**Everlasting options** are an area of interest for traders hoping to maintain an options position indefinitely without manually rolling the position. Dave White's proposal automates the roll via a *funding rate* (simulating rolling a geometric series of options), which achieves two things:

**Consolidated liquidity.**- The
**spread is split**between opposing position holders, rather than accrued to market makers.

This should in theory mitigate the transaction costs of the roll, though one might imagine that we'd want to roll strikes in addition to expiration. The exchange would have to consider how granular to make the strike prices, and when to update. Traders would have to consider how specific to be with strikes and rolling the strike. It's worth thinking about how to further consolidate liquidity so that we can save transaction costs on the roll.

## Gamma

I assume that gamma is what many traders fundamentally want. The mathematically intuitive way to expose gamma is through the square of the underlying price. We therefore consider the implications of an ETH^2-PERP product, operating under the usual `(MARK - SPOT)`

funding rate. Third and higher degrees may have utility as well for trading higher moments (e.g. kurtosis) and crafting more complex P&L graphs (Taylor's theorem, fundamental theorem of algebra), though ETH^2 probably gives the greatest marginal utility as it enables convex and/or defined-loss positions, as well as views on a trading range versus directional views.

## Theta

The deviation of linear perps from the spot price expresses contango/backwardation (in particular, an exponential-decay-weighted integral over dated futures contango). The deviation of quadratic perps from spot expresses contango *plus* willingness to pay for theta (IV, if you will). Indeed, if we denote the price of ETH at time `t`

as `P_t`

, then the non-discounted "premium" for a dated ETH^2 future should be

E[P_expiry^2] - P_now^2 = Var[P_expiry] + E[P_expiry]^2 - P_now^2 = Var[P_expiry] + 2 * P_now * C + C^2

where `C = E[P_expiry] - P_now`

expresses contango. Of course, we have to ask ourselves whether the perpetual price converges - if variance is finite and the exponential decay rate implied by the funding rate is at least twice the exponential growth rates of price and variance, then it should.

**Remark**: It's convenient that we can price this with only mean and variance, the most natural things for a pricing model to include (for example, Black-Scholes).

## Delta

A trader maintaining a particular Greeks exposure will still have to "roll" the position. The equivalent would be re-hedging. For example, to get delta-neutral one could long ETH^2-PERP and short twice the notional in ETH-PERP, simulating a straddle. The advantage of re-hedging in this framework vs rolling strikes would be 1) more consolidated liquidity, and 2) maximally granular control over exposure.

## Notes

**Cross-Margining**: If we are constructing a position using ETH-PERP, ETH^2-PERP, and ETH, we'd like to be able to cross-margin these positions. An established exchange would thus likely be the best place to deploy this product.**Numerical Stability**: It may be better UI to denominate ETH^2 in ETH rather than USDC. This has the additional benefit of normalizing the "contango" and "variance" factors.

## Limitations

**Expiration Agnostic**: All perpetuals have the drawback of not being to express views about the shape of the forward curve (e.g. calendar spreads). If there is demand, dated ETH^2 futures may be a viable product.**Parabola Quirks**: Approximate "puts" would technically include deep OTM call exposure. On the long side, you probably wouldn't be overpaying much, but on the short side, you would be exposed to flash spikes, which you'd theoretically have to protect using a conditional order (e.g. close position if underlying moons above X).**Asymptotics**: Locally, a parabola looks like a ReLU, but asymptotically it dominates, which may look scary for risk management. If liquidation prices are sane, it's not clear that this is worse than shorting OTM puts/calls.

*[next post in this series: Raw Moment Derivatives: Generalizing to ETH^n-PERP]*