Swap Operations (S)

Normal Swap

SP1 - Buy XT with Underlying

Buy λy\lambda_y XT with σu\sigma_u UT

Step1 - Mint FT and XT by UT

According to Eq.C-2\text{Eq.C-2}, mint σx\sigma_x FT and σy\sigma_y XT by σu (input amount)\sigma_u \text{ (input amount)} UT

where

σx=εσu (Eq.S-1)\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-1)}
σy=σu (Eq.S-2)\sigma_y=\sigma_u \text{ (Eq.S-2)}

Step2 - Sell FT to XT

According to Eq.C-5\text{Eq.C-5},

xˆori⋅yˆori=(xˆori+σx)⋅(yˆori−δy)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+\sigma_x)\cdot(\^y_{ori}-\delta_y)

derive

δy=yˆori−xˆori⋅yˆorixˆori+σx (Eq.S-3)\delta_y=\^y_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}+\sigma_x} \text{ (Eq.S-3)}

Step3 - Output Total XT

λy (total ouput XT)=σy+δy=σu+(yˆori−xˆori⋅yˆorixˆori+εσu) (Eq.S-4)\lambda_y \text{ (total ouput XT)}=\sigma_y+\delta_y=\sigma_u+(\^y_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}+\varepsilon \sigma_u}) \text{ (Eq.S-4)}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

SP2 - Buy FT with Underlying

Buy λx\lambda_x FT with σu\sigma_u UT

Step1 - Mint FT and XT by UT

According to Eq.C-2\text{Eq.C-2}, mint σx\sigma_x FT and σy\sigma_y XT by σu (input amount)\sigma_u \text{ (input amount)} UT

where

σx=εσu (Eq.S-5)\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-5)}
σy=σu (Eq.S-6)\sigma_y=\sigma_u \text{ (Eq.S-6)}

Step2 - Sell XT to FT

According to Eq.C-5\text{Eq.C-5},

xˆori⋅yˆori=(xˆori−δx)⋅(yˆori+σy)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-\delta_x)\cdot(\^y_{ori}+\sigma_y)

derive

δx=xˆori−xˆori⋅yˆoriyˆori+σy (Eq.S-7)\delta_x=\^x_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}+\sigma_y} \text{ (Eq.S-7)}

Step3 - Output Total FT

λx (total ouput FT)=σx+δx=εσu+(xˆori−xˆori⋅yˆoriyˆori+δu) (Eq.S-8)\lambda_x \text{ (total ouput FT)}=\sigma_x+\delta_x=\varepsilon\sigma_u+(\^x_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}+\delta_u}) \text{ (Eq.S-8)}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

SP3 - Sell XT for Underlying

To sell σy\sigma_y XT for λu\lambda_u UT, we need to swap some XT to FT first and combine the remaining XT and purchased FT into UT.

Step1 - Sell δy\delta_y XT for δx\delta_x FT

According to Eq.C-5\text{Eq.C-5},

xˆori⋅yˆori=(xˆori−δx)⋅(yˆori+δy) (eq.1)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-\delta_x)\cdot(\^y_{ori}+\delta_y)\text{ (eq.1)}

Step2 - Combine δx\delta_x FT and (σy−δy)(\sigma_y-\delta_y) XT into λu\lambda_u UT

According to Eq.C-2\text{Eq.C-2},

δx=ε(σy−δy) (eq.2)\delta_x=\varepsilon(\sigma_y-\delta_y) \text{ (eq.2)}

Step3 - Calculate output λu\lambda_u UT

According to eq.1eq.1 and eq.2eq.2 with elimination by substitution,

ϵδy2+(xˆori+εyˆori−εσy)⋅δy−εσyyˆori=0\epsilon\delta_y^2+(\^x_{ori}+\varepsilon \^y_{ori}-\varepsilon \sigma_y) \cdot \delta_y-\varepsilon \sigma_y \^y_{ori}=0

derive

δy=−b+b2−4ac2a (Eq.S-9)\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-9)}
δy=−b−b2−4ac2a is invalid, since yˆnew should be greater than 0\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ is invalid, since } \^y_{new} \text{ should be greater than 0}
δx=ε(σy−δy) (Eq.S-10) \delta_x=\varepsilon (\sigma_y-\delta_y) \text{ (Eq.S-10)}
λu=σy−δy (Eq.S-11)\lambda_u = \sigma_y-\delta_y \text{ (Eq.S-11)}

where

{a=εb=xˆori+ε(yˆori−σy) (Eq.S-12)c=−εσyyˆori\begin{cases} a=\varepsilon \\ b=\^x_{ori}+\varepsilon (\^y_{ori}- \sigma_y) \text{ (Eq.S-12)} \\ c= -\varepsilon \sigma_y \^y_{ori} \end{cases}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

SP4 - Sell FT for Underlying

To sell σx\sigma_x FT for λu\lambda_u UT, we need to swap some FT to XT first and combine the remaining FT and purchased XT into UT.

Step1 - Sell δx\delta_x FT for δy\delta_y XT

According to Eq.5\text{Eq.5},

xˆori⋅yˆori=(xˆori+δx)⋅(yˆori−δy) (eq.1)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+\delta_x)\cdot(\^y_{ori}-\delta_y) \text{ (eq.1)}

Step2 - Combine (σx−δx)(\sigma_x-\delta_x) FT and δy\delta_y XT into λu\lambda_u UT

According to Eq.2Eq.2,

σx−δx=εδy (eq.2)\sigma_x-\delta_x=\varepsilon\delta_y \text{ (eq.2)}

Step3 - Calculate output λu\lambda_u UT

According to eq.1eq.1 and eq.2eq.2 with elimination by substitution,

ϵδy2−(xˆori+σx+εyˆori)⋅δy+σxyˆori=0\epsilon\delta_y^2-(\^x_{ori}+\sigma_x+\varepsilon \^y_{ori}) \cdot \delta_y+\sigma_x \^y_{ori}=0

derive

δy=−b−b2−4ac2a (Eq.S-13)\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-13)}
δy=−b+b2−4ac2a (invalid, since ynew′ should be greater than 0)\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ (invalid, since } y_{new}' \text{ should be greater than 0)}
δx=σx−εδy (Eq.S-14)\delta_x=\sigma_x-\varepsilon\delta_y \text{ (Eq.S-14)}
λu=δy (Eq.S-15)\lambda_u = \delta_y \text{ (Eq.S-15)}

where

{a=εb=−(xˆori+σx+εyˆori) (Eq.S-16)c=σxyˆori\begin{cases} a=\varepsilon \\ b=-(\^x_{ori}+\sigma_x+\varepsilon \^y_{ori}) \text{ (Eq.S-16)} \\ c= \sigma_x \^y_{ori} \end{cases}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

Negative Swap

We define Negative Swap operations for applying the same rules of our AMM while withdrawing liquidity and charging tx fees.

SP5 - Buy Negative XT with Negative Underlying

Buy −λy-\lambda_y XT with −σu-\sigma_u UT

Step1 - Mint FT and XT by UT

According to Eq.C-2\text{Eq.C-2}, mint −σx-\sigma_x FT and −σy-\sigma_y XT by −σu (input amount)-\sigma_u \text{ (input amount)} UT

where

σx=εσu (Eq.S-17)\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-17)}
σy=σu (Eq.S-18)\sigma_y=\sigma_u \text{ (Eq.S-18)}

Step2 - Sell FT to XT

According to Eq.C-5\text{Eq.C-5},

xˆori⋅yˆori=(xˆori+(−σx))⋅(yˆori−(−δy))=(xˆori−σx)⋅(yˆori+δy)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+(-\sigma_x))\cdot(\^y_{ori}-(-\delta_y))=(\^x_{ori}-\sigma_x)\cdot(\^y_{ori}+\delta_y)

derive

δy=xˆori⋅yˆorixˆori−σx−yˆori (Eq.S-19)\delta_y=\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}-\sigma_x}-\^y_{ori} \text{ (Eq.S-19)}

Step3 - Output Total XT

−λy (total ouput XT)=−σy−δy=−σu−(xˆori⋅yˆorixˆori−εσu−yˆori) (Eq.S-20)-\lambda_y \text{ (total ouput XT)}=-\sigma_y-\delta_y=-\sigma_u-(\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}-\varepsilon \sigma_u}-\^y_{ori}) \text{ (Eq.S-20)}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

SP6 - Buy Negative FT with Negative Underlying

Buy −λx-\lambda_x FT with −σu-\sigma_u UT

Step1 - Mint FT and XT by UT

According to Eq.C-2\text{Eq.C-2}, mint −σx-\sigma_x FT and −σy-\sigma_y XT by −σu (input amount)-\sigma_u \text{ (input amount)} UT

where

σx=εσu (Eq.S-21)\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-21)}
σy=σu (Eq.S-22)\sigma_y=\sigma_u \text{ (Eq.S-22)}

Step2 - Sell XT to FT

According to Eq.C-5\text{Eq.C-5},

xˆori⋅yˆori=(xˆori−(−δx))⋅(yˆori+(−σy))=(xˆori+δx)⋅(yˆori−σy)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-(-\delta_x))\cdot(\^y_{ori}+(-\sigma_y)) = (\^x_{ori}+\delta_x)\cdot(\^y_{ori}-\sigma_y)

derive

δx=xˆori⋅yˆoriyˆori−σy−xˆori (Eq.S-23)\delta_x = \frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}-\sigma_y}-\^x_{ori} \text{ (Eq.S-23)}

Step3 - Output Total XT

−λx (total ouput XT)=−σx−δx=−εσu−(xˆori⋅yˆoriyˆori−σy−xˆori) (Eq.S-24)-\lambda_x \text{ (total ouput XT)}=-\sigma_x-\delta_x=-\varepsilon\sigma_u-(\frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}-\sigma_y}-\^x_{ori}) \text{ (Eq.S-24)}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

SP7 - Sell Negative XT for Negative Underlying

To sell −σy-\sigma_y XT for −λu-\lambda_u UT, we need to swap some XT to FT first and combine the remaining XT and purchased FT into UT.

Step1 - Sell −δy-\delta_y XT for −δx-\delta_x FT

According to Eq.C-5\text{Eq.C-5},

xˆori⋅yˆori=(xˆori−(−δx))⋅(yˆori+(−δy))=(xˆori+δx)⋅(yˆori−δy) (eq.1)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-(-\delta_x))\cdot(\^y_{ori}+(-\delta_y)) = (\^x_{ori}+\delta_x)\cdot(\^y_{ori}-\delta_y)\text{ (eq.1)}

Step2 - Combine −δx-\delta_x FT and (−σy−(−δy))=(−σy+δy)(-\sigma_y-(-\delta_y))=(-\sigma_y+\delta_y) XT into −λu-\lambda_u UT

According to Eq.C-2\text{Eq.C-2},

−δx=ε(−σy+δy) (eq.2)-\delta_x=\varepsilon(-\sigma_y+\delta_y) \text{ (eq.2)}

Step3 - Calculate output −λu-\lambda_u UT

According to eq.1eq.1 and eq.2eq.2 with elimination by substitution,

ϵδy2−(xˆori+εyˆori+εσy)⋅δy+εσyyˆori=0\epsilon\delta_y^2-(\^x_{ori}+\varepsilon \^y_{ori}+\varepsilon \sigma_y) \cdot \delta_y+\varepsilon \sigma_y \^y_{ori}=0

derive

δy=−b−b2−4ac2a (Eq.S-25)\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-25)}
δy=−b+b2−4ac2a is invalid, since yˆnew should be greater than 0\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ is invalid, since } \^y_{new} \text{ should be greater than 0}
δx=xˆori⋅yˆoriyˆori−δy−xˆori (Eq.S-26)\delta_x=\frac{\^x_{ori}\cdot\^y_{ori}}{\^y_{ori}-\delta_y} - \^x_{ori} \text{ (Eq.S-26)}
−λu=−σy+δy (Eq.S-27)-\lambda_u = -\sigma_y+\delta_y \text{ (Eq.S-27)}

where

{a=εb=−(xˆori+ε(yˆori+σy)) (Eq.S-28)c=εσyyˆori\begin{cases} a=\varepsilon \\ b=-(\^x_{ori}+\varepsilon (\^y_{ori}+ \sigma_y)) \text{ (Eq.S-28)} \\ c= \varepsilon \sigma_y \^y_{ori} \end{cases}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

SP8 - Sell Negative FT for Negative Underlying

To sell −σx-\sigma_x FT for −λu-\lambda_u UT, we need to swap some FT to XT first and combine the remaining FT and purchased XT into UT.

Step1 - Sell −δx-\delta_x FT for −δy-\delta_y XT

According to Eq.5\text{Eq.5},

xˆori⋅yˆori=(xˆori+(−δx))⋅(yˆori−(−δy))=(xˆori−δx)⋅(yˆori+δy) (eq.1)\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+(-\delta_x))\cdot(\^y_{ori}-(-\delta_y)) = (\^x_{ori}-\delta_x)\cdot(\^y_{ori}+\delta_y) \text{ (eq.1)}

Step2 - Combine (−σx−(−δx))=(−σx+δx)(-\sigma_x-(-\delta_x)) = (-\sigma_x+\delta_x) FT and −δy-\delta_y XT into −λu-\lambda_u UT

According to Eq.2Eq.2,

−σx+δx=−εδy (eq.2)-\sigma_x+\delta_x=-\varepsilon\delta_y \text{ (eq.2)}

Step3 - Calculate output −λu-\lambda_u UT

According to eq.1eq.1 and eq.2eq.2 with elimination by substitution,

ϵδy2+(xˆori−σx+εyˆori)⋅δy−σxyˆori=0\epsilon\delta_y^2+(\^x_{ori}-\sigma_x+\varepsilon \^y_{ori}) \cdot \delta_y-\sigma_x \^y_{ori}=0

derive

δy=−b+b2−4ac2a (Eq.S-29)\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-29)}
δy=−b−b2−4ac2a (invalid, since yˆnew should be greater than 0)\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ (invalid, since } \^y_{new} \text{ should be greater than 0)}
δx=xˆori−xˆori⋅yˆoriyˆori+δy (Eq.S-30)\delta_x=\^x_{ori}-\frac{\^x_{ori}\cdot \^y_{ori}}{\^y_{ori}+\delta_y} \text{ (Eq.S-30)}
−λu=−δy (Eq.S-31)-\lambda_u = -\delta_y \text{ (Eq.S-31)}

where

{a=εb=xˆori−σx+εyˆori (Eq.S-32)c=−σxyˆori\begin{cases} a=\varepsilon \\ b=\^x_{ori}-\sigma_x+\varepsilon \^y_{ori} \text{ (Eq.S-32)} \\ c= -\sigma_x \^y_{ori} \end{cases}

Step4 - Update Pool State

Update new APR ( rr ) according to Eq.D-5\text{Eq.D-5}

rnew=Rˆx,newRˆy,new+ε−1θr_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}

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