Bonding Curve Formulas

Mathematical Foundation

The Dynamic Bonding Curve uses the following key parameters:

migration_quote_threshold: The minimum quote amount required for migration
l: Liquidity value for a curve segment
pa: Starting price (sqrt) for a curve segment
pb: Ending price (sqrt) for a curve segment

Example Configuration


sqrt_start_price = 1
curve = [
  (sqrt_price = 2, liquidity = 100),
  (sqrt_price = 4, liquidity = 500)
]

This creates two curve segments:

(l = 100, pa = 1, pb = 2)
(l = 500, pa = 2, pb = 4)

Each segment is defined by its sqrt_price and liquidity values.

Core Formulas

1. Bonding Curve Amount

The bonding curve amount determines how much of the base token is required for a given quote amount within a curve segment.

2. Migration Quote Threshold

The migration quote threshold is the minimum amount of quote tokens that must be in the pool before it can migrate to a DAMM pool.

3. Migration Price Calculation

When a pool reaches the migration threshold, the migration price is calculated based on:

Migration Fee Percentage: The percentage fee taken during migration
Migration Amount: The total amount being migrated

Fee Formulas

Pool Fee Interface


poolFees: {
  baseFee: {
    cliffFeeNumerator: BN // Initial fee numerator (base fee)
    firstFactor: number // feeScheduler: numberOfPeriod, rateLimiter: feeIncrementBps
    secondFactor: BN // feeScheduler: periodFrequency, rateLimiter: maxLimiterDuration
    thirdFactor: BN // feeScheduler: reductionFactor, rateLimiter: referenceAmount
    baseFeeMode: number // 0: FeeSchedulerLinear, 1: FeeSchedulerExponential, 2: RateLimiter
  }
  dynamicFee: {
    // Optional dynamic fee
    binStep: number // u16 value representing the bin step in bps
    binStepU128: BN // u128 value for a more accurate bin step
    filterPeriod: number // Minimum time that must pass between fee updates
    decayPeriod: number // Period after the volatility starts decaying (must be > filterPeriod)
    reductionFactor: number // Controls how quickly volatility decays over time
    variableFeeControl: number // Multiplier that determines how much volatility affects fees
    maxVolatilityAccumulator: number // Caps the maximum volatility that can be accumulated
  } | null
}

Migration Fee

The migration fee is calculated based on:

quote_min: Minimum quote amount
migration_quote_threshold: The threshold for migration


migrationFee: {
  // Optional migration fee (set as 0 for feePercentage and 0 for creatorFeePercentage for no migration fee)
  feePercentage: number // The percentage of fee taken from migration quote threshold (0-50)
  creatorFeePercentage: number // The fee share percentage for the creator from the migration fee (0-100)
}

When using buildCurveWithMarketCap, the initial_market_cap is calculated to ensure the desired market cap at migration.

Formula


let D = desiredMarketCap and M = migrationMarketCap and f = migrationFee and I = initialMarketCap

requiredRatio = sqrt(D / M)
percentageSupplyOnMigration = (requiredRatio * (1 - f) * 100) / (1 + requiredRatio * (1 - f))
x = percentageSupplyOnMigration / ((100 - V - L) - percentageSupplyOnMigration)
I = M * x²

Example


let D = 10000 / SOL_PRICE
let M = 80000 / SOL_PRICE
let f = 0.5
let V = 0
let L = 1 / 1000000000 // leftover needs to be indicated to cover precision loss
 
let requiredRatio = Math.sqrt(D / M)
let percentageSupplyOnMigration =
  (requiredRatio * (1 - f) * 100) / (1 + requiredRatio * (1 - f))
let x =
  percentageSupplyOnMigration /
  (100 - V - L - percentageSupplyOnMigration)
let I = M * x * x
 
const curveConfig = buildCurveWithMarketCap({
  totalTokenSupply: 1000000000,
  initialMarketCap: I,
  migrationMarketCap: M,
  migrationOption: MigrationOption.MET_DAMM_V2,
  tokenBaseDecimal: TokenDecimal.SIX,
  tokenQuoteDecimal: TokenDecimal.NINE,
  lockedVestingParam: {
    totalLockedVestingAmount: 0,
    numberOfVestingPeriod: 0,
    cliffUnlockAmount: 0,
    totalVestingDuration: 0,
    cliffDurationFromMigrationTime: 0,
  },
  baseFeeParams: {
    baseFeeMode: BaseFeeMode.FeeSchedulerLinear,
    feeSchedulerParam: {
      startingFeeBps: 100,
      endingFeeBps: 100,
      numberOfPeriod: 0,
      totalDuration: 0,
    },
  },
  dynamicFeeEnabled: false,
  activationType: ActivationType.Slot,
  collectFeeMode: CollectFeeMode.QuoteToken,
  migrationFeeOption: MigrationFeeOption.FixedBps100,
  tokenType: TokenType.SPL,
  partnerLpPercentage: 0,
  creatorLpPercentage: 0,
  partnerLockedLpPercentage: 100,
  creatorLockedLpPercentage: 0,
  creatorTradingFeePercentage: 0,
  leftover: 1,
  tokenUpdateAuthority: 0,
  migrationFee: {
    feePercentage: 50,
    creatorFeePercentage: 0,
  },
})