This is a cheatsheet of maths related to how concentrated liquidity works, which could be useful for doing analysis on how to provide liquidity on Muffin.

Add / Remove Liquidity

First of all, we define $P$ as the current pool price but being bounded in the price range $[P_\text{lower}, P_\text{upper}]$, i.e.:

$$ P = \text{max}\left\{P_{\text{lower}}, \text{min}\left\{ P_{\text{upper}}, P_{\text{now}}\right\}\right\} $$

What are the underlying token amounts $x$ and $y$ for a position with a liquidity level $L$ in the price range $[P_\text{lower}, P_\text{upper}]$?

$$ \begin{gather} x = L \left( \frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_\text{upper}}} \right) \newline y = L \left( \sqrt{P} - \sqrt{P_\text{lower}} \right) \end{gather} $$

Reversely, what is the liquidity level $L$ if I’m adding $x$ and $y$ tokens into the range $[P_\text{lower}, P_\text{upper}]$?

$$ L = \text{min} \left\{ \frac{x}{\frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_\text{upper}}}}, \frac{y}{\sqrt{P} - \sqrt{P_\text{lower}}} \right\} $$

Position Value

What is the cash value $V$ of a position with a liquidity level $L$ in the price range $[P_\text{lower}, P_\text{upper}]$? Let $c_x$ and $c_y$ are the USD-denominated prices of tokens X and Y respectively.

$$ V = c_x x + c_y y $$

$$ V = L \left[ c_x \left( \frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_\text{upper}}} \right) + c_y \left( \sqrt{P} - \sqrt{P_\text{lower}} \right) \right] $$

Reversely, the liquidity level $L$ if I’m adding liquidity of a value $V$ into the price range $[P_\text{lower}, P_\text{upper}]$:

$$ L = \frac{V}{c_x \left( \frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_\text{upper}}} \right) + c_y \left( \sqrt{P} - \sqrt{P_\text{lower}} \right)} $$

Capitial Efficiency

We quantify capitial efficiency by comparing the value of $L$ of a concentrated liquidity position to that of a full-range liquidity position with the same value $V$.

$$ CE = \frac{L_\text{conc}}{L_\text{full}} $$

where

$$ L_\text{conc} = \frac{V}{P_\text{now} \left( \frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_\text{upper}}} \right) + \left(\sqrt{P} - \sqrt{P_\text{lower}}\right)} $$

$$ L_\text{full} = \frac{V}{P_\text{now}\left(\frac{1}{\sqrt{P_\text{now}}}\right) + \sqrt{P_\text{now}}} = \frac{V}{2 \sqrt{P_\text{now}}} $$

For simplicity, let’s define the price range $[P_\text{lower}, P_\text{upper}] = [\frac{1}{a} P_\text{now}, b P_\text{now}]$ where $\frac{1}{a} < b$ and $a,b>0$.

Then, we simplify the capital efficency formula to:

$$ CE = \begin{cases} \frac{1}{1 - \frac{1}{2 \sqrt{a}} - \frac{1}{2 \sqrt{b}}} & a \ge 1\ \ \text{and}\ \ b \ge 1 & \text{(In range)} \\[4pt] \frac{2}{\sqrt{a}-\frac{1}{\sqrt{b}}} & a < 1\ \ \text{and}\ \ b > 1 & \text{(Below range)} \\[4pt] \frac{2}{\sqrt{b}-\frac{1}{\sqrt{a}}} & a > 1\ \ \text{and}\ \ b < 1 & \text{(Above range)} \end{cases} $$

So, if you create a position of which the current price is in the middle of the price range (i.e. $a = b$), then:

$$ CE = 1 + \frac{1}{\sqrt{a}-1} $$

For example, for a 1-tick position, $a=\sqrt{1.0001}$ and thus $CE=40002.5$. It means you need 40002x more capitial to construct a full-range position with the same liquidity level.

Divergence Loss

Let’s clarify some variables that we’re going to use:

$$ \begin{split} P_{t0} & = \text{Price when creating the position} \\[3pt] P_{t1} & = \text{Current price} \\[3pt] P_0 & = \text{max}\left\{P_{\text{lower}}, \text{min}\left\{ P_{\text{upper}}, P_{t0}\right\}\right\} \\[3pt] P_1 & = \text{max}\left\{P_{\text{lower}}, \text{min}\left\{ P_{\text{upper}}, P_{t1}\right\}\right\} \\[3pt] \newline x_0, y_0 &= \text{Token amounts used to create the position} \\[3pt] x_1, y_1 &= \text{Current token amounts in the position} \\[3pt] \end{split} $$

Divergence loss is the percentage decrease in the value of the position comparing to the value of simply hodling the tokens.

$$ V_\text{LP} = P_{t1} x_1 + y_1 $$

$$ V_\text{Hodl} = P_{t1} x_0+y_0 $$

$$ DL = \frac{V_\text{LP}}{V_\text{Hodl}} - 1 $$

$$ DL = \frac{ P_{t1} \left( \frac{1}{\sqrt{P_1}} - \frac{1}{\sqrt{P_\text{upper}}} \right) + \left(\sqrt{P_1} - \sqrt{P_\text{lower}}\right) }{ P_{t1} \left( \frac{1}{\sqrt{P_0}} - \frac{1}{\sqrt{P_\text{upper}}} \right) + \left(\sqrt{P_0} - \sqrt{P_\text{lower}}\right) } - 1 $$

Let’s assume the pool price was in the middle of the position’s price range when the position was created, i.e. $[P_\text{lower}, P_\text{upper}] = [\frac{1}{a} P_0, a P_0]$ where $a > 1$.

Let’s also define the current price $P_{t1} = u P_{t0}$. Then, we simplify the DL formula to:

$$ DL = \begin{cases} \frac{u \left( \sqrt{a}+1 \right)}{u+1} - 1 & u < \frac{1}{a} & \text{(Below range)} \\[6pt] \frac{\left( \sqrt{a}+1 \right)}{u+1} - 1 & u > a & \text{(Above range)} \\[6pt] \frac{\sqrt{a} \left(\sqrt{u}-1\right)^2} {\left(1-\sqrt{a}\right) (u+1)} & \text{otherwise} & \text{(In range)} \\[6pt] \end{cases} $$

Proportions of Underlying Tokens

Given a current price $P_\text{now}$ and a position’s price range $[P_\text{lower}, P_\text{upper}]$, what are the weights of token X and token Y in the position in terms of cash value?

$$ \begin{gather} w_x = \frac{V_x}{V_x + V_y} \\[2pt] w_x = \left( \frac{V_y}{V_x} + 1 \right)^{-1} \\[2pt] w_x = \left( \frac{y}{P_\text{now} x} + 1 \right)^{-1} \\[2pt] w_x = \left[ \frac {\sqrt{P} - \sqrt{P_\text{lower}}} {P_\text{now} \left( \frac{1}{\sqrt{P}} - \frac{1}{\sqrt{P_\text{upper}}} \right)} + 1 \right]^{-1} \end{gather} $$

$$ w_y = 1 - w_x $$

Tick Conversion

$$ P_i = 1.0001^i $$

$$ i = \log_{1.0001}P_i $$

• $i$: Tick number
• $P$: Price of token0 in terms of token1