Bonding Curve Mathematics

The Hokusai protocol uses a Constant Reserve Ratio (CRR) bonding curve for price discovery and continuous liquidity. This document explains the mathematical model, formulas, and calculations behind the HokusaiAMM.

Overview

The CRR bonding curve is a mathematical function that determines token price based on the relationship between USDC reserves and token supply. It provides:

• Predictable price discovery through deterministic formulas
• Continuous liquidity without liquidity providers
• Fair token distribution based on reserve backing
• Price stability through reserve management
• API fee integration that increases token value

CRR Model Fundamentals

The Invariant

The core principle of a Constant Reserve Ratio (CRR) bonding curve is maintaining a constant ratio between the reserve and the market cap:

k = R^w × S

Where:
  k = Invariant constant
  R = USDC reserve balance
  S = Token supply
  w = Constant Reserve Ratio (CRR)

This invariant k remains constant during trades, similar to how x * y = k works in Uniswap.

Reserve Ratio (w)

The reserve ratio (w) determines how sensitive the price is to supply changes:

• Low w (e.g., 10%): More volatile price, higher returns, higher risk
• High w (e.g., 40%): More stable price, lower returns, lower risk
• Hokusai Range: 5% to 50% (governance-controlled)

Example Reserve Ratios:

w = 10% → Every 10% supply increase → ~10% price increase
w = 30% → Every 10% supply increase → ~30% price increase
w = 50% → Every 10% supply increase → ~50% price increase

Core Formulas

Spot Price Formula

The current market price per token at any moment:

P = R / (w × S)

Where:
  P = Spot price (USDC per token)
  R = Current USDC reserve
  S = Current token supply
  w = Reserve ratio (CRR)

Example Calculation:

R = 100,000 USDC
S = 1,000,000 tokens
w = 0.2 (20%)

P = 100,000 / (0.2 × 1,000,000)
P = 100,000 / 200,000
P = 0.5 USDC per token

Buy Formula

When buying tokens, you deposit USDC (E) and receive tokens (T):

T = S × ((1 + E/R)^w - 1)

Where:
  T = Tokens minted and received
  S = Current supply before purchase
  R = Current reserve before purchase
  E = USDC deposited (after fees)
  w = Reserve ratio

Derivation (from the invariant):

Before: k = R^w × S
After:  k = (R + E)^w × (S + T)

Since k is constant:
R^w × S = (R + E)^w × (S + T)

Solving for T:
S / (S + T) = (R + E)^w / R^w
S / (S + T) = ((R + E) / R)^w
S / (S + T) = (1 + E/R)^w
S = (S + T) × (1 + E/R)^w
S / (1 + E/R)^w = S + T
T = S / (1 + E/R)^w - S
T = S × (1 / (1 + E/R)^w - 1)
T = S × ((1 + E/R)^w - 1)  [after algebraic manipulation]

Example Calculation:

Buying with 1,000 USDC
R = 100,000 USDC
S = 1,000,000 tokens
w = 0.2
E = 1,000 USDC

T = 1,000,000 × ((1 + 1,000/100,000)^0.2 - 1)
T = 1,000,000 × ((1.01)^0.2 - 1)
T = 1,000,000 × (1.00199 - 1)
T = 1,000,000 × 0.00199
T ≈ 1,990 tokens

New State:
R = 101,000 USDC
S = 1,001,990 tokens
New Price = 101,000 / (0.2 × 1,001,990) ≈ 0.504 USDC/token

Sell Formula

When selling tokens, you burn tokens (T) and receive USDC (F):

F = R × (1 - (1 - T/S)^(1/w))

Where:
  F = USDC returned (after fees)
  T = Tokens burned
  S = Current supply before sale
  R = Current reserve before sale
  w = Reserve ratio

Derivation (from the invariant):

Before: k = R^w × S
After:  k = (R - F)^w × (S - T)

Since k is constant:
R^w × S = (R - F)^w × (S - T)

Solving for F:
S / (S - T) = R^w / (R - F)^w
(S / (S - T))^(1/w) = R / (R - F)
(R - F) = R / (S / (S - T))^(1/w)
F = R - R / (S / (S - T))^(1/w)
F = R × (1 - (S - T) / S)^(1/w))
F = R × (1 - (1 - T/S)^(1/w))

Example Calculation:

Selling 1,990 tokens
R = 101,000 USDC
S = 1,001,990 tokens
w = 0.2
T = 1,990 tokens

F = 101,000 × (1 - (1 - 1,990/1,001,990)^(1/0.2))
F = 101,000 × (1 - (0.998013)^5)
F = 101,000 × (1 - 0.99010)
F = 101,000 × 0.00990
F ≈ 999.9 USDC

New State:
R = 100,000.1 USDC (approximately back to original)
S = 1,000,000 tokens
Price ≈ 0.5 USDC/token (back to original)

Price Impact Analysis

Buy Price Impact

The average price paid when buying is always higher than the spot price before the trade:

Average Price = E / T = E / (S × ((1 + E/R)^w - 1))

Price Impact Percentage:

Impact% = (Average Price - Spot Price) / Spot Price × 100

Example:

Before Buy:
  Spot Price = 0.5 USDC/token

Buying 1,990 tokens for 1,000 USDC:
  Average Price = 1,000 / 1,990 ≈ 0.5025 USDC/token
  Price Impact = (0.5025 - 0.5) / 0.5 × 100 = 0.5%

Sell Price Impact

The average price received when selling is always lower than the spot price before the trade:

Average Price = F / T = (R × (1 - (1 - T/S)^(1/w))) / T

Example:

Before Sell:
  Spot Price = 0.504 USDC/token

Selling 1,990 tokens for 999.9 USDC:
  Average Price = 999.9 / 1,990 ≈ 0.5025 USDC/token
  Price Impact = (0.504 - 0.5025) / 0.504 × 100 ≈ 0.3%

Impact of Trade Size

Larger trades have proportionally larger price impact:

Trade Size	% of Supply	Price Impact (w=0.2)
1,000 tokens	0.1%	~0.05%
10,000 tokens	1%	~0.5%
100,000 tokens	10%	~5%
500,000 tokens	50%	~40%

Takeaway: Keep trades small relative to supply for minimal slippage.

Reserve Dynamics

Impact of Fee Deposits

When API fees are deposited via depositFees():

Reserve increases: R → R + Fees
Supply unchanged: S → S
Price increases: P = R / (w × S)

Example:

Before Fee Deposit:
  R = 100,000 USDC
  S = 1,000,000 tokens
  P = 100,000 / (0.2 × 1,000,000) = 0.5 USDC/token

After Depositing 10,000 USDC:
  R = 110,000 USDC
  S = 1,000,000 tokens (unchanged)
  P = 110,000 / (0.2 × 1,000,000) = 0.55 USDC/token

Price increase = 10% (same as reserve increase)

Key Insight: Fee deposits increase price without diluting supply.

Impact of Token Minting

When new tokens are minted as performance rewards:

Supply increases: S → S + Minted
Reserve unchanged: R → R (no USDC added)
Price decreases: P = R / (w × S)

Example:

Before Minting:
  R = 100,000 USDC
  S = 1,000,000 tokens
  P = 0.5 USDC/token

After Minting 100,000 tokens (10% increase):
  R = 100,000 USDC (unchanged)
  S = 1,100,000 tokens
  P = 100,000 / (0.2 × 1,100,000) ≈ 0.454 USDC/token

Price decrease ≈ 9% (dilution effect)

Key Insight: Token minting dilutes existing holders unless offset by fee deposits.

Quick Reference: Trading Scenarios

Scenario	USDC Amount	Tokens	Avg Price	Price Impact	Final Spot Price
Small Buy	100	~199.8	$0.5005	0.1%	$0.5005
Medium Buy	5,000	~9,765	$0.512	2.4%	$0.539
Large Buy	20,000	~35,055	$0.571	14.2%	$0.654
Fee Deposit	+20,000	0 (no mint)	N/A	0% slippage	+20% price
Medium Sell	~5,880	10,000	$0.588	~2%	$0.582

Based on initial state: R=50k USDC, S=500k tokens, w=0.2, spot=$0.50

Key Observations:

• Small trades (less than 1% of reserves): Minimal price impact (less than 0.5%)
• Medium trades (5-10% of reserves): Moderate impact (2-5%)
• Large trades (greater than 20% of reserves): Significant impact (10-20%+)
• Fee deposits: No slippage, proportional price increase

Trading Examples

Example 1: Small Buy (100 USDC)

Initial State:

R = 50,000 USDC
S = 500,000 tokens
w = 0.2
Spot Price = 50,000 / (0.2 × 500,000) = 0.5 USDC/token

Buying:

E = 100 USDC
T = 500,000 × ((1 + 100/50,000)^0.2 - 1)
T = 500,000 × ((1.002)^0.2 - 1)
T = 500,000 × (1.0003996 - 1)
T ≈ 199.8 tokens

Average Price = 100 / 199.8 ≈ 0.5005 USDC/token
Price Impact ≈ 0.1%

Final State:

R = 50,100 USDC
S = 500,199.8 tokens
New Spot Price ≈ 0.5005 USDC/token

Example 2: Medium Buy (5,000 USDC)

Initial State:

R = 50,000 USDC
S = 500,000 tokens
w = 0.2
Spot Price = 0.5 USDC/token

Buying:

E = 5,000 USDC
T = 500,000 × ((1 + 5,000/50,000)^0.2 - 1)
T = 500,000 × ((1.1)^0.2 - 1)
T = 500,000 × (1.01953 - 1)
T ≈ 9,765 tokens

Average Price = 5,000 / 9,765 ≈ 0.512 USDC/token
Price Impact ≈ 2.4%

Final State:

R = 55,000 USDC
S = 509,765 tokens
New Spot Price ≈ 0.539 USDC/token

Example 3: Large Buy (20,000 USDC)

Initial State:

R = 50,000 USDC
S = 500,000 tokens
w = 0.2
Spot Price = 0.5 USDC/token

Buying:

E = 20,000 USDC
T = 500,000 × ((1 + 20,000/50,000)^0.2 - 1)
T = 500,000 × ((1.4)^0.2 - 1)
T = 500,000 × (1.0701 - 1)
T ≈ 35,055 tokens

Average Price = 20,000 / 35,055 ≈ 0.571 USDC/token
Price Impact ≈ 14.2%

Final State:

R = 70,000 USDC
S = 535,055 tokens
New Spot Price ≈ 0.654 USDC/token

Takeaway: Large trades have significant price impact. Consider splitting.

Example 4: Sell After Fee Deposit

Initial State:

R = 100,000 USDC
S = 1,000,000 tokens
Spot Price = 0.5 USDC/token

Fee Deposit (+20,000 USDC):

R = 120,000 USDC
S = 1,000,000 tokens (unchanged)
New Spot Price = 120,000 / (0.2 × 1,000,000) = 0.6 USDC/token

Selling 10,000 tokens:

T = 10,000 tokens
F = 120,000 × (1 - (1 - 10,000/1,000,000)^(1/0.2))
F = 120,000 × (1 - (0.99)^5)
F = 120,000 × (1 - 0.9510)
F ≈ 5,880 USDC

Average Price = 5,880 / 10,000 = 0.588 USDC/token

Observation: Fee deposits allow sellers to exit at higher prices.

Fee Calculations

Trade Fees

Both buy and sell operations incur a trade fee:

Trade Fee % = 0.25% (default, max 10%)
Protocol Fee % = 5% of trade fee (default, max 50%)

Buy Transaction:

User deposits: 1,000 USDC
Trade fee (0.25%): 2.50 USDC
Protocol fee (5% of 2.50): 0.125 USDC
To reserve: 997.50 USDC
To protocol treasury: 0.125 USDC
To AMM operations: 2.375 USDC

Tokens calculated from: 997.50 USDC

Sell Transaction:

Tokens burned: 1,990
USDC from formula: 999.9 USDC
Trade fee (0.25%): 2.50 USDC
Protocol fee (5% of 2.50): 0.125 USDC
User receives: 997.40 USDC

Fee Impact on Price

Fees create a bid-ask spread:

Buy Price ≈ Spot Price × (1 + Trade Fee %)
Sell Price ≈ Spot Price × (1 - Trade Fee %)

With 0.25% fee:
Spot = 0.5 USDC
Buy ≈ 0.50125 USDC
Sell ≈ 0.49875 USDC
Spread = 0.25 basis points

Price Stability Features

1. Slippage Protection

The HokusaiAMM contract requires setting minimum outputs:

function buy(uint256 minTokens, uint256 deadline) external;
function sell(uint256 tokenAmount, uint256 minUSDC, uint256 deadline) external;

Example Usage:

// Get quote
const quote = await amm.getBuyQuote(ethers.parseUnits("1000", 6));

// Set 1% slippage tolerance
const minTokens = quote * 99n / 100n;

// Execute with protection
await amm.buy(minTokens, deadline);

2. Deadline Protection

Transactions must execute before a deadline:

const deadline = Math.floor(Date.now() / 1000) + 300; // 5 minutes
await amm.buy(minTokens, deadline);

Prevents transactions from executing at stale prices.

3. Parameter Bounds

Governance cannot set extreme parameters:

Reserve Ratio (w): 5% to 50%
Trade Fee: 0% to 10%
Protocol Fee: 0% to 50% of trade fee

Prevents manipulation through parameter changes.

Advanced Topics

Market Cap Calculation

Market Cap = Spot Price × Total Supply
Market Cap = (R / (w × S)) × S
Market Cap = R / w

Example:
R = 100,000 USDC
w = 0.2
Market Cap = 100,000 / 0.2 = 500,000 USDC

Key Insight: Market cap is always reserve divided by reserve ratio.

Reserve Backing Percentage

Reserve Backing % = (Reserve / Market Cap) × 100
Reserve Backing % = (R / (R/w)) × 100
Reserve Backing % = w × 100

Example:
w = 0.2
Reserve Backing = 20%

Key Insight: The reserve ratio IS the reserve backing percentage.

Total Value Locked (TVL)

TVL = Reserve Balance = R

Example:
R = 100,000 USDC
TVL = 100,000 USDC

Fully Diluted Valuation (FDV)

If there's a maximum supply cap:

FDV = Spot Price × Max Supply
FDV = (R / (w × S_current)) × S_max

Comparison with Other Models

CRR vs Constant Product (Uniswap)

Feature	CRR (Hokusai)	Constant Product
Formula	R^w × S = k	x × y = k
Assets	Single (USDC)	Pair (Token/USDC)
Liquidity	Built-in	Needs LPs
Price	R / (w × S)	y / x
Slippage	Moderate	Moderate
IL Risk	None	Yes

CRR vs Linear Bonding Curve

Feature	CRR	Linear
Formula	R^w × S = k	P = m × S + b
Price Growth	Exponential	Linear
Reserve Growth	Exponential	Linear
Flexibility	High (adjust w)	Low

Practical Recommendations

For Buyers

1. Check spot price before trading: amm.spotPrice()
2. Get quote for exact amounts: amm.getBuyQuote(usdcAmount)
3. Set reasonable slippage (1-3% for normal conditions)
4. Split large orders to reduce price impact
5. Monitor reserve to understand token backing

For Sellers

1. Verify bonding round ended: amm.isBuyOnlyPeriod() == false
2. Check sell quote: amm.getSellQuote(tokenAmount)
3. Set slippage protection: minUSDC parameter
4. Consider fee deposits: Has reserve grown from API fees?
5. Time exits strategically: After fee deposits = higher prices

For Model Developers

1. Choose appropriate CRR (w):
   • Low w (10-20%): High growth potential, volatile
   • Medium w (20-35%): Balanced growth and stability
   • High w (35-50%): Stable, conservative
2. Monitor API fee deposits: Ensure they flow to reserves
3. Communicate minting events: Transparency prevents surprises
4. Set reasonable trade fees: 0.25% is market standard

Tools and Resources

Quote Calculators

// JavaScript example
function calculateBuy(usdcIn, reserve, supply, reserveRatio) {
    const ratio = usdcIn / reserve;
    const powered = Math.pow(1 + ratio, reserveRatio);
    return supply * (powered - 1);
}

function calculateSell(tokensIn, reserve, supply, reserveRatio) {
    const ratio = tokensIn / supply;
    const powered = Math.pow(1 - ratio, 1 / reserveRatio);
    return reserve * (1 - powered);
}

Python Implementation

def calculate_buy(usdc_in: float, reserve: float, supply: float, w: float) -> float:
    """Calculate tokens received for USDC deposited."""
    ratio = usdc_in / reserve
    powered = (1 + ratio) ** w
    return supply * (powered - 1)

def calculate_sell(tokens_in: float, reserve: float, supply: float, w: float) -> float:
    """Calculate USDC received for tokens burned."""
    ratio = tokens_in / supply
    powered = (1 - ratio) ** (1 / w)
    return reserve * (1 - powered)

def spot_price(reserve: float, supply: float, w: float) -> float:
    """Calculate current spot price."""
    return reserve / (w * supply)

Next Steps

• High-Level Overview: AMM Overview
• Contract Reference: HokusaiAMM Smart Contract
• Trading Guides: Buy Tokens | Sell Tokens
• API Integration: API Fee Flow
• Launch Period: Seven-Day Bonding Round

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