Adding Liquidity in DMM

There are some conditions when adding liquidity to the DMM:

1. After LP contributions, the token price is unchanged.
2. $P_{min}$ and $P_{max}$ are also unchanged after LP contributions.

In the DMM, the pool for pair X-Y needs to maintain 4 parameters:

1. The initial amount of token $X$ that is used for amplification, denoted by $x_0$
2. The initial amount of token $Y$ that is used for amplification, denoted by $y_0$
3. The change in token $X$ amount after trading activities, denoted by $\Delta x_0$
4. The change in token $Y$ amount after trading activities, denoted by $\Delta y_0$

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. You may find more information about the amplification factor here.

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.

Example

• 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: $x = 120$, $y = 120$, $\Delta x = 24$, $\Delta y = −18$.