Introducing Power Perpetuals

Power Perpetuals (Powers) were first come up with by Dave White et al in Aug 2021. Simply put, in the case of power=2, Power Perpetuals are perpetual swaps of the square of some price, e.g. BTCUSD. In other words, a long position in “BTCUSD powers” will have PnL along the curve of BTCUSD^2, while a short position will have the opposite of that. Due to the convexity of the square function, the fair price of a power perpetual is usually slightly higher than the square of the underlying price.

A very simple rule of thumb to remember the PnL of taking long or short positions of power perpetuals is:

• Underlying price goes up, longs make profits while shorts bear losses.

• Underlying price goes down, longs bear losses while shorts make profits.

• Underlying price stays stable, shorts earn funding fees while longs are paying them.

In Jan 2022, Opyn implemented the first power perpetuals of ETH, Squeeth. While mathematically well designed, Opyn’s Squeeth is essentially a synthetic asset (synthesizing the square of ETH price), rather than a derivative. In the Opyn framework, people mint oSQTH (a “spot” synthesizing ETH^2) against collateralized ETH, and trade it on Uniswap just like trading spots. Thanks to its sophisticated design, oSQTH should theoretically work well in terms of tracking ETH^2 (the role of a synthetic asset) but would not sufficiently serve the purposes of derivatives. Among its drawbacks, lack of leverage and hence a low capital efficiency is limiting the adoption of Squeeth as a serious derivative tool.

Boiling down to its essence, Power Perpetuals are yet another instance of the funding-fee-based perpetual derivative family, for which a long contract periodically pays a short contract funding fee as follows:

$$Funding=Mark-I(S)$$

where $Mark$ is the trading price of the derivative and $I(S)$ is a general intrinsic value function of the underlying price $S$. People can define different intrinsic value functions to get different funding-fee-based perpetual derivatives:

• $I(S)=S$ for perpetual futures

• $I(S)=\max(S-K,0)$ for everlasting calls

• $I(S)=\max(K-S,0)$for everlasting puts

• $I(S)=S^p$ for power perpetuals of order $p$

The Power Perpetuals by Deri Protocol are rolled out under this framework. And here is how it works.

First of all, we specifically implemented the case of $p=2$, i.e. the perpetuals of BTC^2 and ETH^2. Since the square of BTC price is a huge number, we downscale it to a smaller unit - milli (prefix for 1/1000). Namely, the object being traded on Deri is mBTC^2 and mETH^2, which correspond to 1/1000 of BTC^2 and ETH^2, respectively. Let’s take mBTC^2 as an example.

Assume BTCUSD =40000, and mBTC^2 is traded at a price of 1631284 (i.e. Mark=1631284). ****People holding a long position of 1 mBTC^2 will pay those holding a short position of 1 mBTC^2 the following amount for every week:

$$Funding=1631284-40000^2/1000=31284$$

That is, a long contract pays a short contract 31284 USD per week. However, please note that the funding fee is accrued on a second basis (and settled whenever there is a trading action), so every second a long contract will accrue a funding fee of $31285/(7*24*3600)=0.0517USD$ to be paid, while every second a short contract will accrue 0.0517USD to be received. Please note that, even with a milli unit, 1 unit of mBTC^2 is still a very large contract, with mark price over 1 million USD.

Let’s take 2 scenarios to see the PnLs of the longs and the shorts upon BTC price moves.

• BTCUSD goes up to 45000, mBTC^2 goes up to 2064594 (a trading result of the market): A long contract will make a profit of $2,064,594-1,631,284=433,310$. Accordingly, a short contract will bear a loss of the same amount.

• BTCUSD goes down to 35000, mBTC^2 goes down to 1248953 (a trading result of the market): A long contract will make a loss of $1,248,952-1,631,284=-382,331$. Accordingly, a short contract will make a profit of 382331.

As you can see from the example above, for a long position, the up-side profit and the down-side loss are asymmetric. That is, the long (short) positions have an advantage (disadvantage) in terms of PnL. That is why the long positions need to pay the short positions a funding fee — to compensate for such an advantage. Such an asymmetric bet is the point to trade Powers:

The longs are paying a funding fee to gain an asymmetric advantage: to boost their up-side profit and mitigate their down-side loss. Whereas, the shorts are offering that advantage to the longs by charging them the funding fee as a yield.

Execution-wise, we set up the powers largely similar to the futures. Namely, one trades mBTC^2 similar to how he/she trades the BTCUSD futures. You just need to choose the long or short side and specify the volume to trade. Please note that under normal circumstances the longs always pay the funding fee and the shorts always receive the funding fee.

The Greeks of Powers

This section is for advanced traders who are familiar with the Greeks of derivatives and are interested in learning the powers from this perspective. A detailed discussion of the mathematics of the Powers can be found in this paper. However, you don’t have to understand this section to trade powers.

The Delta and the Gamma of the powers are as follows:

$$\Delta = \frac{\partial P}{\partial S}= \frac{pS^{p-1}}{1-hT}=\frac{pP}{S}\\ \Gamma = \frac{\partial^2 P}{\partial S^2}= \frac{p(p-1)S^{p-2}}{1-hT}=\frac{p(p-1)P}{S^2}$$

where

• P: the price of the power

• S: the price of the underlying

• T: the funding period. T=1week in our implementation.

• $h=r+\sigma^2/2$, with r the risk-free interest rate and $\sigma$ the volatility

Specifically, for the case $p=2$, we have

$$\Gamma =\frac{2}{1-hT} = \frac{2}{1-(r+\frac{\sigma^2}{2})T}$$

Please note that in the square case $p=2$, the Gamma of powers has a very special and important property: independent of the underlying price.

This property of the powers lead it to a very important use case: hedging the impermanent loss of Constant-Product Marketing Making, e.g. Uniswap and Pancake. We will explain the mechanism in this article.