Uniswap V3 LP Token Analyzer

Uniswap V3 LP token is an NFT token that represents the liquidity provision position. Check Uniswap V3 Whitepaper for detail.

Specifically, the variable $L$ represents the liquidity given the relative price ($P_0$) of asset $X$ in the denomination of asset $Y$ at the time the position is created/last changed, the lower bound ($P_a$) and upper bound ($P_b$) of price in which the user chooses to provide liquidity.

1. Define input parameters. On the page of adding liquidity in Uniswap V3, three factors have to be defined, $P_a$, $P_b$ , and either one of $x_0$ and $y_0$, where $x_0$ and $y_0$ denote the number of $X$ and $Y$ tokens that will be injected into the liquidity provision position.

2. Calculate $L$ and $x_0$ or $y_0$. Depending on which one of $x_0$ and $y_0$ is defined in step 1), the other one will be calculated. The detailed calculation follows formula 2.2 on Uniswap V3 Whitepaper. For example, if $x_0$ is defined, we will first solve $L$ based on (1) and then calculate $y_0$ based on (2) below:

   $$x_0=\max\left(0,L*\left(\frac{1}{\sqrt{P_0}}-\frac{1}{\sqrt{P_b}}\right) \right)$$

   $$y_0=\max\left(0,L*\left({\sqrt{P_0}}-{\sqrt{P_a}}\right)\right)$$

   Note here $\frac{y_0}{x_0}\ne P_0$ in Uniswap V3.

3. Then given price fluctuating to any $P$, $x$ and $y$ that represent the number of $X$ and $Y$ tokens redeemable from the liquidity provision position can be derived following eq. 6.29 and 6.30 in Uniswap V3 Whitepaper, which are copied and pasted below:

   $$y=\begin{cases} 0 & P<P_a \\ L*\left(\sqrt{P}-\sqrt{P_a}\right) &P_a \le P < P_b \\ L*\left(\sqrt{P_b}-\sqrt{P_a}\right) & P\ge P_b \end{cases}$$

   $$x=\begin{cases} L*\left(\frac{1}{\sqrt{P_a}}-\frac{1}{\sqrt{P_b}}\right) & P<P_a \\ L*\left(\frac{1}{\sqrt{P}}-\frac{1}{\sqrt{P_b}}\right) &P_a \le P < P_b \\ 0 & P\ge P_b \end{cases}$$

To get $x$ and $y$ in a given position as well as other information, function positions() can be called. See details here.