Trading tokens on a bonding curve

0xMarko|2025

Bonding curves are a tool used in DeFi to manage how tokens are created and priced. They use a mathematical formula to decide the price of a token based on how many tokens are available. This system is particularly useful in token launches because it makes pricing transparent and predictable.

Here’s a breakdown of what bonding curves are and how they work for token launches:

What Are Bonding Curves?

A bonding curve is a formula that connects the price of a token with its total supply. As more tokens are created (increased supply), the price goes up according to the curve. This encourages people to buy tokens early because the price is lower when fewer tokens exist.

Bonding curves are a mechanism allows for a self-regulating market where the price reflects the asset's current demand and supply dynamics.

There are different types of bonding curves:

Linear Curve

The price increases steadily as supply grows. Caviat is that simple and predictable linear curves may only partially capture the complexities of market demand.

It can be represented by the formula: P(S)=m⋅S+b

Where:

  • P(S) is the price of the token at a given supply S
  • m is the slope of the curve, which determines how quickly the price increases
  • b is the base price, or the starting price when the supply is zero

Exponential Curve

The price increases at an accelerating rate as supply grows. This creates strong incentives for early adoption, but can also lead to steep entry costs for later participants and potential market volatility.

It can be represented by the formula: P(S)=a⋅e^(b⋅S)

Where:

  • P(S) is the price of the token at a given supply S
  • a is the initial price of the token
  • e is Euler's number (approximately 2.71828)
  • b is the rate of increase, which determines how quickly the price increases

Logarithmic Curves

The price follows a logarithmic function, resulting in diminishing price increases as supply grows. This creates an initial steep price appreciation that gradually levels off, making early participation attractive while ensuring long-term price stability. Ideal for projects seeking sustainable token economics with natural price discovery.

It can be represented by the formula: P(S)=a⋅ln(b⋅S+1)

Where:

  • P(S) is the price of the token at a given supply S
  • a is a scaling factor that determines the steepness of the curve
  • ln is the natural logarithm function
  • b is a positive constant that affects how quickly the curve levels off

Sigmoid Curve

The price grows slowly at first, speeds up in the middle, then slows down again as it reaches a maximum. This S-shaped curve can be implemented using a logistic function.

It can be represented by the formula: P(S)=1/(1+e^(-c₁⋅(S-c₂)))

Where:

  • P(S) is the price of the token at a given supply S
  • c₁ indicates how fast the token price increases in the growth phase (steepness parameter)
  • c₂ determines the inflection point of the curve (shift parameter)
  • e is Euler's number (approximately 2.71828)

Constant Product Curve

The constant product curve is the most widely used AMM formula, made famous by Uniswap. It maintains that the product of two token reserves must remain constant after every trade.

It can be represented by the formula: x * y = k

Where:

  • x is the reserve of token X
  • y is the reserve of token Y
  • k is a constant that must remain unchanged after trades

For example, if a pool has:

  • 10 ETH (x)
  • 20,000 USDC (y)
  • k = 200,000 (10 * 20,000)

When someone wants to buy ETH with USDC, the new reserves must maintain the same k. If they want to buy 1 ETH:

  • New ETH reserve: 9 ETH
  • New USDC reserve: 22,222 USDC (approximately)
  • k = 200,000 (9 * 22,222)

This mechanism:

  • Ensures there's always liquidity available
  • Creates price slippage as trade size increases
  • Automatically adjusts prices based on supply and demand
  • Provides a predictable pricing curve

How To Build a Smart Contracts Trading on a Bonding Curve?

Platforms like pump.fun and its derivates are very popular way of launching tokens. These platforms enable users to create tokens with a fixed token supply, perform initial buy and go straight to trading on a bonding curve. After certain conditions are met, token is moved from bonding curve trading to a dex.

To enable trading on a bonding curve, we need to build a set of smart contracts that manage the creation, trading, and transition of tokens from a bonding curve to a dex (token graduation).

Here is the high-level overview of smart contracts that can be implemented:

Bonding Curve

  • Central contract that manages the creation and trading of tokens
  • Implements a bonding curve mechanism where token price increases as supply decreases
  • Handles the transition from bonding curve trading to Uniswap trading when certain conditions are met

Token (ERC20)

  • Custom ERC20 token implementation
  • Includes max transaction limits and exclusions
  • Knows how to add initial liqudiity to dex after graduation

Trading Pair Factory

  • Creates and manages trading pairs
  • Similar to Uniswap's factory but specialized for the bonding curve system
  • Handles tax parameters for buy/sell operations

Trading Pair

  • Manages trading pair (e.g. Token / USDT)
  • Holds token reserves and handles swaps
  • Maintains the constant product formula (k = x * y)

Trading Router

  • Handles all trading operations
  • Similar to Uniswap's router but adapted for the bonding curve system
  • Manages liquidity additions and token swaps

System Flows

Let's explore key points of each flow in this system:

Token Creation Flow

A[User] --> B[Bonding launch]
    B --> C[Create ERC20]
    C --> D[Factory creates pair]
    D --> E[Initial liquidity added]

Key points:

  • Uses a bonding curve mechanism where token price is determined by supply/liquidity ratio
  • Requires an initial purchase amount that covers both the fee and initial liquidity
  • Creates a trading pair with an asset token (e.g. USDT token)
  • Automatically executes an initial purchase for the creator
  • The entire token supply is used for initial liquidity
  • The Router acts as a secure intermediary for the transfers
  • The Trading Pair contract manages the actual liquidity pool

Trading Flow

A[User] -> B[Bonding buy/sell]
    B -> C[Trading Router]
    C -> D[Trading Pair]
    D -> E[Execute swap]

Key points:

  • Tokens must be in trading state to use bonding curve
  • Cannot trade on bonding curve after graduation
  • Initial purchase required during launch
  • Price Mechanics
    • Uses bonding curve for price discovery
    • Price updates after every trade
    • 24-hour volume tracking resets automatically
  • Graduation Process
    • One-way process (cannot revert to bonding curve)
    • Automatic trigger based on reserve threshold

Graduation Flow

A[Trading] --> B[Reach threshold]
    B --> C[Open Uniswap trading]
    C --> D[Lock/Burn LP tokens]

Key points:

  • Graduation is automatically triggered on buy when the reserve threshold is reached
  • Graduation is irreversible
  • Trading moves from bonding curve to Uniswap

Creating liquidity pool

Liquidity pool is created by the Trading Pair contract when the bonding curve is launched.

Key points:

  • Liquidity pool is created when bonding curve graduation is triggered
  • Initial liquidity pool is created as Uniswap V2 pool
  • All funds from bonding curve are added to liquidity pool
  • Liquidity pool LP tokens can be burned, locked for X years or used for other purposes
  • Token can have multiple liquidity pools
  • Token owner or factory can add a new liquidity pool

Building a token launch system with bonding curves requires careful consideration of smart contract architecture, security measures, and integrations with existing platforms.

If you're interested in building a custom token launch platform or need consultation on bonding curve implementations, feel free to reach out to me.