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Bonding Curve

The Hokusai protocol uses a bonding curve for price discovery and token distribution. This document explains the mechanics, mathematics, and examples of our bonding curve system.

Overview​

The bonding curve is a mathematical function that determines token price based on supply. It provides:

  • Predictable price discovery
  • Continuous liquidity
  • Fair token distribution
  • Price stability mechanisms

Mathematical Model​

Basic Formula​

The price (P) of a token is determined by the following function:

P = P₀ * (1 + r)^S

Where:

  • P₀ = Initial price (0.01 USDC)
  • r = Rate of increase (0.1% or 0.001)
  • S = Current supply

Price Impact​

The price impact (ΔP) for a token purchase of size n is:

ΔP = P₀ * ((1 + r)^(S + n) - (1 + r)^S)

Example Calculation​

For a purchase of 1000 tokens when supply is 1,000,000:

P₀ = 0.01 USDC
r = 0.001
S = 1,000,000
n = 1000

P = 0.01 * (1.001)^1,000,000
  ≈ 0.011 USDC per token

Price Bounds​

Minimum Price​

  • Lower bound: 0.001 USDC
  • Prevents token devaluation
  • Maintains minimum liquidity

Maximum Price​

  • Upper bound: 10 USDC
  • Prevents excessive speculation
  • Ensures accessibility

Liquidity Mechanism​

Primary Pool​

  • USDC/Token pair
  • Governed by bonding curve
  • Provides base liquidity

Secondary Pools​

  • Model-specific pairs
  • Additional liquidity sources
  • Risk diversification

Trading Examples​

1. Small Purchase​

Amount: 100 tokens
Current Supply: 500,000
Price Impact: 0.05%
Final Price: 0.0105 USDC

2. Medium Purchase​

Amount: 1,000 tokens
Current Supply: 500,000
Price Impact: 0.5%
Final Price: 0.0105 USDC

3. Large Purchase​

Amount: 10,000 tokens
Current Supply: 500,000
Price Impact: 5%
Final Price: 0.011 USDC

Price Stability Features​

1. Slippage Protection​

  • Maximum price impact: 5%
  • Large order splitting
  • Time-weighted execution

2. Arbitrage Incentives​

  • Price deviation detection
  • Automated arbitrage bots
  • Market efficiency rewards

3. Liquidity Incentives​

  • LP token rewards
  • Trading fee sharing
  • Staking bonuses

Implementation Details​

Smart Contract Integration​

function calculatePrice(uint256 supply) public pure returns (uint256) {
    return initialPrice * (1 + rate) ** supply;
}

function calculatePurchaseAmount(uint256 usdcAmount, uint256 supply) public pure returns (uint256) {
    uint256 price = calculatePrice(supply);
    return usdcAmount / price;
}

Oracle Integration​

  • Price feed updates
  • Supply verification
  • Liquidity monitoring

Monitoring and Analytics​

Key Metrics​

  • Current price
  • Total supply
  • Liquidity depth
  • Price impact

Health Indicators​

  • Price stability
  • Liquidity ratio
  • Trading volume
  • Arbitrage activity

Next Steps​

For additional support, contact our Support Team or join our Community Forum.

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