lp-math Checked

# 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 |

---

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

---

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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