On-Chain Options & Power Perps

Everything so far has been linear: a perp’s PnL is a straight line in the price — up 2%, you make 2% of notional; down 2%, you lose it. Options add the one thing a perp can’t: curvature. An option’s payoff bends, capping your loss while leaving your gain open, and that bend (convexity, a.k.a. gamma) is the whole reason options exist. This lesson brings options on-chain, then pushes to the frontier — power perpetuals, an exotic that packages pure convexity into an everlasting, funding-anchored contract. This is where the DeFi derivatives branch gets genuinely strange, and genuinely powerful.

A two-minute options refresher (the curvature you’ll build on)

You met options in the pricing course; here’s the bend that matters. A call gives the right (not obligation) to buy at a strike $K$; a put, the right to sell at $K$. A long option’s payoff at expiry is a hockey stick:

• Long call: worthless below $K$, then rises 1-for-1 with the price above $K$. Payoff $= \max(S - K,\, 0)$.
• Long put: worthless above $K$, then rises as the price falls below $K$. Payoff $= \max(K - S,\, 0)$.

The kink at the strike is the curvature. Your loss is capped at the premium you paid, but your gain is open (a call) or large (a put). That asymmetry — limited loss, open gain — is impossible with a linear perp, where loss and gain are symmetric. Here’s the long call’s bent payoff:

Bringing options on-chain: AMMs and vaults

A spot AMM prices a token; an options protocol has to price a contract whose value depends on strike, time to expiry, and volatility (the Black–Scholes inputs). On-chain, two designs dominate.

Design A — option AMMs / pricing curves. Liquidity sits in a pool, and a pricing formula (often a Black–Scholes approximation or a bonding curve fed by an implied-volatility parameter) quotes a premium for each strike and expiry. Traders buy and sell options against the pool; the pool’s LPs are the counterparty, much like the peer-to-pool perp design but pricing convex payoffs. The challenge is that option value is sensitive to volatility, so the protocol needs a volatility input (an oracle or a governance parameter) — get it wrong and LPs are systematically mispriced.

Design B — option vaults (DOVs). A DeFi Option Vault lets depositors run a systematic options-selling strategy. The classic is a covered-call vault: depositors hold the asset, and each week the vault sells (writes) out-of-the-money call options on it, collecting premium as yield. Depositors earn the premium income but cap their upside (they’ve sold the calls). The mirror is a put-selling vault (cash-secured puts) that earns premium for taking on downside risk. DOVs (popularized by protocols like Ribbon/Aevo, Dopex, Thetanuts) turned options-writing into a one-click yield product.

Gamma: the curvature, made into a Greek

The single most important second-order quantity for an option is gamma — the curvature of the payoff, formally the rate of change of delta (price sensitivity) as the underlying moves. High gamma means your exposure flips fast as the price moves: a long option gains delta as it goes in your favor and sheds delta as it goes against you, which is exactly the “accelerate into gains, decelerate into losses” property convexity gives you.

The curve below shows how an option’s value bends around the strike — that bend is gamma. A linear perp would be a straight line here; the option’s curve is the convexity you’re paying the premium to own.

The catch with classic options: gamma is tangled up with time and a fixed expiry. An option’s convexity decays as expiry approaches (theta), and you must keep rolling to fixed dates to maintain exposure — the same calendar hassle that made dated futures annoying. Which raises a beautiful question, the same one that birthed the perp: can we strip out the expiry and own pure convexity forever?

Power perpetuals: convexity with the calendar torn off

Here’s the frontier. A power perpetual (the canonical example is Squeeth, “squared ETH”, from Opyn) is a perpetual contract on a power of the price — most famously the square of the price, $S^2$. Because the payoff is $S^2$ rather than $S$, it’s convex by construction: its value curves with the price, giving the holder gamma exposure — and, like a perp, it never expires.

Analogy. A regular perp tracks your altitude. A power perp tracks your altitude squared: climb a little and your reading rises a little; climb a lot and it rises explosively. That accelerating response is convexity — the same bend an option gives you — but baked into a contract you can hold indefinitely, with no strike to choose and no expiry to roll.

The mechanics, by analogy to the plain perp:

• A long power perp (long $S^2$) holds positive gamma — convex, accelerating exposure to big moves in either direction (since $S^2$ rises whether $S$ jumps up or, in relative terms, swings hard). It behaves like being long a strip of options’ convexity, continuously.
• That convexity isn’t free. Just as a long option pays theta (time decay) for its gamma, a long power perp pays a funding rate to the short side. The funding is the everlasting, continuous price you pay to keep owning gamma — the direct analog of an option’s theta, but with no expiry.
• The short side receives that funding for supplying the convexity (and must hedge their negative gamma), exactly as an option seller collects premium/theta for being short gamma.

So a power perp is, in spirit, “an option’s convexity, made perpetual, with funding standing in for theta and the calendar.” It’s the same trick that turned a dated future into a perp — remove the expiry, anchor it with funding — applied to the non-linear payoff of an option.

When to reach for each instrument

A quick practitioner’s map of the whole non-linear toolkit:

The throughline: perps give you direction, options and power perps give you curvature, and on-chain the same funding-rate trick that made futures perpetual makes convexity perpetual too. Once you see funding as “the everlasting cost of carry for any non-spot exposure,” the whole zoo — perps, power perps, even the vaults’ implied financing — falls into one frame.

You’ve now added curvature to your on-chain toolkit — options for defined-risk convexity, DOVs for premium income, and power perps for everlasting gamma, all anchored by the same funding logic. One thread has run through every lesson: the basis, the gap between a derivative and spot, monetized by funding. The final lesson ties it all together into the dominant institutional crypto trade — cash-and-carry, which turns that funding premium into market-neutral yield.