A Bonding Curve is a mathematical function encoded in a smart contract that determines a token's price based on its circulating supply. As more tokens are purchased, the price increases along the curve; as tokens are sold back, the price decreases. This creates an automated, continuous price discovery mechanism without requiring order books or counterparties.

Common curve types

Linear: Price = a × Supply + b — Price increases at a constant rate with supply.

Polynomial: Price = a × Supply^n — Steeper price increases as supply grows, commonly used for token launches.

Logarithmic: Price = a × ln(Supply) — Rapid early price discovery that flattens over time, incentivizing early participants.

Sigmoid: S-shaped curves that start flat, steepen, then flatten again — useful for tokens with natural supply caps.

DeFi applications

• AMMs: Uniswap's constant product formula (x × y = k) is effectively a bonding curve between two token reserves.
• Token launches: Projects use bonding curves for fair launches where price is algorithmically determined by demand.
• Dynamic fee adjustment: Some AMMs modify their effective bonding curve based on moving averages of trading volume or volatility.

Security considerations

• Front-running: Predictable price curves enable MEV extraction through sandwich attacks.
• Manipulation via moving averages: If an AMM adjusts its curve parameters based on moving averages, attackers can manipulate those averages to create favorable trading conditions.
• Slippage: Large trades on steep curves can result in significant price impact, especially in low-liquidity pools.

Bonding curves are a core DeFi primitive that enables automated market making, but their deterministic nature requires careful design to resist manipulation and front-running.