From Uniswap v3 to crvUSD LLAMMA

Original paper: 《From Uniswap v3 to crvUSD LLAMMA.pdf》

By 0xmc,0xjezex,0xstan, paco0x

Preface: The most difficult part of Curve’s stablecoin design is LLAMMA, an AMM for continuous liquidation and deliquidation. LLAMMA builds on several ideas from Uniswap v3, but the price definitions in the Curve whitepaper differ from those used in the Uniswap v3 whitepaper. This article reconciles the two frameworks and uses that comparison to explain the intuition behind the LLAMMA algorithm.

Relation to Uniswap v3

The price definition used in this article is the reciprocal of the one used in Uniswap v3. Therefore, we rewrite the relevant formulas from the Uniswap v3 whitepaper so they are consistent with this article. In short, LLAMMA makes several Uniswap v3 parameters dynamic in order to offer better prices to both crvUSD borrowers and liquidators.

Comparing the Constant Product Formulas

Formula (2.2) from the Uniswap v3 whitepaper:

\begin{aligned} (x+\dfrac{L}{\sqrt{P_b}})(y+L\sqrt{P_a})=L^2 \end{aligned}

Uniswap v3 Simulation of Virtual Liquidity

Formula (1) in the Curve stablecoin whitepaper:

\begin{aligned} I = (x + f) (y + g) \end{aligned}

AMM with an external price source

Here, $P_{cd}$ means $P_{current\_down}$ and $P_{cu}$ means $P_{current\_up}$ .
The corresponding relationships are:

P_{b} = \dfrac{1}{P_{cd}}, P_{a} = \dfrac{1}{P_{cu}}, L=\sqrt{I}

The corresponding constant product formula is:

\begin{aligned} (x+\sqrt{I}\sqrt{p_{cd}})(y+\dfrac{\sqrt{I}}{\sqrt{p_{cu}}})=I \end{aligned}

where:

f=\sqrt{I}\sqrt{p_{cd}}, g=\dfrac{\sqrt{I}}{\sqrt{p_{cu}}}

Correspondence in Liquidity Calculation

Formula (6.7) from the Uniswap v3 whitepaper:

\begin{aligned} L=\dfrac{\Delta{y}}{\Delta{\sqrt{p}}} \end{aligned}

Because the two price definitions are reciprocals, this corresponds to:

\begin{aligned} \sqrt{I} = \dfrac{\Delta{y}} {\Delta{\sqrt{\dfrac{1}{p}}}} \end{aligned}

One specific application of this formula is:

\begin{aligned} \sqrt{I} = &\dfrac{(y_0-0)}{(\dfrac{1}{\sqrt{p_{cd}}}-\dfrac{1}{\sqrt{p_{cu}}})} \\ = &\dfrac{\sqrt{p_{cd}p_{cu}}}{\sqrt{p_{cu}}-\sqrt{p_{cd}}} y_0 \end{aligned}

Expanding the square gives:

\begin{aligned} I= \dfrac{{p_{cd}p_{cu}}}{p_{cu}+p_{cd}-2\sqrt{p_{cd}p_{cu}}} y_0^2 \end{aligned}

From the formulas above, when $y_0$ remains constant, the closer $p_{cd}$ and $p_{cu}$ are to each other, the larger the corresponding liquidity $I$ becomes.

In other words:

\lim\limits_{p_{cd} \to p_{cu}} \sqrt{I}= +\infty

Liquidity cannot be infinite. In Uniswap v3, the minimum tick spacing limits the size of $L$ .

By analogy, LLAMMA also needs an indicator that measures the minimum price difference.

Correspondence in the Minimum Price Difference

\dfrac{p\downarrow}{p\uparrow} = \dfrac{A-1}{A}

From the definition of $A$ , the closer $p\downarrow$ and $p\uparrow$ are to each other, the larger $A$ becomes and the more concentrated the liquidity is:

A = \dfrac{p\uparrow}{p\uparrow - p\downarrow}

In Uniswap v3, only ticks whose indexes are divisible by tickSpacing can be initialized. Thus, tickSpacing determines the minimum price range in which LPs can allocate their liquidity. The smaller the tickSpacing, the tighter and more precise the price ranges. In Uniswap v3, different fee tiers correspond to different tickSpacing values.

However, crvUSD LLAMMA does not need many different tickSpacing values. Since LLAMMA is designed for ETH-crvUSD, each band can use a $tickSpacing$ of 100 basis points. Formula (6.1) from Uniswap v3:

p_i=1.0001^i

In LLAMMA, $A=100$ . Formula (11) from the Curve stablecoin whitepaper is:

\begin{aligned} p\uparrow (n) = (\dfrac{A-1}{A})^n p_{base} \end{aligned}

\begin{aligned} p\downarrow (n) = (\dfrac{A-1}{A})^{n+1} p_{base} \end{aligned}

Setting $n = -i$ and $A=100$ , we get:

\begin{aligned} p\uparrow (-i) = (\dfrac{100}{100-1})^i p_{base}\ \ \ \ \end{aligned}

Design $p_{cd}$ and $p_{cu}$

We want LLAMMA to have the following behavior: when the ETH price rises, the pool buys ETH; when the ETH price falls, the pool sells ETH. Given this goal, we define $p_{cd}$ and $p_{cu}$ as functions of $p_o$ that are steeper than linear functions, so their growth rates are faster than that of $p_o$ . At the same time, the figure shows that the two curves $p_{cu}$ and $p_{cd}$ pass through $(p\downarrow,p\downarrow)$ and $(p\uparrow,p\uparrow)$ , respectively. Many possible curves can satisfy these requirements. The general form is:

p_{cd} = \dfrac{p_o^{n+1}}{p\uparrow^n}, p_{cu} = \dfrac{p_o^{m+1}}{p\downarrow^{m}}

where $m<n$ .

Let’s start with the simplest case:

p_{cd} = \dfrac{p_o^2}{p\uparrow}, p_{cu} = \dfrac{p_o^2}{p\downarrow}

Substituting $p_{cu}$ and $p_{cd}$ into the expanded formula for $I$ gives:

\begin{aligned} I= &\dfrac{{p_{cd}p_{cu}}}{p_{cu}+p_{cd}-2\sqrt{p_{cd}p_{cu}}} y_0^2 \\ = &\dfrac{\dfrac{p_o^2}{p\uparrow}\dfrac{p_o^2}{p\downarrow} y_0^2}{\dfrac{p_o^2}{p\uparrow} + \dfrac{p_o^2}{p\downarrow} - 2\sqrt{\dfrac{p_o^2}{p\uparrow}\dfrac{p_o^2}{p\downarrow}}}\\ =&\dfrac{p_o^2 y_0^2}{p\downarrow+p\uparrow - 2\sqrt{p\downarrow p\uparrow}}\\ =&\dfrac{p_o^2 y_0^2}{(\sqrt{p\uparrow}-\sqrt{p\downarrow})^2} \end{aligned}

Then $f^2$ can be calculated as:

\begin{aligned} f^2=&Ip_{cd}\\ =&\dfrac{p_o^2 y_0^2}{(\sqrt{p\uparrow}-\sqrt{p\downarrow})^2}·\dfrac{p_o^2}{p\uparrow} \end{aligned}

Under this assumption, $f^2$ is difficult to interpret and calculate. Now consider the case where $p_{cd}$ and $p_{cu}$ are cubic functions of $p_o$ :

p_{cd} = \dfrac{p_o^3}{p\uparrow^2}, p_{cu} = \dfrac{p_o^3}{p\downarrow^2}

Substituting $p_{cu}$ and $p_{cd}$ into the expanded formula for $I$ gives:

\begin{aligned} I= &\dfrac{{p_{cd}p_{cu}}}{p_{cu}+p_{cd}-2\sqrt{p_{cd}p_{cu}}} y_0^2 \\ = &\dfrac{\dfrac{p_o^3}{p\uparrow^2}\dfrac{p_o^3}{p\downarrow^2} y_0^2}{\dfrac{p_o^3}{p\uparrow^2} + \dfrac{p_o^3}{p\downarrow^2} - 2\sqrt{\dfrac{p_o^3}{p\uparrow^2}\dfrac{p_o^3}{p\downarrow^2}}}\\ =&\dfrac{p_o^3 y_0^2}{p\downarrow^2+p\uparrow^2-2p\downarrow p\uparrow}\\ =&\dfrac{p_o^3 y_0^2}{(p\uparrow-p\downarrow)^2} \end{aligned}

Recalculating $f^2$ :

\begin{aligned} f^2=&Ip_{cd}\\ =&\dfrac{p_o^3 y_0^2}{(p\uparrow-p\downarrow)^2}·\dfrac{p_o^3}{p\uparrow^2}\\ =&\dfrac{p_o^6 y_0^2}{(p\uparrow-p\downarrow)^2 p\uparrow^2} \end{aligned}

When $p_{cd}$ and $p_{cu}$ are cubic functions of $p_o$ , the mathematical form becomes much simpler. The square-root term disappears, making the calculation more convenient. If we use a higher-order function, the AMM price and $p_o$ will diverge more significantly. As a result, the cost of buying ETH when the price rises becomes much higher, leading to greater liquidation losses. Therefore, defining $p_{cd}$ and $p_{cu}$ as cubic functions of $p_o$ is a better choice.

Derivation of Other Parameters

Assuming that $p_{cd}$ and $p_{cu}$ are cubic functions of $p_o$ , take the special case $p_o=p\uparrow$ . In this case, we get $y=y_0$ and $x=0$ , so:

\begin{aligned} I=&\dfrac{p_o^3 y_0^2}{(p\uparrow-p\downarrow)^2}\\ =&p_o y_0^2 · \dfrac{p_o^2}{(p\uparrow-p\downarrow)^2}\\ =&p_o y_0^2 · (\dfrac{p\uparrow}{p\uparrow-p\downarrow})^2\\ =&p_oy_0^2A^2 \end{aligned}

Given the formula for $I$ , we can calculate $f$ and $g$ :

\begin{aligned} f=&\sqrt{I}\sqrt{p_{cd}} \\ =&\sqrt{p_oy_0^2A^2}\sqrt{\dfrac{p_o^3}{p\uparrow^2}}\\ =&\dfrac{p_o^2}{p\uparrow}Ay_0 \end{aligned}

\begin{aligned} g =& \dfrac{\sqrt{I}}{\sqrt{p_{cu}}}\\ =& \dfrac{\sqrt{p_oy_0^2A^2}}{\sqrt{\dfrac{p_o^3}{p\downarrow^2}}}\\ =&\dfrac{p\downarrow}{p_o}Ay_0\\ =&\dfrac{p\uparrow}{p_o}(A-1)y_0 \end{aligned}

From this, we obtain the complete constant product formula:

(\dfrac{p_o^2}{p\uparrow}Ay_0 + x)(\dfrac{p\uparrow}{p_o}(A-1)y_0 + y) = p_oA^2y_0^2

Transforming the equation above into a quadratic equation in $y_0$ gives:

p_oAy_0^2- y_0(\dfrac{p\uparrow}{p_o}(A-1)x + \dfrac{p_o^2}{p\uparrow}Ay) - xy = 0

Solving this quadratic equation for $y_0$ gives:

y_0 = \dfrac{(\dfrac{p\uparrow}{p_o}(A-1)x + \dfrac{p_o^2}{p\uparrow}Ay)+\sqrt{(\dfrac{p\uparrow}{p_o}(A-1)x + \dfrac{p_o^2}{p\uparrow}Ay)^2+4p_oA·xy}}{2p_oA}

If the price moves slowly enough for the oracle price $p_o$ to follow it, then given $x$ and $y$ , we can use the Uniswap v3 formulas to calculate how much ETH, denoted by $y\uparrow$ , will eventually remain in the band if the price rises, or how much USD, denoted by $x\downarrow$ , will eventually remain in the band if the price falls:

\begin{aligned} y\uparrow =& y + \Delta y \\ =& y + \sqrt{I}(\dfrac{1}{\sqrt{p\uparrow}} - \dfrac{1}{\sqrt{p}}) \\ =& y + \dfrac{\sqrt{I}\sqrt{p} - \sqrt{I}\sqrt{p\uparrow}}{\sqrt{p\uparrow p}} \\ =& y + \dfrac{(f +x) - f}{\sqrt{p\uparrow p}} \\ =& y + \dfrac{x}{\sqrt{p\uparrow p}} \end{aligned}

\begin{aligned} x\downarrow =& x + \sqrt{I}(\sqrt{p\downarrow}- \sqrt{p}) \\ =& x + \sqrt{I}(\dfrac{1}{\sqrt{p}} - \dfrac{1}{\sqrt{p\downarrow}})\sqrt{p\downarrow p}\\ =& x + ((g+y) - g)\sqrt{p\downarrow p}\\ =& x + y\sqrt{p\downarrow p} \end{aligned}

Reference

[1] Adams, Hayden, et al. “Uniswap v3 Core.” Technical report, Uniswap, 2021. https://uniswap.org/whitepaper-v3.pdf

[2] Egorov, Michael, and Curve Finance. “Curve Stablecoin Design.” Technical report, Curve Finance, 2022.
https://github.com/curvefi/curve-stablecoin/blob/master/doc/curve-stablecoin.pdf