Liquidity Concentration

Let's understand how liquidity deposited in DEX is used for swaps, and what it means to concentrate liquidity to specific price ranges.

Distribution of DEX liquidity by price range

Let's define price P = (Y/X)^0.5 .

When a trader sells X tokens to buy Y tokens, the price of the pool goes down, and when a trader sells Y tokens to buy X tokens, the price of the pool goes up.

In other words, when a trade occurs in a direction that makes price go up, the pool must receive X tokens and give Y tokens to the trader, and when a trade occurs in the direction that makes price go down, the pool must receive Y tokens and give X tokens to the trader.

Therefore, the pool needs Y tokens to process transactions in the downward direction from the current price, and X tokens to process transactions in the upward direction from the current price.

This is similar to the mechanism that happens in stock trading where you “deposit cash (≈Y tokens) to place a buy order at a price lower than the current market price” and “deposit stocks (≈X tokens) to place a sell order at a price higher than the current market price”.

Liquidity Concentration

Based on the graph above, the quantity of X and Y tokens required to provide liquidity in a given price range is derived as follows.

$$amountY = L \cdot (P_c - P_a) \\ amountX = L \cdot (\frac{1}{P_c} - \frac{1}{P_b})$$

Based on the formula above, the quantity of X and Y tokens required to provide L amount of liquidity for the entire price range (0~∞) in Uniswap V2 manner is derived as follows.

$$amountY_{full range} = L \cdot (P_c - 0) = L P_c\\ amountX_{full range} = L \cdot (\frac{1}{P_c} - \frac{1}{\infty}) = \frac{L}{P_c}$$

However, as mentioned in the previous chapter(Lazy Liquidity), the actual price of the two tokens does not vary from 0 to infinity, except for extremely exceptional cases. If the price of the two tokens changes between 50% and 200% (=-50%~+100%) from the current price, how much of the liquidity provided to the entire price range will actually be used for the swap? Let's look at the formula below.

$$amountY_{50\% \sim 100\%} = L \cdot (P_c - \frac{1}{\sqrt 2} P_c) = \frac{2 -\sqrt2}{2} LP_c \simeq 0.29 * amountY_{fullrange} \\ amountX_{100\% \sim 200\%} = L \cdot (\frac{1}{P_c} - \frac{1}{\sqrt 2P_c} ) = \frac{2 -\sqrt2}{2} \frac{L}{P_c} \simeq 0.29 * amountX_{fullrange}$$

If the price changes between 50% and 200% from the current price, only about 29% of the supplied liquidity is used for the actual swap (Active Liquidity), and the remaining 71% of the liquidity is not used for the swap (Lazy Liquidity).

In other words, it can be understood as follows.

• Concentrating liquidity to the 50%~200% price range with the V3 method yields the same effect with only 29% of the liquidity compared to the V2 method.

• Concentrating liquidity to the 50%~200% price range with the V3 method is 3.41 times (1/(29%)) more effective than the V2 method.

Concentration Ratio

Concentration Ratio is a number that indicates the efficiency that increases compared to the V2 method of liquidity provision by concentrating liquidity to a specific price range(In the service UI, the term Fee Boost is used to help users understand. Fee Boost = Concentration Ratio). Concentration Ratio increases as the price range that is provided liquidity to is more concentrated, and can be expressed by the following formula.

• reference : uniswapv3#capital-efficiency

$$Concentration Ratio = \frac{1}{1-{\frac{P_l}{P_u}^{0.25}}}$$

Concentration Ratio by price range is as follows :

Price Range	Concentration Ratio
Full Range	1.00
10% ~ 1,000%	1.46
25% ~ 400%	2.00
50% ~ 200%	3.41
90% ~ 111%	21.49
95% ~ 105%	41.49