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DLMM’s math is built around a simple product idea: each pool is a ladder of price bins, and each bin behaves like its own tiny market at a fixed price.
The program performs integer arithmetic with fixed-point values and explicit rounding. This page shows the conceptual formula first, then calls out the important rounding or unit behavior where it matters.

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Units

DLMM uses a few important units:

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Bin Price

Every DLMM pool has a bin step, measured in basis points. The bin step defines the percentage difference between one bin and the next. Pi=(1+bin step10,000)iP_i = \left(1 + \frac{\text{bin step}}{10{,}000}\right)^i Where:
  • P_i is the price of bin i
  • i is the bin ID
  • bin step is the pool’s configured step size
The program computes this as get_price_from_id(bin_id, bin_step) and returns the result in Q64.64 fixed-point form. Example: if the bin step is 25 basis points, each neighboring bin is approximately 0.25% apart. Moving from bin 0 to bin 1 multiplies the price by 1.0025.

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Bin Liquidity

DLMM bin liquidity diagram
Inside a bin, DLMM uses a constant-sum style liquidity value: L=P⋅x+yL = P \cdot x + y Where:
  • L is the bin liquidity value
  • P is the bin price
  • x is the amount of token X
  • y is the amount of token Y
This fixed-price bin design is what creates zero price impact inside a single bin. Price changes when the available output liquidity in the active bin is consumed and the swap moves to another bin.

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Swap Amounts

For a swap from token X to token Y, the output amount is: amount outY=⌊amount inX⋅P⌋\text{amount out}_Y = \left\lfloor \text{amount in}_X \cdot P \right\rfloor For a swap from token Y to token X, the output amount is: amount outX=⌊amount inYP⌋\text{amount out}_X = \left\lfloor \frac{\text{amount in}_Y}{P} \right\rfloor The inverse formulas are used when the program needs to know the required input for a desired output: amount inX=⌈amount outYP⌉\text{amount in}_X = \left\lceil \frac{\text{amount out}_Y}{P} \right\rceil amount inY=⌈amount outX⋅P⌉\text{amount in}_Y = \left\lceil \text{amount out}_X \cdot P \right\rceil The program rounds output amounts down and rounds required input amounts according to the swap path. This protects the pool from giving out more token than the bin can support.

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Total Trading Fee

DLMM trading fees have a base fee and an optional variable fee: total fee rate=min⁡(base fee rate+variable fee rate,MAX_FEE_RATE)\text{total fee rate} = \min(\text{base fee rate} + \text{variable fee rate}, \text{MAX\_FEE\_RATE}) MAX_FEE_RATE is 100,000,000 in 1e9 precision, or 10%.

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Base Fee

The base fee is configured through pool fee parameters: base fee rate=base factor×bin step×10×10base fee power factor\text{base fee rate} = \text{base factor} \times \text{bin step} \times 10 \times 10^{\text{base fee power factor}} This value is stored in 1e9 precision. Product intuition: a larger bin step often pairs with a larger base fee because each bin represents a larger price movement. A smaller bin step is usually used for tighter markets where LPs expect smaller price changes.

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Variable Fee

The variable fee is DLMM’s volatility-aware fee. In the program, it is calculated as: variable fee rate=⌈variable fee control×(volatility accumulator×bin step)2100,000,000,000⌉\text{variable fee rate} = \left\lceil \frac{ \text{variable fee control} \times (\text{volatility accumulator} \times \text{bin step})^2 }{ 100{,}000{,}000{,}000 } \right\rceil If variable fee control is 0, the variable fee is 0. The volatility accumulator updates from bin movement: volatility accumulator=min⁡(volatility reference+∣index reference−active id∣×10,000,max volatility accumulator)\text{volatility accumulator} = \min( \text{volatility reference} + |\text{index reference} - \text{active id}| \times 10{,}000, \text{max volatility accumulator} ) The volatility reference decays based on elapsed time: This lets fees increase when swaps move across bins and cool down after quieter periods.

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Fee Amounts

The program supports two common fee calculations. When the fee is part of the amount already provided by the trader: fee amount=⌈amount with fee×total fee rate1,000,000,000⌉\text{fee amount} = \left\lceil \frac{\text{amount with fee} \times \text{total fee rate}}{1{,}000{,}000{,}000} \right\rceil amount excluding fee=amount with fee−fee amount\text{amount excluding fee} = \text{amount with fee} - \text{fee amount} When the fee must be added on top of an amount that does not include fees: fee amount=⌈amount excluding fee×total fee rate1,000,000,000−total fee rate⌉\text{fee amount} = \left\lceil \frac{ \text{amount excluding fee} \times \text{total fee rate} }{ 1{,}000{,}000{,}000 - \text{total fee rate} } \right\rceil amount including fee=amount excluding fee+fee amount\text{amount including fee} = \text{amount excluding fee} + \text{fee amount} Which form is used depends on the swap direction and the pool’s Collect Fee Mode.

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Fee Split

How a bin’s trading fee is distributed depends on the pool’s function mode: Both modes use the same protocol share for MM liquidity and the same referral host fee rule. Limit-order pools add an extra split when limit-order liquidity contributes to the fill.
These rates apply to how the trading fee is shared among recipients. They are separate from the pool’s base fee, variable fee, and total fee cap.

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MM protocol and LP shares

For any fill that comes from MM positions, the MM slice of the trading fee is split by the pool’s protocol share: protocol fee=⌊trading fee×protocol share10,000⌋\text{protocol fee} = \left\lfloor \frac{\text{trading fee} \times \text{protocol share}}{10{,}000} \right\rfloor LP fee=trading fee−protocol fee\text{LP fee} = \text{trading fee} - \text{protocol fee} On limit-order pools, replace trading fee with total MM fee when the fill also includes limit-order liquidity (see below).

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Referral host fee

When a swap passes a referral account, the host receives 20% of the applicable protocol fee (HOST_FEE_BPS = 2,000 bps). The rest stays as protocol treasury fee. host fee=⌊protocol fee×2,00010,000⌋\text{host fee} = \left\lfloor \frac{\text{protocol fee} \times 2{,}000}{10{,}000} \right\rfloor protocol fee remainder=protocol fee−host fee\text{protocol fee remainder} = \text{protocol fee} - \text{host fee} Without a referral account, host fee = 0 and protocol fee remainder = protocol fee. On limit-order pools with mixed fills, host fee can also include a capped share from the limit-order protocol slice (see below).

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Liquidity mining pools

Liquidity mining pools do not support limit orders. Every swap fill is MM liquidity, so the entire trading fee uses the MM formulas above with no liquidity-source split. trading fee=LP fee+protocol fee remainder+host fee\text{trading fee} = \text{LP fee} + \text{protocol fee remainder} + \text{host fee}

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Limit-order pools

Limit-order pools can fill from MM positions, limit orders, or both in the same bin. The program first assigns the trading fee by liquidity source, then applies the recipient splits above. By liquidity source (rounding favors MM): total MM fee=⌈trading fee×MM amount intotal amount in⌉\text{total MM fee} = \left\lceil \text{trading fee} \times \frac{\text{MM amount in}}{\text{total amount in}} \right\rceil total limit order fee=trading fee−total MM fee\text{total limit order fee} = \text{trading fee} - \text{total MM fee} MM slice — split with protocol share as in the table above: MM protocol fee=⌊total MM fee×protocol share10,000⌋\text{MM protocol fee} = \left\lfloor \frac{\text{total MM fee} \times \text{protocol share}}{10{,}000} \right\rfloor MM LP fee=total MM fee−MM protocol fee\text{MM LP fee} = \text{total MM fee} - \text{MM protocol fee} Limit-order slice — always 50% / 50% via LIMIT_ORDER_FEE_SHARE (5,000 bps), independent of pool type: limit order participant fee=⌊total limit order fee×5,00010,000⌋\text{limit order participant fee} = \left\lfloor \frac{\text{total limit order fee} \times 5{,}000}{10{,}000} \right\rfloor limit order protocol fee=total limit order fee−limit order participant fee\text{limit order protocol fee} = \text{total limit order fee} - \text{limit order participant fee} Referral host fee — 20% of MM protocol fee, plus a capped share from the limit-order protocol slice: host fee on limit order=min⁡(⌊⌊total limit order fee×protocol share10,000⌋×2,00010,000⌋,limit order protocol fee)\text{host fee on limit order} = \min \left( \left\lfloor \frac{ \left\lfloor \frac{\text{total limit order fee} \times \text{protocol share}}{10{,}000} \right\rfloor \times 2{,}000 }{ 10{,}000 } \right\rfloor, \text{limit order protocol fee} \right) host fee=⌊MM protocol fee×2,00010,000⌋+host fee on limit order\text{host fee} = \left\lfloor \frac{\text{MM protocol fee} \times 2{,}000}{10{,}000} \right\rfloor + \text{host fee on limit order} protocol fee remainder=MM protocol fee+limit order protocol fee−host fee\text{protocol fee remainder} = \text{MM protocol fee} + \text{limit order protocol fee} - \text{host fee} When a swap fills only MM liquidity, total limit order fee = 0 and the limit-order paths above are zero — the result matches a liquidity mining pool. trading fee=MM LP fee+limit order participant fee+protocol fee remainder+host fee\text{trading fee} = \text{MM LP fee} + \text{limit order participant fee} + \text{protocol fee remainder} + \text{host fee}

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Liquidity Shares

When a user deposits into a bin, the program first computes the liquidity value of the deposit: Lin=P⋅xin+yinL_\text{in} = P \cdot x_\text{in} + y_\text{in} If the bin has no existing liquidity supply, the new liquidity share equals the deposit liquidity: liquidity share=Lin\text{liquidity share} = L_\text{in} If the bin already has liquidity, the new share is: liquidity share=⌊Lin×existing liquidity supplyLbin⌋\text{liquidity share} = \left\lfloor \frac{ L_\text{in} \times \text{existing liquidity supply} }{ L_\text{bin} } \right\rfloor Where:
  • L_in is the liquidity value of the user’s deposit
  • L_bin is the existing liquidity value of the bin before the deposit
  • existing liquidity supply is the bin’s current share supply
Those shares determine the position’s proportional claim on bin liquidity, fees, and eligible rewards.

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Withdrawal Amounts

When liquidity is removed from a bin, token amounts are returned pro rata: amount outX=⌊liquidity share removed×bin amountXbin liquidity supply⌋\text{amount out}_X = \left\lfloor \frac{ \text{liquidity share removed} \times \text{bin amount}_X }{ \text{bin liquidity supply} } \right\rfloor amount outY=⌊liquidity share removed×bin amountYbin liquidity supply⌋\text{amount out}_Y = \left\lfloor \frac{ \text{liquidity share removed} \times \text{bin amount}_Y }{ \text{bin liquidity supply} } \right\rfloor

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Composition Fee

When adding liquidity to the active bin of a bin that already has liquidity, the program may charge a composition fee if the deposit changes the bin composition in a way that resembles a swap. The composition fee is based on the token delta and the current total fee rate: composition fee=⌊Δamount×total fee rate×(1,000,000,000+total fee rate)1,000,000,0002⌋\text{composition fee} = \left\lfloor \frac{ \Delta \text{amount} \times \text{total fee rate} \times (1{,}000{,}000{,}000 + \text{total fee rate}) }{ 1{,}000{,}000{,}000^2 } \right\rfloor The protocol share is then taken from the composition fee using the same protocol-fee formula: composition protocol fee=⌊composition fee×protocol share10,000⌋\text{composition protocol fee} = \left\lfloor \frac{\text{composition fee} \times \text{protocol share}}{10{,}000} \right\rfloor No composition fee is charged when depositing into an empty bin or a non-active bin.

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Liquidity Mining Rewards

For reward-enabled pools, rewards are tracked per token of liquidity for eligible bins. The reward emission rate is funded over a reward duration. If a reward is funded while a previous reward period is still active, the program includes the remaining unvested reward amount in the new rate calculation: reward rate=new funded amount+remaining reward amountreward duration\text{reward rate} = \frac{\text{new funded amount} + \text{remaining reward amount}}{\text{reward duration}} If the funder chooses to carry forward ineligible rewards from empty-liquidity time, that carry-forward amount is also included in the new funded amount. In the program, reward_rate is stored in fixed-point precision. On active-bin reward updates, the per-token reward increment is: Δreward per token=⌊seconds elapsed×reward rateactive bin liquidity supply⌋\Delta \text{reward per token} = \left\lfloor \frac{ \text{seconds elapsed} \times \text{reward rate} }{ \text{active bin liquidity supply} } \right\rfloor If the active bin has no liquidity, the program records the elapsed empty-liquidity time instead of assigning rewards to LPs. During v2 swaps, accumulated rewards can also be split across crossed bins with liquidity, up to the program’s reward-bin split limit.

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Price Impact Guard

Some swap flows can include a maximum price impact limit. The program computes a guard price from the active bin price: For X to Y swaps: minimum price=Pactive×10,000−max price impact bps10,000\text{minimum price} = P_\text{active} \times \frac{10{,}000 - \text{max price impact bps}}{10{,}000} For Y to X swaps: maximum effective price=Pactive×10,00010,000−max price impact bps\text{maximum effective price} = P_\text{active} \times \frac{10{,}000}{10{,}000 - \text{max price impact bps}} This protects users from swaps that would move too far from their expected active-bin price.