000_000  # Total L in your tick range
your_share = your_liquidity / total_liquidity  # 5%
your_daily_fees = daily_fees * your_share  # $75

6. Solana Pool Types

Raydium V4 (Constant Product)

Model: Standard xy = k
Fee: 0.25% (0.22% to LP, 0.03% to RAY buyback)
Best for: New token launches, volatile pairs
Note: Integrated with OpenBook for limit order flow

Orca Whirlpool (Concentrated Liquidity)

Model: Concentrated liquidity with tick spacing
Fee tiers: 0.01%, 0.05%, 0.3%, 1%
Position: Represented as NFT (each position is unique)
Best for: Major pairs (SOL/USDC), stablecoin pairs

Raydium CLMM

Model: Concentrated liquidity (similar to Uniswap V3)
Tick spacing: 1, 10, 60, 200
Fee tiers: 0.01%, 0.05%, 0.25%, 1%
Best for: Pairs with predictable ranges

Meteora DLMM (Dynamic Liquidity Market Maker)

Model: Discrete bins instead of continuous ticks
Strategies: Spot (uniform), Curve (concentrated), Bid-Ask (around current price)
Fees: Dynamic, adjusting based on volatility
Best for: Active LPs who rebalance frequently

See

references/pool_mechanics.md

for detailed mechanics and comparison.

7. Practical Decision Framework

Should You LP?

1. Calculate expected fee APR
2. Estimate impermanent loss for expected price movement
3. Net return = fee APR - IL
4. Compare to simply holding the assets

Which Pool Type?

Stablecoin pair     → CLMM with tight range (±0.5%)
Major pair (SOL/USDC) → CLMM with moderate range (±10-25%)
New/volatile token  → Constant product (full range)
Active management   → Meteora DLMM with dynamic rebalancing

Position Sizing for LP

python

# Never LP more than you can afford to lose to IL
max_lp_allocation = portfolio_value * 0.20  # 20% max in any single pool

# For volatile pairs, reduce further
volatility_adjustment = 1 - (annualized_vol / 2)  # Scale down for vol
adjusted_allocation = max_lp_allocation * max(0.1, volatility_adjustment)

Files

References

```
references/amm_formulas.md
```
— Complete mathematical derivations for constant product and concentrated liquidity AMMs
```
references/pool_mechanics.md
```
— Solana-specific pool mechanics for Raydium, Orca, and Meteora

Scripts

```
scripts/amm_calculator.py
```
— Constant product AMM calculator with trade simulation, LP shares, and fee accrual
```
scripts/clmm_calculator.py
```
— Concentrated liquidity calculator with position valuation, capital efficiency, and range comparison

Quick Reference

Formula	Expression
Constant product	`x * y = k`
Spot price	`P = y / x`
Trade output	`Δy = y * Δx / (x + Δx)`
Required input	`Δx = x * Δy / (y - Δy)`
Price impact	`Δx / (x + Δx)`
Initial LP shares	`sqrt(x * y)`
Subsequent shares	`min(Δx/x, Δy/y) * total`
Fee APR	`(daily_fees * 365) / TVL`
CLMM efficiency	`sqrt(P_u/P_l) / (sqrt(P_u/P_l) - 1)`
Tick to price	`1.0001^tick`

lp-math

NPX Install

Tags

SKILL.md Content

LP Math — AMM Liquidity Provision Mathematics

1. Constant Product AMM (xy = k)

Core Invariant

Spot Price

Trade Execution

Inverse Calculation

Price After Trade

Worked Example

2. Concentrated Liquidity (CLMM)

Key Concepts

Capital Efficiency

Position Value

Range Strategy Comparison

3. Price Impact

Constant Product Impact

Multi-Hop Impact

Impact Thresholds

4. LP Share Calculations

Initial Deposit (Empty Pool)

Subsequent Deposits

Withdrawal

Share Value

5. Fee Accrual

6. Solana Pool Types

Raydium V4 (Constant Product)

Orca Whirlpool (Concentrated Liquidity)

Raydium CLMM

Meteora DLMM (Dynamic Liquidity Market Maker)

7. Practical Decision Framework

Should You LP?

Which Pool Type?

Position Sizing for LP

Files

References

Scripts

Quick Reference