Continuous Token Model

A bonding curve is a mathematical function, typically stored in a smart contract, that defines a direct relationship between a token's supply and its price. As more tokens are purchased (minted), the price increases along the curve; as tokens are sold (burned), the price decreases. This creates a predictable, on-chain price discovery mechanism independent of traditional order books. Common curve types include linear, polynomial, and logarithmic, each offering different liquidity and price volatility characteristics.

Unlike Automated Market Makers (AMMs) that require paired liquidity, CTMs provide single-sided liquidity directly from the bonding curve reserve. A buyer's funds are deposited into the contract's reserve, and new tokens are minted. A seller returns tokens to the contract, which are burned, and receives assets from the reserve. This ensures 24/7 liquidity for the token at the price defined by the curve, eliminating the need for counterparties or liquidity providers.

Token supply in a CTM is non-fixed and changes with every buy or sell transaction, governed by the bonding curve.

• Minting: When a user buys, the contract accepts payment (e.g., ETH) into its reserve and mints new tokens to the buyer, increasing total supply.
• Burning: When a user sells, they send tokens to the contract, which are burned (permanently removed from supply), and the user receives payment from the reserve. This creates a direct, algorithmic link between capital inflow/outflow and token supply.

The reserve is the pool of underlying assets (e.g., ETH, DAI, USDC) held by the bonding curve contract that backs the minted tokens. The reserve ratio is a key parameter representing the fraction of the token's market cap held in the reserve. A higher ratio means each token is backed by more collateral, typically leading to lower price volatility. The reserve provides the liquidity for redemptions and determines the token's intrinsic floor value.

CTMs are highly configurable through key parameters set at deployment, often controlled by governance tokens. Critical parameters include:

• Bonding Curve Formula: The mathematical function (e.g., P = k * S^2).
• Reserve Ratio: The target fraction of value held in reserve.
• Fees: Protocol fees on buys/sells can be directed to a treasury.
• Curve Halts: Mechanisms to pause minting/burning under certain conditions. These parameters dictate the economic behavior and security of the model.

Different bonding curve implementations serve specific use cases:

• Continuous Organizations (COs): Use a CTM as a fundraising and community alignment tool (e.g., Fairmint's Continuous Securities Offering).
• Rebasing Tokens: Adjust all holders' balances based on reserve changes to maintain a stable price target (e.g., Ampleforth).
• Liquidity-Enabled Tokens: Combine a CTM with a Uniswap V2-style AMM pool for deeper, two-sided liquidity (a hybrid model).

Protocols use bonding curves to create continuous, on-chain liquidity for new tokens without traditional market makers. The curve's formula (e.g., linear, polynomial) defines the price-supply relationship.

• Purpose: Bootstrap liquidity and enable continuous, permissionless trading.
• Mechanism: A smart contract holds reserves (e.g., ETH) and mints/burns the project's token based on the curve.
• Example: Early decentralized autonomous organizations (DAOs) used bonding curves for initial funding and token distribution.

Many Automated Market Makers (AMMs) are built on a Continuous Token Model, where liquidity pool (LP) tokens represent a continuous share of the pool's reserves.

• Constant Product Formula: The foundational x * y = k model used by Uniswap V2 is a specific type of bonding curve.
• LP Tokens: Minted when users deposit assets; burned upon withdrawal. Their value changes continuously with the pool's composition.
• Dynamic Pricing: Swap prices are determined continuously by the on-chain reserve ratios, not an order book.

A Continuous Organization is a legal entity that uses a CTM to create a direct, fluid link between its economic performance and a tradable token.

• Mechanism: The organization issues bonding curve shares on-chain. Revenue is sent to a reserve pool, which drives the price of the shares up the curve via buybacks.
• Token Utility: Represents a claim on future revenues and/or governance rights.
• Goal: Align investor and contributor incentives through continuous, transparent funding and exit liquidity.

CTMs can manage access to exclusive lists or NFT memberships where the minting cost follows a bonding curve.

• Application: A curated registry of verified entities (e.g., token-curated registries).
• Pricing: The cost to add an entry increases as the registry grows, discouraging spam and capturing the rising value of curation.
• Example: The Kleros TCR uses a model where challenging or adding entries involves deposits and pricing influenced by the number of existing entries.

Protocols use CTM mechanics for rebasing tokens or elastic supplies that adjust continuously based on an oracle price.

• Goal: Maintain a stable peg (e.g., to a dollar) or a target price range.
• Process: The total token supply expands or contracts for all holders based on a predefined formula triggered by market price deviations.
• Contrast: While similar in continuous adjustment, these often focus on stabilization rather than providing direct buy/sell liquidity from a reserve.

Implementing a CTM involves critical design choices that impact security and economics.

• Curve Choice: Linear, polynomial, or logarithmic curves create different liquidity and volatility profiles.
• Reserve Asset Risk: The protocol's solvency depends on the value and security of the assets in its bonding curve reserve.
• Manipulation Risks: Large deposits/withdrawals can significantly move price on the curve, requiring safeguards.
• Exit Liquidity: Provides inherent liquidity but may create sell pressure if not coupled with sustained demand drivers.

A bonding curve is a mathematical function that defines the relationship between a token's price and its total supply. It is the core engine of a CTM.

• Price Discovery: The token price increases as the supply is bought and decreases as it is sold, providing continuous liquidity.
• Automated Market Making: The smart contract acts as a constant-function market maker (CFMM), removing the need for traditional order books.
• Example: A linear bonding curve might set price = k * supply, where 'k' is a constant.

Unlike traditional models requiring counterparties, CTMs provide programmatic liquidity.

• Instant Liquidity: Users can buy or sell tokens directly to/from the bonding curve contract at any time.
• Slippage: The price impact of a trade is determined by the bonding curve's slope; large trades move the price more.
• Exit Mechanism: This creates a predictable, non-custodial exit path, reducing reliance on secondary markets like DEXs.

The bonding curve serves as a perpetual funding mechanism for a project's treasury.

• Mint-and-Sell: When a user buys, new tokens are minted, and the payment (e.g., ETH) is deposited into the project treasury.
• Buyback-and-Burn: When a user sells, the treasury's funds are used to buy back tokens, which are then burned, reducing supply.
• Sustainable Funding: This creates a direct, automated link between token demand and project resources.

Key parameters of the bonding curve are governance decisions that define the token's economic policy.

• Curve Shape: Governance decides the function (linear, polynomial, logarithmic) which affects volatility and capital efficiency.
• Reserve Ratio: Determines the fraction of treasury funds backing the token's redeemable value.
• Fee Structure: May include protocol fees on buys/sells, which are directed to the treasury or token holders.

CTMs can incorporate mechanisms to align long-term holding with protocol success.

• Fee Distribution: Transaction fees generated by the underlying protocol can be distributed to token holders staking in the bonding curve.
• Staked Liquidity: Users can stake their tokens within the curve contract, often earning a share of buy/sell fees or newly minted tokens.
• Voting Power: Staked tokens typically confer governance rights, linking economic stake to decision-making.

While innovative, CTMs introduce unique economic risks.

• Ponzi-like Dynamics: Early buyers profit from later buyers' capital if not coupled with real utility, creating unsustainable inflation.
• Treasury Volatility: The project treasury's value fluctuates with the token price, impacting budgeting.
• Manipulation Vulnerability: The predictable pricing can be exploited via flash loans or coordinated attacks if not properly guarded.
• Exit Liquidity: Large sell-offs can rapidly deplete the treasury and crash the price along the curve.

A bonding curve is a mathematical function that defines the relationship between a token's price and its total supply. It is the core mechanism behind continuous token models, algorithmically setting the buy and sell price for every transaction.

• Key Feature: Price increases as supply is purchased and decreases as supply is sold.
• Function Types: Common implementations use linear, exponential, or logarithmic curves.
• Purpose: Provides continuous, on-chain liquidity without relying on traditional order books.

An Automated Market Maker (AMM) is a decentralized exchange protocol that uses liquidity pools and a constant function (e.g., x*y=k) to price assets. While related, it differs from a bonding curve model.

• Similarity: Both provide automated, on-chain liquidity.
• Key Difference: AMMs facilitate trading between two or more assets in a pool, while a bonding curve defines the price of a single token against a reserve currency.
• Use Case: Uniswap and Curve are AMMs; they are often used in conjunction with tokens issued via bonding curves for secondary market liquidity.

An Initial DEX Offering (IDO) is a fundraising event where a project's tokens are initially made available for purchase on a decentralized exchange (DEX). Continuous token models offer an alternative or complementary launch mechanism.

• Contrast: Traditional IDOs often involve a fixed-price sale or liquidity pool seeding, followed by volatile market discovery.
• CTM Advantage: A bonding curve launch provides smooth price discovery from the first purchase, potentially reducing front-running and initial volatility.
• Hybrid Models: Some projects use a CTM for the initial continuous sale, then deposit remaining tokens into an AMM pool for secondary trading.

A token sink is a mechanism that permanently or temporarily removes tokens from circulating supply, often to create deflationary pressure or utility. In continuous token models, the buy/sell mechanism can be designed to function as a sink.

• Burning on Sell: Some models burn a percentage of tokens on each sale, permanently reducing supply.
• Treasury Allocation: A fee on transactions may be directed to a treasury or staking pool, temporarily locking supply.
• Objective: Sinks help align long-term incentives by rewarding holders and counteracting sell-side pressure from the bonding curve's decreasing price function.

Rebasing tokens are assets where the circulating supply automatically adjusts (expands or contracts) across all holders' wallets to maintain a target price peg or index value. This contrasts with the price-change mechanism of bonding curves.

• Mechanism Difference: Rebasing changes holder balances to stabilize price, while bonding curves change the token's price based on supply changes.
• Supply Elasticity: Both models feature elastic supply, but the adjustment method differs fundamentally.
• Example: Ampleforth (AMPL) is a well-known rebasing token that targets adjustment to the value of the 2019 US Dollar.
• Interaction: It is theoretically possible to combine mechanisms, e.g., a token traded via a bonding curve that also employs rebases.