Uniswap v2 and all derivative platforms had a serious problem - they were not capital efficient.
- Users were forced to lock up large amounts of capital so that there would be liquidity at every trading price, from 0 to infinity. However, most prices stay within a small range and almost never use all the liquidity in the pool.
- This made it hard for fees to compensate for the effects of impermanent loss.
- Was there a way to concentrate the liquidity around the trading price while ensuring that the pool still had liquidity to cover a worst-case imbalance?
Concentrated Liquidity
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Uniswap v3 allows liquidity providers to concentrate their liquidity between two prices, instead of the entire curve.
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Within a single pool, LP’s can combine any number of concentrated liquidity positions.
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Example: Alice deposits $1M worth of capital into a v2 ETH-DAI pool, and Bob deposits $183.5k of capital in a v3 ETH-DAI pool between the price of 1000 DAI to 2500 DAI. This means that while the price of ETH is between 1000 DAI to 2500 DAI, Bob is earning around 6.3 times the amount of transaction fees.
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However, if ETH moves outside of that price range, Bob is making no money from transaction fees.
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Liquidity positions are represented by NFT’s.
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Fees in concentrated liquidity are much higher than normal liquidity
Ticks and Visualizing Concentrated Liquidity
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Uniswap v3 breaks apart the entire price curve into sections called ticks.
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Each tick is identified by an integer $i$. The price at each tick is defined as $p(i) = 1.0001^{i/2}$. This gives the nice property that each tick is the square root of 0.01% away from the next tick.
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Price moves nonlinearly but sqrt(price) moves linearly with respect to liquidity. For liquidity to remain constant across a tick, it must be defined in sqrt(price) space.
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When specifying concentrated liquidity, users will place their liquidity between two ticks.
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You can imagine the space between two ticks as a bucket where liquidity can fall. The height of the bucket represents how much liquidity is in each bucket.
- Under a normal Uniswap v2 pool, each bucket would have the same amount of liquidity (the user’s liquidity is evenly distributed across the entire protocol).
- With concentrated liquidity, each bucket can have different amounts of liquidity.
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Back to our Alice and Bob example, how is Bob returning around 6.3x more value in v3 compared to v2 (in his current price range)?
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Capital efficiency between v2 and v3 can be realized by the ratio of range widths
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Uniswap devs empirically analyzed that the full range width (v2 range) is around 115.5 sqrt-price units.
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115.5 is essentially the entire realistic price domain in which ETH trades converted into v3’s sqrt(price) system.
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So now calculating capital efficiency:
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sqrt(2500) - sqrt(1000) = 18.38
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Capital efficiency = 115.5 / 18.38 = 6.3
Providing Liquidity in Uniswap v3
- When specifying concentrated liquidity, users will place their liquidity between two ticks.
- The composition of the ticks is VERY important. At each time, only the bucket that contains the true price has a mixture of assets. ALL OTHER BUCKETS contain either all of one asset or all of the other asset for the liquidity provider.
- This means that the ratio of overall liquidity provided is not the price. Notice how the user below is submitting $850 worth of ETH and $1000 worth of USDC, which is not the price ratio between USDC and ETH
- The price range determines the asset ratio, not the price itself.
- In v3, the asset ratio is not 50/50, unlike in v2.
- As the price approaches the lower bound, v3 will shift more liquidity to ETH.
- As the price approaches the upper bound, v3 will shift more liquidity to USDC.
Swapping with Uniswap v3
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Within a tick, the swap acts like the virtual liquidity function. If a trade crosses a tick, then the pool parameter is updated and then trading continues into the new tick.
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By definition, if a trade crosses a tick, then all of the liquidity in that tick has been exhausted. For example, the trader may want to receive asset A but this tick no longer has any asset A to give.
- Luckily, one of the adjacent ticks is entirely full of asset A, so the trader will continue to trade from that new tick. This new tick becomes active, and will receive asset B while giving asset A to the trader.
Real and Virtual Reserves
- How do we represent a concentrated liquidity position as a liquidity curve?
- We can imagine a concentrated liquidity curve as having a similar shape to an $x*y = k$ curve, except that it is not solvent past the prices specified
- Therefore, we need to shift the $x*y = k$ equation somehow
- Imagine that the liquidity in the concentrated position exists everywhere in the curve. Let us call this curve the virtual reserves.
- Virtual reserves make the concentrated position look like it fits on an x * y = k curve.
- Virtual reserves allow us to “pretend” the LP’s curve continues smoothly outside the range, even though it's dead there and technically non-existent.
- This helps us with the math for swaps within the range itself.
- Let $L^2$ be the amount of liquidity in the position. Let $x$ and $y$ be the amount of assets x and y in the liquidity position, and let $x_v$ and $y_v$ be the amount of assets x and y in the virtual liquidity curve.
- Note that $x_v * y_v = k$ and $x*y = L^2$
- Let $p_a$ and $p_b$ be the two prices the liquidity is concentrated between. Let’s measure all prices as the price of $x$ in terms of $y$, so $P = \frac{y}{x}$. In other words, we can get $P$ of $y$ for 1 of $x$.
- Notice that $x$ at $p_b$, $x_{p_b} = 0$ and $y$ at $p_a$, $y_{p_a} = 0$
- Let’s try to solve for the virtual liquidity at $p_a$ and $p_b$. We know that $L = \sqrt{x_v} \sqrt{y_v}$ and $p_b = \frac{y_v}{x_v}$. Then, $x_v = L \frac{\sqrt{x_v}}{\sqrt{y_v}} = \frac{L}{\sqrt{p_b}}$. Similarly, $y_v = L\sqrt{p_a}$
- Now, we can write the real reserves $x$ and $y$ in terms of the virtual reserves. We know that at $p_b$, $x = 0$ and $x_v = \frac{L}{\sqrt{p_b}}$. Therefore, we can argue that within the price range, $x_v = x + \frac{L}{\sqrt{p_b}}$. Similarly, $y_v = y + L\sqrt{p_a}$.
- Plugging back into $x_v * y_v = L^2$, we have $(x + \frac{L}{\sqrt{p_b}})(y + L\sqrt{p_a}) = L^2$
