Overview
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.
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
S y m b o l = E T H 2 − P E R P − U S D C U n d e r l y i n g A s s e t = E T H I n d e x p r i c e = S 2 ∗ 1 10 , 000 , ( S = E T H p r i c e f r o m C h a i n l i n k ) Δ D e l t a = δ V δ S ≈ δ S 2 δ S ∗ 1 10 , 000 = 2 S ∗ 1 10 , 000 Γ G a m m a = δ 2 V δ S 2 ≈ δ 2 S 2 δ S 2 ∗ 1 10 , 000 = 2 ∗ 1 10 , 000 = C o n s t a n t w h e r e , V = I n d e x p r i c e , T r a d e p r i c e s h o u l d b e u s e d b y s t r i c t d e f i n i t i o n T r a d e p r i c e = I n d e x p r i c e ∗ ( 1 + F u n d i n g R a t e + T r a d i n g F e e R a t e ) T r a d i n g F e e R a t e = 0.1 % V a r i e n c e E T H = ( V o l a t i l i t y E T H ) 2 = σ t 2 = λ ∗ σ t − 1 2 + ( 1 − λ ) u t − 1 2 u t = l n S t S t − 1 ≈ 1 − S t S t − 1 λ = 0.94 ( T h e R i s k M e t r i c s d a t a b a s e p r o d u c e d b y J P M o r g a n ) F u n d i n g R a t e = σ t 2 ∗ ( 1 + β ∗ f ( T r a d e a m o u n t , A M M l i q u i d i t y S t a t u s ) ) β = 5.5 ( D e p l o y e d p a r a m = 3.5 ) β = 3.0 o n V e r s i o n 2.0.2 \begin{align*}
&Symbol = ETH2-PERP-USDC &\\
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&Underlying Asset = ETH &\\
&Index_{price} ={S^2} * \frac{1}{10,000}, (S = ETH_{price} \ from \ Chainlink) &\\
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&\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 &\\
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&where, \ V = Index_{price} \ , \ Trade_{price} \ should \ be \ used \ by \ strict \ definition &\\
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&Trade_{price} =Index_{price} * (1+FundingRate+TradingFeeRate) &\\
&TradingFeeRate = 0.1 \% & \\
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&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)&\\
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&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*} S y mb o l = ET H 2 − PERP − U S D C U n d er l y in g A sse t = ET H I n d e x p r i ce = S 2 ∗ 10 , 000 1 , ( S = ET H p r i ce f ro m C hain l ink ) Δ De lt a = δ S δ V ≈ δ S δ S 2 ∗ 10 , 000 1 = 2 S ∗ 10 , 000 1 Γ G amma = δ S 2 δ 2 V ≈ δ S 2 δ 2 S 2 ∗ 10 , 000 1 = 2 ∗ 10 , 000 1 = C o n s t an t w h ere , V = I n d e x p r i ce , T r a d e p r i ce s h o u l d b e u se d b y s t r i c t d e f ini t i o n T r a d e p r i ce = I n d e x p r i ce ∗ ( 1 + F u n d in g R a t e + T r a d in g F ee R a t e ) T r a d in g F ee R a t e = 0.1% Va r i e n c e ET H = ( V o l a t i l i t y ET H ) 2 = σ t 2 = λ ∗ σ t − 1 2 + ( 1 − λ ) u t − 1 2 u t = l n S t − 1 S t ≈ 1 − S t − 1 S t λ = 0.94 ( T h e R i s k M e t r i cs d a t aba se p ro d u ce d b y J P M or g an ) F u n d in g R a t e = σ t 2 ∗ ( 1 + β ∗ f ( T r a d e am o u n t , A M M l i q u i d i t y St a t u s )) β = 5.5 ( De pl oye d p a r am = 3.5 ) β = 3.0 o n V ers i o n 2.0.2 f ( T r a d e a m o u n t , A M M l i q u i d i t y S t a t u s ) = ∫ L L + Δ L ∫ m m + Δ m ( x y ) 3 d x d y Δ L Δ m = m 3 + 3 2 m 2 Δ m + m Δ m 2 + Δ m 3 4 L ∗ L ∗ ( L + Δ L ) 2 ∗ ( L + Δ L 2 ) ≈ ( m L ) 3 = ( U t i l i z a t i o n R a t e A M M ) 3 m = L i q u i d i t y L o c k e d b e f o r e T h e T r a d e Δ m = L i q u i d i t y L o c k e d f o r T h e T r a d e L = L i q u i d i t y t o t a l b e f o r e T h e T r a d e Δ L = L i q u i d i t y c h a n g e d f o r T h e T r a d e ( x y ) 3 = > k ∗ ( x y ) + ( 1 − k ) ∗ ( x y ) 3 w h e r e , k = 0.3 , 0 < k < 1 o n V e r s i o n 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&\\
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&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&\\
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&(\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*} f ( T r a d e am o u n t , A M M l i q u i d i t y St a t u s ) = Δ L Δ m ∫ L L + Δ L ∫ m m + Δ m ( y x ) 3 d x d y = L ∗ L ∗ ( L + Δ L ) 2 m 3 + 2 3 m 2 Δ m + m Δ m 2 + 4 Δ m 3 ∗ ( L + 2 Δ L ) ≈ ( L m ) 3 = ( U t i l i z a t i o n R a t e A MM ) 3 m = L i q u i d i t y L oc k e d b e f ore T h e T r a d e Δ m = L i q u i d i t y L oc k e d f or T h e T r a d e L = L i q u i d i t y t o t a l b e f ore T h e T r a d e Δ L = L i q u i d i t y c han g e d f or T h e T r a d e ( y x ) 3 => k ∗ ( y x ) + ( 1 − k ) ∗ ( y x ) 3 w h ere , k = 0.3 , 0 < k < 1 o n V ers i o n 2.0.2 