lp-math

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.

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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:

# 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.

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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))
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.

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3. Price Impact

Constant Product Impact

# 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:

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

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4. LP Share Calculations

Initial Deposit (Empty Pool)

shares = sqrt(x_deposited * y_deposited)

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

Subsequent Deposits

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

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

share_value = pool_tvl / total_shares
your_value = your_shares * share_value

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5. Fee Accrual

Fees accumulate inside the pool, increasing k:

# 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

# 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

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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.

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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

# 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)

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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

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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