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DeFi Reference

Power perpetuals and everlasting options

1. TL;DR

Section titled “1. TL;DR”

An everlasting option is a perpetual future’s funding-payment trick applied to an option instead of a future: it pays funding equal to (mark − current payoff) instead of (mark − index), which lets a trader hold options exposure forever without ever rolling into a new expiry. A power perpetual pushes the same idea further by targeting indexp^p instead of a strike-based payoff at all, giving pure convex (“optionality”) exposure with no strike and no expiry — the p=2p=2 case, nicknamed squeeth, was the first to launch live, on Ethereum via Opyn, though it later shut down. No production power-perpetual or everlasting-option venue was found on Solana in the sources for this page; both instruments remain a small, largely Ethereum-native corner of DeFi derivatives relative to linear perpetuals.

2. Explain it simply

Section titled “2. Explain it simply”

Analogy

Section titled “Analogy”

An ordinary option is like a home insurance policy you have to renew every year, paying a new premium and re-signing paperwork each time, with the risk that you forget to renew right when you need the coverage most. An everlasting option is like a policy that auto-renews forever with a small recurring charge automatically deducted, so you never have to think about a renewal date. A power perpetual, meanwhile, is less like insurance and more like a lever that amplifies itself: pull it twice as hard and it doesn’t just move twice as far, it moves four times as far, because it’s rigged to respond to the square of how hard you pull.

ELI5

Section titled “ELI5”

Normal options eventually expire, so keeping the same bet going means selling the old contract and buying a new one — paying a small fee each time to whoever is on the other side. An everlasting option never expires: instead of re-buying, you make a small ongoing payment, sized so the contract always behaves like the option you’d get from constantly, automatically re-buying it. A power perpetual is a cousin instrument that skips the strike price and expiry date altogether — a bet that scales up faster than the underlying price: if the price doubles, a power-two bet is worth four times as much, and it keeps paying an ongoing fee (a “premium yield”) to whoever is on the safer side.

Step-by-step walkthrough

Section titled “Step-by-step walkthrough”

Scenario: Alice holds one contract of a $3,000-strike everlasting ETH put, with funding paid once daily at midnight; the funding fee is (mark − current payoff).

  1. Position opened (state: ETH at $3,000, put payoff = $0, Alice buys at a mark price of, say, $180 reflecting embedded time value). Alice pays $180 for permanent downside protection below $3,000, with no expiry date to track or renew.
  2. ETH falls (state: index $2,900, so the put’s payoff is now $3,000 − $2,900 = $100). Alice’s position gains intrinsic value; if the mark price is $150 that night, funding is (mark − payoff) = $150 − $100 = $50, which Alice — long the put — pays to whoever is short. This is smaller than before because the option is now more clearly “in the money” and closer to its true worth.
  3. ETH keeps falling (state: index $2,700, payoff = $300). Alice’s hedge is doing its job: whatever she loses holding ETH elsewhere, the put’s rising value offsets. Because she never had to roll a monthly contract to keep this protection alive, she paid no bid-ask spread beyond her original entry and exit.
  4. ETH rallies back above the strike (state: index $3,100, payoff = $0). Funding that night is (mark − 0), a small positive number reflecting only the option’s remaining time value; Alice keeps paying this modest ongoing cost for as long as she wants to keep the hedge live, exactly as she would pay an insurance premium.

Common misconceptions

Section titled “Common misconceptions”
  • Myth: An everlasting option is the same thing as a “perpetual American option” that can be exercised any time. Reality: A perpetual American option requires a market maker to take on unlimited up-front risk and is “effectively never traded”; an everlasting option instead reuses the funding-payment trick from perpetual futures, which is why White and Bankman-Fried named it “everlasting” rather than “perpetual” to distinguish it (Paradigm, 2021-05).
  • Myth: Power perpetuals need a strike price like an option does. Reality: A power perpetual has no strike at all — it targets indexp^p directly, which is what lets it “consolidate much of options market liquidity into a single instrument” instead of fragmenting across strikes (Paradigm, 2021-08).
  • Myth: Because it has convex (“positive gamma”) payoff, holding a power perpetual long-term is free money. Reality: Longs pay a continuous “premium yield” to shorts specifically because of that convexity — it is priced, not free, exactly the way a call option’s time value is not free.
  • Myth: Any power and funding period combination can be priced and traded safely. Reality: The paper shows that “poorly formulated power perpetuals may have prices which fail to converge” — for high powers, high volatility, or long funding periods, the theoretical price becomes infinite, so implementers must choose a sufficiently short funding period (Paradigm, 2021-08).
  • Myth: Squeeth (the ETH² power perp) is still a live, widely-traded product. Reality: Opyn shut Squeeth down; the protocol had attracted “over 18,000 unique users” who traded “more than $1.8 billion in ETH exposure” before closing (Opyn, secondary, as of 2024-11).

If you only remember one thing

Section titled “If you only remember one thing”

Everlasting options and power perpetuals both replace an option’s expiry-and-roll cycle with a continuous funding payment — the same trick that makes a perpetual future work, just applied to a payoff curve instead of a straight line.

3. How it works

Section titled “3. How it works”

From rolling options to everlasting options

Section titled “From rolling options to everlasting options”

A European option’s payoff is max⁡(spot−strike,0)\max(\text{spot}-\text{strike},0) for a call or max⁡(strike−spot,0)\max(\text{strike}-\text{spot},0) for a put, and it is worth strictly more than that payoff before expiry because of remaining time value. Anyone who wants continuous options exposure has to roll: sell the expiring contract, buy the next one, and pay a market maker’s spread each time — a cost that compounds with liquidity fragmentation, since market makers must split capital across every strike and every expiry. An everlasting option removes the expiry by paying a funding fee, once per period, of

Funding=Mark−Payoff\text{Funding} = \text{Mark} - \text{Payoff}

where Payoff is the option’s current intrinsic value (its payoff if exercised right now) — the only change from a plain perpetual future’s Mark−Index\text{Mark}-\text{Index} funding. Setting the strike to zero on a call recovers ordinary perpetual-future funding exactly, since a zero-strike call’s payoff is just the spot price itself.

Pricing by equivalent portfolio

Section titled “Pricing by equivalent portfolio”

White and Bankman-Fried price everlasting options with a no-arbitrage argument: an everlasting option that pays funding once per period is worth exactly the same as a portfolio of ordinary expiring options with the same strike, one expiring at each future funding date, in geometrically shrinking size:

Price=∑i=1∞12i C(ti)\text{Price} = \sum_{i=1}^{\infty} \frac{1}{2^i}\, C(t_i)

where C(ti)C(t_i) is the Black-Scholes price of a regular option with the same strike expiring at the ii-th funding time tit_i — one half of a contract expiring at the next funding date, one quarter at the date after that, one eighth after that, and so on, a series that sums to exactly one contract. The intuition: at each funding payment, half of the “virtual” portfolio expires and pays out, and the funding fee is precisely the cost of buying a fresh half-contract’s worth of exposure to replace it — “unlike in the case of manual rolling, these new contracts are distributed across multiple expiries, no spreads have to be paid, and no execution risk is incurred.” This framework generalizes to price any funding-fee-based perpetual, including plain perpetual futures and even “floating strike” everlasting options whose strike tracks a moving average of the spot price.

Power perpetuals

Section titled “Power perpetuals”

A power perpetual targets indexp^p directly: funding is Mark−Indexp\text{Mark}-\text{Index}^p, paid by longs to shorts, and Paradigm calls this a premium yield to emphasize that it compensates shorts for writing convex exposure. A short position is opened by locking collateral in a vault and minting (borrowing) the power perpetual to sell, subject to

Collateral ratio=(collateral qty)×(collateral price)(perp qty)×(index price)p>1\text{Collateral ratio} = \frac{(\text{collateral qty})\times(\text{collateral price})}{(\text{perp qty})\times(\text{index price})^{p}} > 1

Because an expiring power claim’s expected value under Black-Scholes assumptions has a clean closed form — for spot SS, power pp, volatility vv, and time tt, E[ST p]=Sp e p(p−1)v2t/2E[S_T^{\,p}] = S^{p}\,e^{\,p(p-1)v^2t/2} (ignoring drift) — pricing a power perpetual just applies the same geometric-series logic as everlasting options to this simpler building block:

Perpp≈Sp∑i=1∞12i e p(p−1)v2(if)/2\text{Perp}_p \approx S^{p}\sum_{i=1}^{\infty} \frac{1}{2^i}\, e^{\,p(p-1)v^2 (i f)/2}

for a funding period ff. This is a geometric series with ratio r=12ep(p−1)v2f/2r=\tfrac12 e^{p(p-1)v^2 f/2}, and it only converges — i.e., the power perpetual has a finite, well-defined price at all — when r<1r<1; the paper is explicit that “the higher the power and volatility, the more valuable longer-dated expiring power futures get, and the longer the funding period, the more of the value of the power perpetual is concentrated in longer-dated power futures,” to the point that “the equivalent portfolio can become infinitely valuable” for badly chosen parameters. In practice this problem is “easily avoided… by choosing a sufficiently small funding period.”

Squeeth: the p = 2 case

Section titled “Squeeth: the p = 2 case”

The ETH2\text{ETH}^2 power perpetual, nicknamed squeeth (“squared ETH”), is the paper’s worked example: “it will 16X as ETH 4Xs,” and it has “the convenient property of having a constant gamma… meaning it offers constant optionality regardless of the price of ETH” — unlike a vanilla option, whose sensitivity to price changes (gamma) is highest near the strike and fades away from it. The ETH3\text{ETH}^3 power perp is more convex still, 64x-ing when ETH 4x’s, and trades at a visibly larger premium to ETH3\text{ETH}^3 because of the extra optionality embedded in the cube.

4. Worked numeric example

Section titled “4. Worked numeric example”

Everlasting put, continuing §2. ETH is trading at $2,900 against a $3,000-strike everlasting put with daily funding. Current payoff = $3,000 − $2,900 = $100. If the put’s mark price that day is $150 (reflecting $50 of remaining time value on top of the $100 intrinsic value), Alice — long — pays shorts Mark−Payoff=$150−$100=$50\text{Mark}-\text{Payoff}=\text{\textdollar}150-\text{\textdollar}100=\text{\textdollar}50 that day. If instead ETH rallies to $3,100 (above the strike), the payoff is $0, and if mark is $50 (pure time value, no intrinsic value left), Alice pays shorts the full $50. In both cases, whether ETH is above or below the strike, Alice’s daily cost is exactly the option’s time value — the same as what she’d implicitly forfeit rolling a conventional option, except she never has to execute a trade to do it.

Squeeth-style power perp, illustrating convexity. Suppose an ETH2\text{ETH}^2 power perpetual is trading with ETH at $2,000 as the reference index. If ETH rises 41.4% to $2,828 (a 2\sqrt2 multiple), the ETH² index rises by a factor of (2)2=2(\sqrt2)^2=2 — a 100% gain from a 41.4% move in the underlying, the convexity the instrument is built to deliver. If ETH instead doubles outright to $4,000, the index quadruples (22=42^2=4); if ETH 4x’s to $8,000, the index 16x’s, matching the paper’s stated result exactly. This asymmetric upside is why longs must continuously pay the premium yield: shorts are compensated in the same way an option writer is compensated for the convex risk they underwrite, and the fair size of that ongoing payment is what the geometric-series pricing formula in §3 computes.

5. Where it’s used

Section titled “5. Where it’s used”

Ethereum

Section titled “Ethereum”
  • Opyn Squeeth — the first live power perpetual, the ETH2\text{ETH}^2 instrument described above; live from January 2022 until Opyn shut it down, having attracted “over 18,000 unique users” who traded “more than $1.8 billion in ETH exposure,” with its associated Crab delta-neutral strategy reportedly delivering “50%+ returns” over its life (Opyn, secondary, as of 2024-11). Opyn subsequently announced plans for a successor product line, Opyn Markets, though this page found no confirmation of its live launch status.
  • Everlasting options — the paper’s core proposal has not been found in any sources as a currently-live, widely-used production venue; it remains primarily an influential pricing framework that later shaped power perpetuals and the “Everything Is A Perp” unification (see /derivatives/perpetual-futures/) rather than a standalone product line with confirmed adoption.
  • Floor perpetuals — a related, NFT-specific extension proposed by the same author, letting NFT holders mint perpetuals against locked NFTs to hedge floor-price risk without selling; the paper is explicit this depends on solving “index reliability and spot NFT floor liquidity” first, and this page found no evidence of a widely-adopted live implementation (Paradigm, 2021-08).

Solana

Section titled “Solana”
  • n/a — no power-perpetual or everlasting-option protocol was found in the sources reviewed for this page. Solana’s live perpetual venues (Drift, Jupiter Perps/JLP; see /derivatives/perpetual-futures/) all trade ordinary, linear (p=1p=1) perpetual futures rather than power-indexed or option-style payoffs.

6. Risks, attacks, and incidents

Section titled “6. Risks, attacks, and incidents”
  • Non-convergence risk is a design flaw, not just an edge case. Because a power perpetual’s fair price is an infinite geometric series, choosing too high a power, too high a volatility assumption, or too long a funding period can make the theoretical price literally undefined — a protocol that doesn’t respect this constraint could end up systematically mispricing the instrument it’s meant to fund correctly.
  • Funding drag is easy to misunderstand. A naive long-squeeth holder is not simply “long convex ETH exposure” — every funding period bleeds premium yield to shorts, so a position that never moves still loses money over time purely to funding, the same negative-theta dynamic a long options position has, which is precisely why delta-neutral strategies like Opyn’s Crab strategy exist to harvest that yield instead of paying it.
  • Vault liquidation risk concentrates on the short side. Because indexp^p moves faster than the index itself for p>1p>1, a short power-perpetual vault’s liabilities can spike far more violently than an equivalent linear-perp short’s during a large index move, requiring more conservative collateralization and faster liquidation than a standard 1x perp position.
  • Product discontinuation as a realized outcome. Squeeth’s shutdown after roughly two and a half years live — despite genuine usage ($1.8B in cumulative traded ETH exposure, 18,000+ users) — is itself evidence that power perpetuals have so far struggled to find durable product-market fit relative to plain perpetual futures and vanilla options, whatever the underlying mechanism’s theoretical elegance. No sources reviewed for this page describe a security exploit, oracle manipulation, or liquidation-cascade incident specific to a live everlasting-option or power-perpetual venue — the primary documented risk in the record is business/product discontinuation rather than a hack.

7. Open problems

Section titled “7. Open problems”
  • Whether a real market ever forms. The Everlasting Options paper’s own “Future Work” section leaves this explicitly open: “Is there a market for everlasting options, or for other new funding-fee-based perpetuals? Which types will be most useful? How can they best be parameterized?” — questions the sources for this page found still largely unanswered given Squeeth’s eventual shutdown.
  • Liquidation criteria on margin. The same paper flags “what are appropriate liquidation criteria for those trading on margin?” as unresolved for funding-fee-based perpetuals generally, a question that matters more, not less, for convex power perpetuals whose liabilities can move nonlinearly.
  • Index reliability for non-financial underliers. Floor Perps identifies “index reliability and spot NFT floor liquidity” as the “largest barriers” to extending this framework beyond liquid, continuously-priced assets like ETH — a constraint that applies to any attempt to build a power perpetual or everlasting option on a thin or manipulable market.
  • Liquidity consolidation versus fragmentation. Power Perpetuals argues a single power-perpetual instrument could “consolidate much of options market liquidity” that would otherwise fragment across strikes and expiries, but whether that consolidation actually outcompetes vanilla options and linear perpetuals in practice remains an open empirical question given the limited track record so far.

8. Ethereum vs Solana

Section titled “8. Ethereum vs Solana”
AspectEthereumSolana
Live everlasting optionsNone found (research framework only)n/a — none found
Live power perpetualsOpyn Squeeth (ETH2\text{ETH}^2), live 2022–2024, since shut downn/a — none found
Funding basisMark − payoff (everlasting options) or Mark − indexp^p (power perps)Mark − index (linear perps only; see /derivatives/perpetual-futures/)
Dominant derivative shape actually tradedOrdinary options (vaults, see /derivatives/options/) and linear perpetuals (GMX, dYdX)Linear perpetuals only (Drift, Jupiter Perps)
Underlying reasonEthereum’s research culture (Paradigm, Opyn) produced and tested the mechanism firstNo sources indicate active development of a power-perp or everlasting-option primitive on Solana as of this writing

Both power perpetuals and everlasting options originated as Ethereum-native research, built and tested there because that is where the relevant options infrastructure (Opyn) and research funding (Paradigm) were concentrated; Solana’s derivatives ecosystem to date has focused on scaling ordinary linear perpetuals (see /derivatives/perpetual-futures/) rather than porting this specific non-linear payoff design, and this page found no sources suggesting that is changing.

9. Reference doc

Section titled “9. Reference doc”

The reference

Section titled “The reference”

Everlasting Options — Dave White, Sam Bankman-Fried (Paradigm), 11 May 2021. paradigm.xyz/writing/everlasting-options

Summary of the reference

Section titled “Summary of the reference”

The paper opens with option basics — call and put payoffs, the concept of time value, and Black-Scholes pricing — before introducing the core problem it solves: rolling positions. An investor hedging with options must repeatedly close expiring contracts and open new ones, paying a market-maker spread each time because market makers cannot distinguish informed from uninformed order flow and must charge for that risk on every trade; the same problem afflicts expiring futures. Perpetual futures, introduced by BitMEX in 2016, solved this for futures via a funding fee of (mark − index); the paper’s contribution is extending exactly this idea to options, defining an everlasting option whose funding fee is (mark − payoff) instead. It proves, via a no-arbitrage argument, that an everlasting option is priced identically to a portfolio of ordinary expiring options with the same strike — one half expiring at the next funding date, one quarter at the date after, and so on in a geometric series summing to one contract — and that the funding payment corresponds exactly to the cost of rolling that virtual portfolio forward, without the spread or execution risk of manual rolling. A worked formal proof (Appendix B) shows that if the everlasting option’s market price ever diverged from this equivalent-portfolio price, an arbitrageur could construct a risk-free profit by buying one side and shorting the appropriately-weighted basket, then unwinding it piece by piece at each funding date. The paper closes by noting the framework generalizes to price any funding-fee-based perpetual derivative with a definable expiring payoff, including plain perpetual futures, binary options (useful as protocol-failure hedges), and “floating strike” everlasting options whose strike itself tracks a moving average of the spot price.

Key quotes

Section titled “Key quotes”

“Everlasting options give traders long-term options exposure without the effort, risk, or expense of rolling positions.” (Introduction)

“Because market makers don’t know who is informed and who is uninformed, they must charge a fee, called a spread, on every trade.” (Rolling Positions § Problems)

“Everlasting options work exactly the same way as perpetual futures, with one difference: the funding fee is calculated as the difference of the mark price and the current payoff of the option.” (Everlasting Options § Mechanism)

“This framework can be used to price any funding-fee based perpetual derivative for which we can price the expiring equivalents, not just European calls and puts.” (Further Applications)

How to read the original

Section titled “How to read the original”

Background needed: European call/put payoff definitions and a passing familiarity with Black-Scholes pricing and perpetual-future funding (see /derivatives/perpetual-futures/ for the latter). Skip the full formal proof in Appendix B on a first pass — the “Equivalent Portfolio Intuition” section a few paragraphs earlier gives the same result informally and is easier to follow. The hardest part is the “Price Dynamics Around Funding Payment” discussion: the key is to always reason about the instant immediately before or immediately after funding is paid, never “at” the funding instant itself, because — like a dividend-paying stock — the price necessarily jumps discontinuously at that moment, and conflating the two sides of the jump is the single most common source of confusion when reasoning about any funding-fee-based perpetual.

What changed since

Section titled “What changed since”
  • The follow-up paper “Power Perpetuals” (2021-08, covered below) applies this exact machinery to indexp^p instead of option payoffs, producing a strike-free, expiry-free convex instrument.
  • “Everything Is A Perp” (2024-03) folds both instruments into a single unifying framework alongside stablecoins and AMM liquidity positions (see /derivatives/perpetual-futures/).
  • The one live implementation of this family of ideas, Opyn’s Squeeth, shut down in November 2024 after roughly $1.8 billion in cumulative traded exposure — the clearest real-world data point on adoption to date (Opyn, secondary).

Secondary references

Section titled “Secondary references”
  1. Opyn, “Squeeth has Shutdown” — read if you want the concrete usage numbers and stated reasons behind the one live power perpetual’s discontinuation.
  2. “Everything Is A Perp” (Paradigm, 2024-03) — read if you want to see everlasting options and power perpetuals placed side by side with stablecoins and AMM liquidity in one funding-payment framework.
  3. “Floor Perps” (Paradigm, 2021-08) — read if you want to see the same funding mechanism proposed for NFT-collateralized, non-financial underliers.

The reference

Section titled “The reference”

Power Perpetuals — Dan Robinson, Dave White, Zubin Koticha, Andrew Leone, Alexis Gauba, Aparna Krishnan (Paradigm), 17 August 2021. paradigm.xyz/writing/power-perpetuals

Summary of the reference

Section titled “Summary of the reference”

Building explicitly on Everlasting Options, this paper defines a power perpetual: a perpetual indexed to indexp^p for a chosen power pp, funded by a premium yield of (mark − indexp^p) paid from longs to shorts. Shorting requires locking collateral in a vault and minting perps against it, subject to a collateral ratio that compares collateral value to perp-quantity-times-indexp^p; falling below a safe ratio triggers liquidation. The pricing section notes powers greater than one give the instrument positive convexity (“gamma” in options language), so it trades at a premium to the raw indexp^p just as an option trades at a premium to its intrinsic value; the authors price an expiring power derivative under Black-Scholes assumptions (noted to be “significantly simpler” than pricing a vanilla option because a power claim’s expectation has closed form) and then apply the same equivalent-portfolio geometric-series logic from Everlasting Options to derive the perpetual’s price and its premium yield in closed form. A convergence section is the paper’s most important caveat: because the theoretical price is an infinite sum, sufficiently high combinations of power, volatility, and funding period make that sum diverge, meaning “poorly formulated power perpetuals may have prices which fail to converge” — a problem the authors say is “easily avoided in practice by choosing a sufficiently small funding period.” Worked examples for the ETH2\text{ETH}^2 (“squeeth,” with constant gamma) and ETH3\text{ETH}^3 power perpetuals close the paper, alongside a public Python pricing implementation and empirical correctness tests.

Key quotes

Section titled “Key quotes”

“If the price of ETH doubles, the ETH^2 power perp 4Xs, the ETH^3 power perp 8Xs, and the ETH^5 power perp 32Xs.” (Introduction)

“Power perpetuals provide global options-like exposure without the need for either strikes or expiries, giving them the potential to consolidate much of options market liquidity into a single instrument.” (Introduction)

“Unlike stock Everlasting Options, which we can always price, poorly formulated power perpetuals may have prices which fail to converge.” (Convergence)

“It has the convenient property of having a constant gamma… meaning it offers constant optionality regardless of the price of ETH.” (Examples § ETH² Power Perp)

How to read the original

Section titled “How to read the original”

Background needed: the Everlasting Options equivalent-portfolio pricing method above, plus the concept of options “gamma” (sensitivity of an instrument’s price-sensitivity to further price moves). Skip the linked Python notebooks and StackExchange derivation on a first pass — they support the pricing claims but are not needed to follow the paper’s argument. The hardest part is the Convergence section: the intuition to hold onto is that a power perpetual’s price is a weighted sum over infinitely many future funding dates, and because E[STp]E[S_T^p] grows with TT for p>1p>1 under Black-Scholes assumptions (higher powers amplify volatility’s upward drift in expectation), that infinite sum only has a finite value when the funding period is short enough relative to the chosen power and assumed volatility to keep the series’ growth rate below its 1/2-per-term shrinkage rate.

What changed since

Section titled “What changed since”
  • Opyn implemented the p=2p=2 case as Squeeth in January 2022, the first (and to date only widely-used) live power perpetual; it shut down in November 2024 (Opyn, secondary).
  • “Everything Is A Perp” (2024-03) reframes power perpetuals as one point on a continuum that also includes stablecoins (p=0p=0) and margined futures (p=1p=1), arguing the whole DeFi derivatives landscape is a family of power perpetuals (see /derivatives/perpetual-futures/).
  • No successor power-perpetual product with confirmed live volume was found in the sources reviewed for this page as of 2026-08.

Secondary references

Section titled “Secondary references”
  1. Everlasting Options (Paradigm, 2021-05) — read first; Power Perpetuals assumes its pricing machinery.
  2. Everything Is A Perp (Paradigm, 2024-03) — read if you want power perpetuals unified with stablecoins, futures, and AMMs.
  3. Opyn’s public Squeeth pricing/implementation notebooks (linked from the paper) — read if you want a runnable, empirically-tested pricing implementation rather than the closed-form derivation alone.

10. Sources

Section titled “10. Sources”