2.5 — Bonding Curves

Bonding Curves — Deterministic Primary Markets

A bonding curve is a smart contract that mints and sells tokens along a deterministic price schedule: the price is a known function of the cumulative supply already sold. There is no order book, no auction, no bookbuilding syndicate. The curve is the market.

This is the primary-market analogue of the AMM. Where a constant-product pool provides secondary-market liquidity for an existing token pair, a bonding curve provides primary-market liquidity for a token that is being created on the fly. The intellectual link is the same: replace a human process (IPO bookbuilding, Dutch auction) with a closed-form pricing function on-chain.

The General Framework

Let $S$ denote the cumulative supply of tokens minted so far. The bonding curve defines a price function $p(S)$ that gives the marginal cost of the next infinitesimal token.

The total cost to mint from supply $S_0$ to $S_1$ is:

$$C(S_0,S_1)={\int }_{S_0}^{S_1}p(s)ds$$

This integral is the reserve the contract holds after those tokens are minted. The curve is fully collateralized by construction: the reserve always equals the cost integral.

Selling (burning) tokens reverses the process — the seller receives $C(S_1,S_0)$ back from the reserve (possibly minus a fee). This guarantees a bid at every supply level, unlike a traditional market where the bid can vanish.

Common Curve Shapes

Linear: $p(S)=a+bS$

$$C(0,S)=aS+\frac{b}{2}{S}^2$$

Price starts at $a$ and grows linearly. Early buyers get a lower price. The cost integral is a quadratic — buying $n$ tokens from zero costs $O(n^2)$.

Quadratic: $p(S)=a{S}^2$

$$C(0,S)=\frac{a}{3}{S}^3$$

Steeper price acceleration. Strongly rewards early participants. The cubic cost function means late buyers pay dramatically more.

Exponential: $p(S)=a\cdot e^{bS}$

$$C(0,S)=\frac{a}{b}(e^{bS}-1)$$

The fastest-growing schedule. Practically, this makes very large purchases impossible, bounding the total supply that will ever be economically minted.

Virtual AMM (Pump.fun’s approach)

Instead of defining $p(S)$ explicitly, Pump.fun uses a virtual constant-product AMM. The contract initializes virtual reserves $(x_v,y_v)$ with $x_v{y}_v=k$ and then processes buys and sells using the standard constant-product formula from constant-product-amm.

The “virtual” qualifier means the initial reserves are not real deposited tokens — they are parameters set by the contract creator. The effect is a bonding curve whose marginal price function is:

$$p(S)=\frac{y_v-C(S)}{x_v+S}$$

where $C(S)$ is the SOL (or ETH) paid in so far. This produces a price trajectory that follows the hyperbolic shape of the constant-product invariant — concave in supply, with diminishing price growth as supply increases. It is smoother than a polynomial curve and naturally bounded.

Pump.fun: Anatomy of a Token Launch

Pump.fun (launched January 2024 on Solana) operationalizes the virtual AMM bonding curve for meme token issuance. The lifecycle:

Phase 1: Bonding Curve (Primary Market)

1. A creator launches a token with no upfront cost.
2. The contract initializes a virtual AMM with predetermined parameters.
3. Buyers purchase tokens along the curve, paying SOL.
4. The SOL accumulates in the bonding curve’s reserve.
5. Early buyers get lower prices; each purchase pushes the price up.
6. Sellers can sell back at the current curve price (minus fees).

Phase 2: Graduation

When the bonding curve’s reserve reaches a threshold — approximately 85 SOL — the token “graduates.” At graduation:

1. The bonding curve is permanently closed (no more primary minting).
2. The accumulated SOL and remaining token supply are deposited into a full AMM pool (originally Raydium V4, now PumpSwap).
3. The token transitions from primary market (bonding curve) to secondary market (constant-product AMM).

The 85 SOL threshold is a design choice that sets the “IPO size.” At typical SOL prices, this means a token graduates with roughly $10K--$15K of liquidity, implying a fully-diluted market cap in the low hundreds of thousands.

Phase 3: Secondary Market

Post-graduation, the token trades in a standard constant-product pool. All the machinery from constant-product-amm, impermanent-loss, and lp-profitability applies.

Cost Structure

At the bonding curve stage, the pricing is deterministic — every buyer knows exactly what they will pay for a given quantity, and the contract enforces it. Compare this with a TradFi IPO:

Feature	Pump.fun bonding curve	TradFi IPO bookbuilding
Price discovery	Deterministic function	Human negotiation
Allocation	First-come, first-served	Discretionary
Transparency	Fully on-chain	Opaque
Minimum size	~0.01 SOL	Institutional minimums
Time to market	~2 minutes	3—6 months
Aftermarket	Automatic graduation	Separate listing process

Connection to Yield Curve Fitting

There is an unexpected parallel between bonding curves and the yield curve work in fixed income. Both are smooth, deterministic pricing functions that map a single variable (supply or maturity) to a price (token cost or yield). The Nelson-Siegel or Svensson parametric families for yield curves play the same structural role as the polynomial or virtual-AMM parametrizations for bonding curves: a small number of parameters that determine the entire pricing surface.

The cost integral $C(0,S)={\int }_0^{S}p(s)ds$ is analogous to the discount function $D(T)=\exp \left(-{\int }_0^{T}r(s)ds\right)$ in fixed income — both integrate a marginal rate to get a cumulative price. The mathematical machinery transfers directly.

Manipulation and Risks

Bonding curves are not without pathology:

• Front-running: On a public blockchain, a pending buy transaction is visible in the mempool. A bot can insert a buy before it and a sell after, capturing the price difference. This is a specific form of MEV.
• Creator extraction: The creator can buy early at low prices, promote the token, and sell after others have pushed the price up. The bonding curve does not prevent this — it only makes the pricing transparent.
• Graduation sniping: Bots monitor bonding curves approaching the 85 SOL threshold and buy large positions in the AMM pool immediately after graduation, before other participants react.

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Companion notebook: notebooks/defi/05-bonding-curves.py — Altair plots of linear, quadratic, and virtual-AMM bonding curves; cost integrals; simulation of a Pump.fun token lifecycle from launch through graduation.

Questions to sit with:

1. The virtual AMM bonding curve is concave in supply (diminishing marginal price growth). A quadratic bonding curve is convex (accelerating growth). What are the economic implications of concavity vs. convexity for early vs. late buyers?
2. The 85 SOL graduation threshold is fixed. How would you design an adaptive threshold that accounts for SOL price volatility?
3. A bonding curve guarantees a bid at every price level. An order book does not. What is the cost of this guarantee — where does the risk go?