Swap Operations (S)

Normal Swap

SP1 - Buy XT with Underlying

Buy $\lambda_y$ XT with $\sigma_u$ UT

Step1 - Mint FT and XT by UT

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

where

$$\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-1)}$$

$$\sigma_y=\sigma_u \text{ (Eq.S-2)}$$

Step2 - Sell FT to XT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+\sigma_x)\cdot(\^y_{ori}-\delta_y)$$

derive

$$\delta_y=\^y_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}+\sigma_x} \text{ (Eq.S-3)}$$

Step3 - Output Total XT

$$\lambda_y \text{ (total ouput XT)}=\sigma_y+\delta_y=\sigma_u+(\^y_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}+\varepsilon \sigma_u}) \text{ (Eq.S-4)}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$

SP2 - Buy FT with Underlying

Buy $\lambda_x$ FT with $\sigma_u$ UT

Step1 - Mint FT and XT by UT

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

where

$$\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-5)}$$

$$\sigma_y=\sigma_u \text{ (Eq.S-6)}$$

Step2 - Sell XT to FT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-\delta_x)\cdot(\^y_{ori}+\sigma_y)$$

derive

$$\delta_x=\^x_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}+\sigma_y} \text{ (Eq.S-7)}$$

Step3 - Output Total FT

$$\lambda_x \text{ (total ouput FT)}=\sigma_x+\delta_x=\varepsilon\sigma_u+(\^x_{ori}-\frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}+\delta_u}) \text{ (Eq.S-8)}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$

SP3 - Sell XT for Underlying

To sell $\sigma_y$ XT for $\lambda_u$ UT, we need to swap some XT to FT first and combine the remain XT and the bought FT into UT.

Step1 - Sell $\delta_y$ XT for $\delta_x$ FT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-\delta_x)\cdot(\^y_{ori}+\delta_y)\text{ (eq.1)}$$

Step2 - Combine $\delta_x$ FT and $(\sigma_y-\delta_y)$ XT into $\lambda_u$ UT

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

$$\delta_x=\varepsilon(\sigma_y-\delta_y) \text{ (eq.2)}$$

Step3 - Calculate output $\lambda_u$ UT

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

$$\epsilon\delta_y^2+(\^x_{ori}+\varepsilon \^y_{ori}-\varepsilon \sigma_y) \cdot \delta_y-\varepsilon \sigma_y \^y_{ori}=0$$

derive

$$\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-9)}$$

$$\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ is invalid, since } \^y_{new} \text{ should be greater than 0}$$

$$\delta_x=\varepsilon (\sigma_y-\delta_y) \text{ (Eq.S-10)}$$

$$\lambda_u = \sigma_y-\delta_y \text{ (Eq.S-11)}$$

where

$$\begin{cases} a=\varepsilon \\ b=\^x_{ori}+\varepsilon (\^y_{ori}- \sigma_y) \text{ (Eq.S-12)} \\ c= -\varepsilon \sigma_y \^y_{ori} \end{cases}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$

SP4 - Sell FT for Underlying

To sell $\sigma_x$ FT for $\lambda_u$ UT, we need to swap some FT to XT first and combine the remain FT and the bought XT into UT.

Step1 - Sell $\delta_x$ FT for $\delta_y$ XT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+\delta_x)\cdot(\^y_{ori}-\delta_y) \text{ (eq.1)}$$

Step2 - Combine $(\sigma_x-\delta_x)$ FT and $\delta_y$ XT into $\lambda_u$ UT

According to $Eq.2$,

$$\sigma_x-\delta_x=\varepsilon\delta_y \text{ (eq.2)}$$

Step3 - Calculate output $\lambda_u$ UT

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

$$\epsilon\delta_y^2-(\^x_{ori}+\sigma_x+\varepsilon \^y_{ori}) \cdot \delta_y+\sigma_x \^y_{ori}=0$$

derive

$$\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-13)}$$

$$\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ (invalid, since } y_{new}' \text{ should be greater than 0)}$$

$$\delta_x=\sigma_x-\varepsilon\delta_y \text{ (Eq.S-14)}$$

$$\lambda_u = \delta_y \text{ (Eq.S-15)}$$

where

$$\begin{cases} a=\varepsilon \\ b=-(\^x_{ori}+\sigma_x+\varepsilon \^y_{ori}) \text{ (Eq.S-16)} \\ c= \sigma_x \^y_{ori} \end{cases}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$

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 $-\lambda_y$ XT with $-\sigma_u$ UT

Step1 - Mint FT and XT by UT

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

where

$$\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-17)}$$

$$\sigma_y=\sigma_u \text{ (Eq.S-18)}$$

Step2 - Sell FT to XT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+(-\sigma_x))\cdot(\^y_{ori}-(-\delta_y))=(\^x_{ori}-\sigma_x)\cdot(\^y_{ori}+\delta_y)$$

derive

$$\delta_y=\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}-\sigma_x}-\^y_{ori} \text{ (Eq.S-19)}$$

Step3 - Output Total XT

$$-\lambda_y \text{ (total ouput XT)}=-\sigma_y-\delta_y=-\sigma_u-(\frac{\^x_{ori} \cdot \^y_{ori}}{\^x_{ori}-\varepsilon \sigma_u}-\^y_{ori}) \text{ (Eq.S-20)}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$

SP6 - Buy Negative FT with Negative Underlying

Buy $-\lambda_x$ FT with $-\sigma_u$ UT

Step1 - Mint FT and XT by UT

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

where

$$\sigma_x=\varepsilon \sigma_u \text{ (Eq.S-21)}$$

$$\sigma_y=\sigma_u \text{ (Eq.S-22)}$$

Step2 - Sell XT to FT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-(-\delta_x))\cdot(\^y_{ori}+(-\sigma_y)) = (\^x_{ori}+\delta_x)\cdot(\^y_{ori}-\sigma_y)$$

derive

$$\delta_x = \frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}-\sigma_y}-\^x_{ori} \text{ (Eq.S-23)}$$

Step3 - Output Total XT

$$-\lambda_x \text{ (total ouput XT)}=-\sigma_x-\delta_x=-\varepsilon\sigma_u-(\frac{\^x_{ori} \cdot \^y_{ori}}{\^y_{ori}-\sigma_y}-\^x_{ori}) \text{ (Eq.S-24)}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$

SP7 - Sell Negative XT for Negative Underlying

To sell $-\sigma_y$ XT for $-\lambda_u$ UT, we need to swap some XT to FT first and combine the remain XT and the bought FT into UT.

Step1 - Sell $-\delta_y$ XT for $-\delta_x$ FT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}-(-\delta_x))\cdot(\^y_{ori}+(-\delta_y)) = (\^x_{ori}+\delta_x)\cdot(\^y_{ori}-\delta_y)\text{ (eq.1)}$$

Step2 - Combine $-\delta_x$ FT and $(-\sigma_y-(-\delta_y))=(-\sigma_y+\delta_y)$ XT into $-\lambda_u$ UT

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

$$-\delta_x=\varepsilon(-\sigma_y+\delta_y) \text{ (eq.2)}$$

Step3 - Calculate output $-\lambda_u$ UT

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

$$\epsilon\delta_y^2-(\^x_{ori}+\varepsilon \^y_{ori}+\varepsilon \sigma_y) \cdot \delta_y+\varepsilon \sigma_y \^y_{ori}=0$$

derive

$$\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-25)}$$

$$\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ is invalid, since } \^y_{new} \text{ should be greater than 0}$$

$$\delta_x=\varepsilon (\sigma_y-\delta_y) \text{ (Eq.S-26)}$$

$$-\lambda_u = -\sigma_y+\delta_y \text{ (Eq.S-27)}$$

where

$$\begin{cases} a=\varepsilon \\ b=-(\^x_{ori}+\varepsilon (\^y_{ori}+ \sigma_y)) \text{ (Eq.S-28)} \\ c= \varepsilon \sigma_y \^y_{ori} \end{cases}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$

SP8 - Sell Negative FT for Negative Underlying

To sell $-\sigma_x$ FT for $-\lambda_u$ UT, we need to swap some FT to XT first and combine the remain FT and the bought XT into UT.

Step1 - Sell $-\delta_x$ FT for $-\delta_y$ XT

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

$$\^x_{ori} \cdot \^y_{ori}=(\^x_{ori}+(-\delta_x))\cdot(\^y_{ori}-(-\delta_y)) = (\^x_{ori}-\delta_x)\cdot(\^y_{ori}+\delta_y) \text{ (eq.1)}$$

Step2 - Combine $(-\sigma_x-(-\delta_x)) = (-\sigma_x+\delta_x)$ FT and $-\delta_y$ XT into $-\lambda_u$ UT

According to $Eq.2$,

$$-\sigma_x+\delta_x=-\varepsilon\delta_y \text{ (eq.2)}$$

Step3 - Calculate output $-\lambda_u$ UT

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

$$\epsilon\delta_y^2+(\^x_{ori}-\sigma_x+\varepsilon \^y_{ori}) \cdot \delta_y-\sigma_x \^y_{ori}=0$$

derive

$$\delta_y=\frac{-b+\sqrt{b^2-4ac}}{2a} \text{ (Eq.S-29)}$$

$$\delta_y=\frac{-b-\sqrt{b^2-4ac}}{2a} \text{ (invalid, since } \^y_{new} \text{ should be greater than 0)}$$

$$\delta_x=\sigma_x-\varepsilon\delta_y \text{ (Eq.S-30)}$$

$$-\lambda_u = -\delta_y \text{ (Eq.S-31)}$$

where

$$\begin{cases} a=\varepsilon \\ b=\^x_{ori}-\sigma_x+\varepsilon \^y_{ori} \text{ (Eq.S-32)} \\ c= -\sigma_x \^y_{ori} \end{cases}$$

Step4 - Update Pool State

Update new APR ( $r$ ) according $\text{Eq.D-5}$

$$r_{new}=\frac{\frac{\^R_{x,new}}{\^R_{y,new}}+\varepsilon-1}{\theta}$$