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(ETH)²-Perpetual

The (ETH)² Perpetual is a power perpetual, described by the Paradigm team in their 2021 paper. 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.

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.

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.

Symbol=ETH2PERPUSDCUnderlyingAsset=ETHIndexprice=S2110,000,(S=ETHprice from Chainlink)Δ Delta=δVδSδS2δS110,000=2S110,000Γ Gamma=δ2VδS2δ2S2δS2110,000=2110,000=Constantwhere, V=Indexprice , Tradeprice should be used by strict definitionTradeprice=Indexprice(1+FundingRate+TradingFeeRate)TradingFeeRate=0.1%VarienceETH=(VolatilityETH)2=σt2=λσt12+(1λ)ut12ut=lnStSt11StSt1λ=0.94(The RiskMetrics database produced by JP Morgan)FundingRate=σt2(1+βf(Tradeamount,AMMliquidityStatus))β=5.5 (Deployed param=3.5)β=3.0 on Version 2.0.2\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*}
f(Tradeamount,AMMliquidityStatus)=LL+ΔLmm+Δm(xy)3dxdyΔLΔm=m3+32m2Δm+mΔm2+Δm34LL(L+ΔL)2(L+ΔL2)(mL)3=(UtilizationRateAMM)3m=LiquidityLocked before The TradeΔm=LiquidityLocked for The TradeL=Liquiditytotal before The TradeΔL=Liquiditychanged for The Trade(xy)3 => k(xy)+(1k)(xy)3where,k=0.3, 0<k<1 on Version 2.0.2\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<k<1 \ on \ Version \ 2.0.2 & \\ \end{align*}
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