> For the complete documentation index, see [llms.txt](https://docs.predy.finance/predy-v2-documentation/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.predy.finance/predy-v2-documentation/predy-1/eth2.md). # (ETH)²-Perpetual ### Overview The (ETH)² Perpetual is a power perpetual, described by the Paradigm team in their [2021 paper](https://www.paradigm.xyz/2021/08/power-perpetuals). The delta of the (ETH)² Perpetual is constant, meaning that as the underlying price increases, the contract increases at a faster rate, and as the underlying price decreases, the contract decreases at a slower rate. The downside of the (ETH)² Perpetual is the higher funding rate compared to the standard ETH Perpetual. ### How to Use the Strategy #### Long The (ETH)² Perpetual can be longed anytime you're bullish on the ETH price. The funding rate needs to be taken into consideration with the positioning. #### Short The (ETH)² Perpetual can be shorted anytime you're bearish on the ETH price. The funding rate needs to be taken into consideration with the positioning. ### Technical Details $$ \begin{align\*} \&Symbol = ETH2-PERP-USDC &\\ \\ \&Underlying Asset = ETH &\\ \&Index\_{price} ={S^2} \* \frac{1}{10,000}, (S = ETH\_{price} \ from \ Chainlink) &\\ \\ &\Delta \ Delta =\frac {\delta {V}}{\delta S} \approx \frac {\delta {S^2}}{\delta S} \* \frac{1}{10,000} = 2S \* \frac{1}{10,000} &\\ &\Gamma \ Gamma =\frac {\delta^2 {V}}{{\delta} S^2} \approx \frac {\delta^2 {S^2}}{{\delta} S^2} \* \frac{1}{10,000} = 2 \* \frac{1}{10,000} = Constant &\\ \\ \&where, \ V = Index\_{price} \ , \ Trade\_{price} \ should \ be \ used \ by \ strict \ definition &\\ \\ \&Trade\_{price} =Index\_{price} \* (1+FundingRate+TradingFeeRate) &\\ \&TradingFeeRate = 0.1 % & \\ \\ \&Varience\_{ETH} = (Volatility\_{ETH})^2 = \sigma\_t^2 = \lambda \* \sigma\_{t-1}^2+(1-\lambda)u\_{t-1}^2&\\ \&u\_{t} = ln\frac{S\_t}{S\_{t-1}}\approx1- \frac{S\_t}{S\_{t-1}}&\\ &\lambda = 0.94 (The \ RiskMetrics\ database\ produced \ by \ JP \ Morgan)&\\ \\ \&FundingRate =\sigma\_t^2\*(1+\beta*f( Trade\_{amount},AMM\_{liquidityStatus})) &\\ &\beta = 5.5 \ (Deployed \ param = 3.5) & \\ &\beta = 3.0 \ on \ Version \ 2.0.2 & \\\ \end{align*} $$ $$ \begin{align\*} f( Trade\_{amount},AMM\_{liquidityStatus}) &= \frac{\int\_L^{L+\Delta L}\int\_m^{m+\Delta m}(\frac{x}{y})^3dxdy}{\Delta L \Delta m} &\\ &=\frac{m^3+\frac{3}{2}m^2\Delta m + m\Delta m^2+\frac{\Delta m^3}{4}}{L*L*(L+\Delta L)^2}\*(L+\frac{\Delta L}{2}) &\\ &\approx ( \frac{m}{L} )^3 = (UtilizationRate\_{AMM})^3&\\ \\ \&m = Liquidity\_{Locked} \ before \ The \ Trade &\\ &\Delta m = Liquidity\_{Locked} \ for \ The \ Trade &\\ \&L = Liquidity\_{total} \ before \ The \ Trade &\\ &\Delta L = Liquidity\_{changed} \ for \ The \ Trade&\\ \\ &(\frac{x}{y})^3 \ => \ k\*(\frac{x}{y}) + (1-k)\*(\frac{x}{y})^3&\\ \&where, k=0.3, \ 0\