Bonding curves and token launches

1. TL;DR

Launching a brand-new token with no pre-existing market is a mechanism-design problem on its own: price it too low and insiders snipe the difference, too high and nobody shows up. A bonding curve solves this by making a smart contract itself the counterparty from token zero, deriving a price from a formula rather than an order book that doesn’t exist yet. Gradual and Variable Rate Gradual Dutch Auctions (GDA/VRGDA) target a chosen issuance schedule by raising or lowering price relative to how far ahead of or behind schedule sales are running; DAICOs gate a fundraising team’s spending behind an ongoing token-holder vote; and Pump.fun’s constant-product curve, running almost entirely on synthetic reserves until it “graduates” a token to a real pool, became the dominant venue for permissionless memecoin issuance on Solana, with only about 0.63% of the 655,770 tokens created on it in a one-month 2025 sample ever reaching that graduation point.

2. Explain it simply

Analogy

Imagine a bakery that hasn’t opened yet raising money by pre-selling loaf vouchers: instead of needing someone to stock a shop with both bread and cash before day one, the bakery sells vouchers from a vending machine whose price rises the faster people buy and falls if nobody’s buying, so the machine sets its own fair-ish price using nothing but a formula and the pace of sales.

ELI5

When someone invents a brand-new token, there is no shop yet where people can buy or sell it — no one has put money and tokens together to trade against. A bonding curve fixes this by turning a small computer program into that shop from the very first purchase: the program itself holds some starting supply of both a base currency and the new token, and every purchase or sale moves along a fixed formula that decides the price. As more of the base currency piles up inside the program, the token usually gets more expensive; if the pile of base currency crosses a set target, the program can hand its holdings over to an ordinary trading pool so the token keeps trading normally from then on.

Step-by-step walkthrough

Scenario: a creator launches a new coin on Pump.fun with no funds of their own, using the program’s standard starting parameters.

1. Creation (state: bonding curve initialized with 30 SOL of synthetic reserves and 1,073,000,000 synthetic token reserves; creator pays nothing). The curve is live and tradeable immediately — no liquidity was deposited by the creator.
2. First buy (state: a buyer sends 1 SOL; curve reserves move from 30/1,073,000,000 to roughly 30.99/1,038,722,168, per the constant-product formula net of a 1% fee). The buyer receives newly-issued tokens from the curve’s real (non-synthetic) supply, and the implied price per token ticks up slightly.
3. More buys accumulate (state: total virtual SOL reserves climb toward 115 SOL as buying continues). Every trade moves along the same $x\cdot y=k$ curve; the price keeps rising as the token reserve is depleted.
4. Graduation (state: virtual SOL reserves reach 115 SOL, meaning 85 SOL of real SOL has been raised; the curve’s complete flag is set). The remaining 20.69% of total supply plus the 85 real SOL migrate to a new PumpSwap pool; the curve’s LP tokens are burned, permanently locking that liquidity.
5. After (state: token trades on an ordinary constant-product AMM pool, no longer on the bonding curve). Price discovery continues exactly like any other AMM pair from this point on.

Common misconceptions

• Myth: A bonding-curve token has real, team-provided liquidity from the start. Reality: Pump.fun’s curve begins with entirely synthetic (“virtual”) reserves; real SOL only accumulates as buyers actually pay in (Pump program README).
• Myth: VRGDA guarantees a token sells at its target price. Reality: VRGDA only nudges the price up when sales run ahead of a chosen schedule and down when they run behind it — the actual clearing price still depends entirely on real demand (Paradigm, 2022-08).
• Myth: DAICOs prevent scams. Reality: A DAICO only bounds how fast a team can withdraw funds and lets token holders vote to cut it off; the vote itself remains subject to the same 51%-style attacks as any other on-chain governance (Buterin, ethresear.ch, 2018-01).
• Myth: Getting sniped at launch is just bad luck for retail buyers. Reality: Sniping is a structural, quantifiable MEV strategy — one launch-defense fund alone generated over $7.5 million in front-running profit across a single launchpad’s history (Nethermind, ethresear.ch, 2025-08).

If you only remember one thing

A bonding curve is a formula that turns a smart contract into a token’s very first counterparty, so a market exists before anyone has agreed to be a market maker.

3. How it works

Gradual Dutch Auctions (GDA)

A GDA sells an asset by running many virtual Dutch auctions (starting high, decaying over time) at once, cleared efficiently as a batch. In the discrete version (for NFTs), auction $n$‘s price at time $t$ since it started is $p_n(t) = k\cdot\alpha^n e^{-\lambda t}$, where $k$ is the initial price, $\alpha>1$ a scale factor raising each successive auction’s starting price, and $\lambda$ a decay constant; buying $q$ units when $m$ have sold, $T$ seconds in, costs $P(q) = \sum_{n=m}^{m+q-1} p_n(T) = \dfrac{k\,\alpha^m(\alpha^q-1)}{e^{\lambda T}(\alpha-1)}$. The continuous version (fungible tokens, emission rate $r$) uses $p(t)=k\cdot e^{-\lambda t}$ and integrates: $P(q) = \int_{T-q/r}^{T} p(t)\,dt = \dfrac{k}{\lambda}\cdot\dfrac{e^{\lambda q/r}-1}{e^{\lambda T}}$.

Variable Rate GDAs (VRGDA)

VRGDA generalizes GDA to target an arbitrary issuance schedule $f(t)$ (cumulative units meant to be sold by time $t$) rather than a fixed rate. If $f^{-1}(n)$ is the time the $n$th unit should sell, its price at time $t$ is $\texttt{vrgda}_n(t) = p_0(1-k)^{\,t-f^{-1}(n)}$, where $p_0$ is the target price and $k$ the per-unit-time decay rate: selling on schedule prices it at $p_0$; ahead of schedule, higher; behind schedule, lower. A linear schedule $f(t)=rt$ recovers $\texttt{linear\_vrgda}_n(t)=p_0(1-k)^{t-n/r}$ (isomorphic to a continuous GDA); front-loaded issuance $f(t)=\sqrt t$ gives $\texttt{sqrt\_vrgda}_n(t)=p_0(1-k)^{t-n^2}$; and a logistic schedule asymptoting to a max supply uses $f(t)=\tfrac{2L}{1+e^{-st}}-L$.

DAICOs

Buterin’s DAICO splits a fundraise into a contribution phase followed by a spending phase gated by a tap variable (wei/second the team may withdraw). Token holders can only vote to raise the tap or permanently self-destruct the contract (returning remaining funds pro rata) — never to lower it unilaterally against the team. This is deliberately asymmetric: a malicious raise-tap vote can be ignored by an honest team, and a malicious self-destruct vote only forces a relaunch — the two most damaging attacks (redirecting funds, or freezing them forever) are structurally disallowed rather than merely discouraged.

Pump.fun’s virtual-reserve constant-product curve

Pump.fun mints a fixed 1,000,000,000-token supply per coin and initializes a bonding curve with virtual reserves $x_0=30$ SOL and $y_0 = 1{,}073{,}000{,}000$ tokens (793,100,000 real/sellable, 279,900,000 synthetic and present only to shape the curve), governed by the ordinary constant-product invariant $x\cdot y=k$. A buy of $\Delta x$ SOL (after a protocol fee) moves $x^{\text{tot}} = x_0^{\text{tot}} + r\Delta x$ and $y^{\text{tot}}=k/x^{\text{tot}}$, exactly as in any $xy=k$ pool (see /exchange/cfmm-math/). The curve “graduates” once total virtual SOL reserve reaches 115 SOL — 85 real SOL raised — migrating the remaining token supply and real SOL into a new PumpSwap pool, with the migration’s LP tokens burned so that liquidity can never be withdrawn.

Anti-sniper defenses

Nethermind’s 2025 analysis identifies two properties every “sniper” strategy depends on: speed of execution and the ability to exit a position within milliseconds to minutes. Proposed defenses each break one property directly: partial execution of orders over time (removes the exit-speed advantage), selling time-based bidding (commit to an earliest resale time before knowing the price), a trusted sequencer, or a commit-reveal batch auction (hides order contents until after commitment).

4. Worked numeric example

Pump.fun graduation math, derived from the program’s own on-chain constants. Constant-product invariant $k = x_0\,y_0 = 30\times1{,}073{,}000{,}000 = 32{,}190{,}000{,}000$. Graduation requires selling all 793,100,000 real tokens, leaving virtual token reserves at $1{,}073{,}000{,}000-793{,}100{,}000=279{,}900{,}000$; solving $k/y=x$ gives the total virtual SOL reserve at that point: $32{,}190{,}000{,}000/279{,}900{,}000 = 115.00$ SOL — confirming the well-known “85 SOL raised” figure (115 total minus the 30 SOL the curve started with).

Implied graduation market cap. The marginal price at graduation is $115/279{,}900{,}000=4.108\times10^{-7}$ SOL per token; multiplying by the full 1,000,000,000-token supply gives a graduation market cap of $\approx410.9$ SOL. At the roughly $168/SOL prevailing when this “$69k graduation” figure became folklore in the memecoin community (implying $410.9\times168\approx\text{\textdollar}69{,}000$), the number checks out arithmetically against the protocol’s own published constants — though the actual USD figure moves with SOL’s price and is not itself a protocol parameter.

A single buy, worked through the curve. A buyer sending 1 SOL, net of Pump.fun’s 1% fee ($r=0.99$), moves $x^{\text{tot}}=30+0.99=30.99$ SOL; the new token reserve is $k/x^{\text{tot}}=32{,}190{,}000{,}000/30.99=1{,}038{,}722{,}168$, so the buyer receives $1{,}073{,}000{,}000-1{,}038{,}722{,}168\approx34{,}277{,}832$ tokens (about 4.32% of the curve’s real token supply) for that 1 SOL — a large share early on, shrinking sharply as more SOL accumulates and the curve gets steeper.

Real-world scale (Marino et al., 2026-02). In a one-month sample (1 September – 1 October 2025), Pump.fun saw 655,770 tokens created by 243,123 distinct creator addresses and 2,600,790 distinct trader addresses; only 4,338 coins reached graduation — a frequency of about 0.63%. Among tokens that did graduate, the median number of bonding-curve trading steps to reach graduation was about 457, and the median time from creation to graduation was about 4.4 minutes.

5. Where it’s used

Ethereum

• Gradual Dutch Auctions (GDA) and VRGDA — designed by Paradigm for NFT and token issuance without a pre-existing market; VRGDA was built for and used in Art Gobblers and the 0xMonaco game (Paradigm, 2022-04 and 2022-08).
• DAICOs — proposed by Vitalik Buterin as an ICO-era fundraising primitive; no source in this research confirms a major production DAICO deployment matching the original design exactly, though the tap-and-vote pattern influenced later milestone-based fundraising designs (ethresear.ch, 2018-01).
• Virtuals Protocol “Genesis Launch” — mints 1 billion Agent Tokens per launch on a Uniswap v2 pool with a starting ratio of 42,000 $VIRTUAL : 125,000,000 Agent Tokens; the project’s own anti-sniper defense fund front-runs its own launches, submitting the first 42,000-$VIRTUAL buy itself and airdropping the resulting 62.5 million Agent Tokens to stakers rather than letting a sniper capture them (Nethermind, ethresear.ch, 2025-08).

Solana

• Pump.fun — the dominant permissionless memecoin launchpad; free coin creation, constant-product virtual-reserve bonding curve, graduation to PumpSwap at 85 real SOL raised, burned migration LP tokens (Pump program README, fetched 2026-08-29).
• Meteora’s dynamic bonding curve — used to launch the $LIBRA token on 14 February 2025, which reached a peak market cap of roughly $4.5 billion before crashing over 89% within three hours of its promotional post, with on-chain analysis showing insiders extracted tens of millions of dollars in SOL from the pool before the crash (TRM Labs, secondary, accessed 2026-08-29; see §6).
• LetsBonk.fun — a named competitor to Pump.fun using a similar bonding-curve launch model, cited in the same anti-sniper research as sharing the identical sniping problem (Nethermind, ethresear.ch, 2025-08).

6. Risks, attacks, and incidents

• Virtuals launch sniping, ongoing through 2025-08. The Virtuals Defense Fund’s own front-running strategy — submitting the first buy itself to preempt snipers — has generated an estimated “T+1min expected profit of over $7.5M” and “T+1day expected profit of over $6M” across all Virtuals launches to date; the same research documents the IRIS token launch, where price peaked at “almost 40,000% above the opening price” (Nethermind, ethresear.ch, 2025-08).
• $LIBRA rug pull, 14 February 2025. After Argentine President Javier Milei posted promoting the token on Solana, $LIBRA’s market cap rose to roughly $4.5 billion before crashing more than 89% within three hours and ultimately over 96% from its peak; on-chain investigators traced roughly $90–107 million in SOL and USDC to addresses linked to the project’s team and insiders, and reported investor losses ranged across estimates from about $87 million to over $260 million (TRM Labs; DL News; secondary sources, accessed 2026-08-29).
• Pump.fun’s near-universal failure rate. Only 4,338 of 655,770 tokens created in the sampled month (0.63%) ever reached graduation, consistent with a broader pattern the same authors cite of roughly 88% of Uniswap v2 token launches in late 2024 being classifiable as manipulation schemes (Marino et al., arXiv 2602.14860, 2026-02).
• DAICO governance attacks remain possible. Buterin’s design concedes any vote is “subject to 51% attacks, bribe attacks and other game-theoretic vulnerabilities” — it only bounds the damage such an attack can do (ethresear.ch, 2018-01).

7. Open problems

• Fee model for permissionless launch mechanisms. Neither the GDA nor VRGDA papers specify how a launch platform should be compensated beyond the auction’s own price schedule, leaving platform-level fees to individual implementations (Paradigm, 2022-04/2022-08).
• Can anti-sniper defenses avoid simply relocating the advantage? Nethermind concedes “strategies may still exist that benefit professionals over retail buyers” even under their proposed defenses, just with a “significantly less” advantage than today’s sniping profits — mitigated, not solved (ethresear.ch, 2025-08).
• Whether bonding-curve graduation rates can be predicted, and what that implies. Marino et al. build predictive models conditioning graduation probability on bonding-curve state and behavioral variables, finding markets dominated by informed participants graduate at different rates than those dominated by bot-like activity — but describe this as improving prediction, not as a solved question of how to design curves that filter out purely extractive activity (arXiv 2602.14860, 2026-02).
• Governance-vote security at scale. The DAICO model was proposed in 2018 before large token-voting oracle attacks (like the Polymarket/UMA incident on /exchange/prediction-market-amms/) were well documented; whether tap-and-vote governance is robust against a similarly concentrated voting bloc remains untested at scale in this research’s sources.

8. Ethereum vs Solana

Aspect	Ethereum	Solana
Dominant launch primitive	GDA/VRGDA (NFTs, targeted schedules); ordinary Uniswap v2/v3 pools for token launches	Pump.fun-style constant-product virtual-reserve bonding curve
Liquidity at creation	Creator or protocol typically funds an initial pool (e.g. Virtuals’ 42,000 $VIRTUAL)	Fully synthetic at creation — zero real capital required from the creator (Pump program README)
Graduation / migration concept	Not standard — most launches go straight to a permanent AMM pool	Explicit “complete” state transition to PumpSwap once 85 SOL is raised
Documented sniping scale	Virtuals Defense Fund: >$7.5M (T+1min) front-running profit captured internally rather than by outside snipers (as of 2025-08)	0.63% graduation rate across 655,770 tokens in one month (2025-09), implying the overwhelming majority of launches fail outright (as of 2026-02)
Largest documented single-launch incident	Not identified in this research	$LIBRA, Feb 2025: ~$4.5B peak market cap, >89% crash within three hours

Both ecosystems converge on the same structural problem — a brand-new token needs a counterparty before any market exists — but solve it differently. Ethereum’s GDA/VRGDA lineage is a deliberate, schedule-targeting pricing mechanism aimed mostly at NFTs and curated launches, with sniping addressed through bespoke defenses layered atop an otherwise ordinary AMM pool. Solana’s Pump.fun took the opposite path: near-zero-friction, fully permissionless creation with one standardized curve, trading enormous volume for an extremely high failure rate, and repeatedly the venue for the ecosystem’s largest launch-related losses.

9. Reference doc

The reference

Predicting the success of new crypto-tokens: the Pump.fun case — Giulio Marino, Manuel Naviglio, Francesco Tarantelli, Fabrizio Lillo, 16 February 2026. arxiv.org/abs/2602.14860

Summary of the reference

The paper studies Pump.fun as a natural experiment in permissionless token issuance, motivated by the observation that “crypto markets are characterized by an extreme proliferation of new assets and by a vanishingly small fraction of ‘successful’ launches.” It documents the exact mechanics of Pump.fun’s two-phase architecture and builds predictive models of graduation probability.

Mechanism. Each coin mints a fixed $10^9$-token supply, split into 793.1 million real tokens for the virtual bonding curve and 206.9 million reserved for a future real pool. The virtual AMM initializes at $x_0=30$ SOL, $y_0^{\text{virt,mint}}=793.1$M real tokens plus $y_0^{\text{synt}}=279.9$M synthetic tokens (total $y_0=1.073$B), governed by $xy=k$. A 1.25% fee applies to swaps (0.3% to the token creator, 0.95% to the protocol, per the paper’s stated fee schedule at time of study). Graduation triggers when total virtual SOL reserves reach 115 SOL (85 SOL of real inflow), migrating the real SOL and remaining token supply to a new PumpSwap constant-product pool with continuous pricing across the transition.

Dataset. Covering 1 September–1 October 2025, the paper records 655,770 tokens created by 243,123 distinct creators and 2,600,790 distinct trader addresses; only 4,338 tokens (≈0.63%) graduated. Among graduated tokens, the median number of bonding-curve steps to graduation is ≈457 and the median time to graduation is ≈4.4 minutes, with both distributions showing pronounced heavy tails.

Findings. Conditioning graduation probability on bonding-curve state (accumulated virtual SOL) and on behavioral/structural variables meaningfully improves predictive power over the state alone; markets with more organic, human-driven participation show different (generally higher, at intermediate stages) graduation likelihood than markets dominated by bot-like activity. The paper also introduces an economic breakeven framework linking graduation probability to trader incentives, and highlights how the virtual-to-real liquidity transition at graduation creates incentives for pre-graduation liquidation by sophisticated participants.

Context. The paper situates Pump.fun within a broader pattern of extractive token launches, citing the authors’ own separate finding that approximately 88% of tokens launched on Uniswap v2 in late 2024 could be classified as manipulation schemes — framing Pump.fun’s near-universal failure rate as part of a cross-chain phenomenon rather than a Solana-specific pathology.

Key quotes

“This technological ease of token creation has a direct statistical consequence: crypto markets are characterized by an extreme proliferation of new assets and by a vanishingly small fraction of ‘successful’ launches.” (Introduction)

“By definition, graduation occurs when the total reserve x_tot,virt reaches 115 SOL.” (Virtual AMM: Pump.fun Launchpad)

“Among these, only 4,338 coins reached the graduation point, corresponding to a graduation frequency of approximately 0.63%.” (The Dataset)

“The median time to graduation is ≃ 4.4 minutes.” (Results)

How to read the original

Background needed: the constant-product AMM formula (see /exchange/cfmm-math/) and a general sense of what “graduation” means for a bonding-curve launchpad. Skip the predictive-modeling machinery (logistic/survival-style regressions) on a first pass and focus on Section III (the exact mechanism, worth reading in full since it’s the primary source for Pump.fun’s on-chain parameters) and Section IV (the dataset). The hardest part to internalize is the virtual-vs-real reserve distinction: the “30 SOL” and roughly 280 million synthetic tokens never actually exist as deposited capital — they only shape the curve’s math — which is why graduation transfers just the real 85 SOL and 206.9 million tokens, not the full virtual totals.

What changed since

• Pump.fun’s own fee schedule has changed over time: the program’s current on-chain Global account (fetched 2026-08-29) shows fee_basis_points = 100 (1%), while this paper (covering September 2025 data) describes a 1.25% fee split 0.3%/0.95% between creator and protocol — a discrepancy this page notes rather than resolves, since both are directly sourced.
• Nethermind’s anti-sniper research (2025-08) postdates most of the GDA/VRGDA/DAICO literature and represents the most recent proposed mitigation for the sniping problem this page discusses.
• The $LIBRA incident (February 2025) postdates Pump.fun’s own design and occurred on a different Solana launchpad (Meteora’s dynamic bonding curve), showing the same virtual/synthetic-liquidity launch pattern carries similar risk even outside Pump.fun specifically.

Secondary references

1. Paradigm, “Gradual Dutch Auctions” (2022-04) — read for the discrete/continuous GDA price-function derivations underlying VRGDA.
2. Paradigm, “Variable Rate GDAs” (2022-08) — read for the production Solidity implementation and worked issuance-schedule examples (linear, sqrt, logistic).
3. Buterin, “Explanation of DAICOs” (ethresear.ch, 2018-01) — read for the original tap-and-vote governance design and its game-theoretic security argument.
4. Nethermind (The-CTra1n et al.), “Anti-Sniper Defenses: Token Launches & Beyond” (ethresear.ch, 2025-08) — read for concrete, quantified sniping profit figures and proposed mitigations.

10. Sources

• Predicting the success of new crypto-tokens: the Pump.fun case — Giulio Marino, Manuel Naviglio, Francesco Tarantelli, Fabrizio Lillo — 2026-02-16 — https://arxiv.org/abs/2602.14860
• Gradual Dutch Auctions — Frankie, Dan Robinson, Dave White, Andy8052 (Paradigm) — 2022-04-04 — https://www.paradigm.xyz/writing/gda
• Variable Rate GDAs — transmissions11, Frankie, Dave White (Paradigm) — 2022-08-24 — https://www.paradigm.xyz/writing/vrgda
• Explanation of DAICOs — Vitalik Buterin — 2018-01-06 — https://ethresear.ch/t/explanation-of-daicos/465
• Anti-Sniper Defenses: Token Launches & Beyond — Conor McMenamin, Nikola Mratinic, Angel Dominguez (Nethermind) — 2025-08-06 — https://ethresear.ch/t/anti-sniper-defenses-token-launches-beyond/22882
• Pump program README — pump-fun/pump-public-docs (GitHub) — fetched 2026-08-29 — https://raw.githubusercontent.com/pump-fun/pump-public-docs/main/docs/PUMP_PROGRAM_README.md
• The $LIBRA Affair: Tracking the Memecoin That Launched a Scandal in Argentina (secondary) — TRM Labs — accessed 2026-08-29 — https://www.trmlabs.com/resources/blog/the-libra-affair-tracking-the-memecoin-that-launched-a-scandal-in-argentina
• Javier Milei’s $4.5bn Libra memecoin scandal rocks Argentina: a timeline (secondary) — DL News — accessed 2026-08-29 — https://www.dlnews.com/articles/markets/how-did-milei-libra-memecoin-crash/