> ## Documentation Index
> Fetch the complete documentation index at: https://docs.meteora.ag/llms.txt
> Use this file to discover all available pages before exploring further.

# DAMM v2 Formulas

> Understand the product math behind DAMM v2 prices, liquidity, fees, compounding, and rewards.

DAMM v2 is designed around constant-product liquidity. The formulas below explain the product math and the implementation details users most often need to reason about. On-chain values use integer math, so rounding matters.

## Constant Product

The basic AMM idea is simple:

```math theme={"system"}
x \times y = k
```

`x` is the token A reserve, `y` is the token B reserve, and `k` is the pool's liquidity invariant. In compounding mode this is reserve-based. In concentrated mode, the same idea is bounded by the pool's configured square-root price range.

## Price and Square-Root Price

DAMM v2 stores price as a square-root price:

```math theme={"system"}
\sqrt{P} = \sqrt{\frac{\text{token B amount}}{\text{token A amount}}}
```

This format makes liquidity math more precise and efficient on-chain. Product users can still think of it as the pool price expressed in a form that is easier for the program to update during swaps and liquidity changes.

## Concentrated Liquidity

In concentrated liquidity mode, a DAMM v2 pool has a fixed lower and upper square-root price:

```math theme={"system"}
\sqrt{P_{\min}} \le \sqrt{P} \le \sqrt{P_{\max}}
```

For a liquidity amount `L`, the token amounts are:

```math theme={"system"}
\Delta A = L \times \left(\frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_{\max}}}\right)
```

```math theme={"system"}
\Delta B = L \times (\sqrt{P} - \sqrt{P_{\min}})
```

The program rounds required deposit amounts up and withdrawal or swap output amounts down. Token A calculations also guard against values larger than `u64::MAX`.

## Compounding Liquidity

In compounding mode, DAMM v2 uses full-range constant-product liquidity and prices the pool from reserves:

```math theme={"system"}
\text{token A reserve} \times \text{token B reserve} = k
```

```math theme={"system"}
P = \frac{\text{token B reserve}}{\text{token A reserve}}
```

At initialization, compounding mode derives reserves from liquidity and square-root price:

```math theme={"system"}
A_0 = \left\lceil \frac{L}{\sqrt{P}} \right\rceil
```

```math theme={"system"}
B_0 = \left\lceil \frac{L \times \sqrt{P}}{2^{128}} \right\rceil
```

The first position receives:

```math theme={"system"}
L_{\text{position}} = L - (100 \ll 64)
```

because `100 << 64` is kept as dead liquidity in the pool.

## Trading Fees

DAMM v2 fee rates are stored as numerators over `1,000,000,000`:

```math theme={"system"}
\text{Total Fee Numerator} = \min(\text{Base Fee Numerator} + \text{Dynamic Fee Numerator}, \text{Pool Fee Cap})
```

For fees taken from an amount that already includes the fee, the program rounds the fee up:

```math theme={"system"}
\text{Trading Fee} = \left\lceil \frac{\text{Included Amount} \times \text{Fee Numerator}}{1{,}000{,}000{,}000} \right\rceil
```

For fees added to an amount that excludes the fee:

```math theme={"system"}
\text{Included Amount} = \left\lceil \frac{\text{Excluded Amount} \times 1{,}000{,}000{,}000}{1{,}000{,}000{,}000 - \text{Fee Numerator}} \right\rceil
```

After the trading fee is calculated, DAMM v2 splits it:

```math theme={"system"}
\text{Protocol Fee} = \left\lfloor \text{Trading Fee} \times \frac{\text{Protocol Fee Percent}}{100} \right\rfloor
```

```math theme={"system"}
\text{LP Fee Before Compounding} = \text{Trading Fee} - \text{Protocol Fee}
```

If the pool uses compounding mode, the LP fee can be split again:

```math theme={"system"}
\text{Compounding Fee} = \left\lfloor \text{LP Fee Before Compounding} \times \frac{\text{Compounding Fee Bps}}{10{,}000} \right\rfloor
```

```math theme={"system"}
\text{Claimable Fee} = \text{LP Fee Before Compounding} - \text{Compounding Fee}
```

If a referral is present, the referral fee is taken from the protocol fee:

```math theme={"system"}
\text{Referral Fee} = \left\lfloor \text{Protocol Fee} \times \frac{\text{Referral Fee Percent}}{100} \right\rfloor
```

DAMM v2's default protocol fee percent is `20`, and the referral fee percent is `20` of the protocol fee when a referral account is used.

## Dynamic Fee

Dynamic fees use a volatility accumulator. When price changes quickly, the accumulator rises. When the market calms down, it decays.

```math theme={"system"}
\text{Dynamic Fee Numerator} =
\left\lceil
\frac{(\text{Volatility Accumulator} \times \text{Bin Step})^2 \times \text{Variable Fee Control}}{100{,}000{,}000{,}000}
\right\rceil
```

In the current implementation, dynamic fee `bin_step` must be `1` bps and `bin_step_u128` must match the default 1 bps fixed-point value. `filter_period` must be lower than `decay_period`, and both `variable_fee_control` and `max_volatility_accumulator` are capped at `0xffffff`.

## Liquidity Mining Rewards

DAMM v2 liquidity mining distributes reward tokens by position liquidity:

```math theme={"system"}
\text{Reward Per Liquidity} = \frac{\text{Elapsed Time} \times \text{Reward Rate}}{\text{Total Pool Liquidity}}
```

```math theme={"system"}
\text{Position Reward} = \text{Position Liquidity} \times \text{Reward Per Liquidity Since Last Checkpoint}
```

This means LPs earn rewards in proportion to their share of pool liquidity. Unlocked, vesting, and permanently locked liquidity can all count toward rewards because the position's total liquidity is used for reward accounting.

The reward rate is stored as a fixed-point value scaled by `2^64`, while reward-per-liquidity accounting is accumulated with a `2^128` liquidity scale. Integer division truncates fractional rewards.