To guarantee the safety of the pool, there are some conditions which need to be fulfilled when adding liquidity to KyberSwap Classic pools:
- After LP contributions, the token price remains unchanged.
-
$$P_{min}$$ and $$P_{max}$$ are also unchanged after LP contributions.
In KyberSwap, the pool for pair X-Y needs to maintain 4 parameters:
- The initial amount of token $$X$$ that is used for amplification, denoted by $$x_0$$
- The initial amount of token $$Y$$ that is used for amplification, denoted by $$y_0$$
- The change in token $$X$$ amount after trading activities, denoted by $$Δx_0$$
- The change in token $$Y$$ amount after trading activities, denoted by $$Δy_0$$
Therefore, the real balances and virtual balances of the reserves are:
Real Balances
$$
x = x_0 + \Delta x_0 \\
y = y_0 + \Delta y_0
$$
Virtual Balances
$$
x' = a \cdot x_0 + \Delta x_0 \\
y' = a \cdot y_0 + \Delta y_0
$$
where $$a$$ is the amplification factor. \
The constant product $$x' \cdot y' = (a \cdot x_0 + \Delta x_0) \cdot (a \cdot y_0 + \Delta y_0) = k'$$. Note that $$P_{min}$$ and $$P_{max}$$ at this time are:
$$
\begin{cases}
P_{min} = \cfrac{(y_0 \cdot a - y_0)^2}{k'} \\
\\
P_{max} = \cfrac{k'}{(x_0 \cdot a - x_0)^2}
\end{cases}
$$
The current price: $$P = \cfrac{y'}{x'} = \cfrac{a \cdot y_0 + \Delta y_0}{a \cdot x_0 + \Delta x_0}$$
Liquidity Providers have to contribute in the same proportion for all 4 amount types. We denote the contribution ratio to be $$b$$. LPs have to contribute $$x_1 + \Delta x_1$, $y_1 + \Delta y_1$$in which:
$$
\begin{cases}
x_1 = b \cdot x_0 \\
\Delta x_1 = b \cdot \Delta x_0 \\
y_1 = b \cdot y_0 \\
\Delta y_1 = b \cdot \Delta y_0
\end{cases}
$$
The real balances and virtual balances of the reserve after contribution are:
Real Balances
$$
x = (x_0 + x_1) + (\Delta x_0 + \Delta x_1) = (b + 1) \cdot (x_0 + \Delta x_0) \\
y = (y_0 + y_1) + (\Delta y_0 + \Delta y_1) = (b + 1) \cdot (y_0 + \Delta y_0)
$$
Virtual Balances
$$
x' = a \cdot (x_0 + x_1) + (\Delta x_0 + \Delta x_1) = (b + 1) \cdot (a \cdot x_0 + \Delta x_0) \\
y' = a \cdot (y_0 + y_1) + (\Delta y_0 + \Delta y_1) = (b + 1) \cdot (a \cdot y_0 + \Delta y_0)
$$
The constant product, after the LP contribution, becomes:
$$
x' \cdot y' = (b + 1)^2 \cdot (a \cdot x_0 + \Delta x_0) \cdot (a \cdot y_0 + \Delta y_0)
= (b + 1)^2 \cdot k'
$$
$$P_{min}$$ and $$P_{max}$$ at this time are:
$$
\begin{cases}
P_{min} = \cfrac{((y_0 + y_1) \cdot a - (y_0 + y_1))^2}{(b + 1)^2 \cdot k'} = \cfrac{(y_0 \cdot a - y_0)^2}{k'} \\
P_{max} = \cfrac{(b + 1)^2 \cdot k'}{((x_0 + x_1) \cdot a - (x_0 + x_1))^2} = \cfrac{(x_0 \cdot a - x_0)^2}{k'}
\end{cases}
$$
The current price is updated to be $$P = \cfrac{y'}{x'} = \cfrac{(a \cdot y_0 + \Delta y_0) \cdot (b + 1)}{(a \cdot x_0 + \Delta x_0) \cdot (b + 1)} = \cfrac{a \cdot y_0 + \Delta y_0}{a \cdot x_0 + \Delta x_0}$$
We see that after LP contributes, the current price, $$P_{min}$$ and $$P_{max}$$ are unchanged. It is similar in the case of LPs withdrawals, where the ratio $$b$$ is negative.
-
Initially, the first LP put 100 $$X$$ and 100 $$Y$$ to the reserve, we have: $$x = 100, y = 100, \Delta x = 0, \Delta y = 0$$.
-
A user trades 20 $$X$$ for 15 $$Y$$, so we have the updated parameters: $$x = 100, y = 100, \Delta x = 20, \Delta y = −15$$.
-
Suppose an LP wants to contribute 20% of the current token amounts in the pool, so he should deposit:
$$
0.2 · 100 + 0.2 · 20 = 24 (X) \
0.2 · 100 + 0.2 · (−15) = 17 (Y)
$$
ie. deposit 24X and 17Y tokens.
The parameters are then updated to be: $$x = 120$, $y = 120$, $\Delta x = 24$, $\Delta y = −18$$.