 Crack the math magic of Michael Egorov’s design

by 0xmc, 0xJezex, 0xstan@0xreviews.xyz, paco@perp.com

Preface

The most difficult part of Curve stable coin is LLAMMA (AMM for continuous

liquidation/deliquidation). LLAMMA refers to some of the principles in

Uniswap v3. However, the price in the white paper is different from the

mathematics in the Uniswap v3 white paper. We will unify these two projects

and try to figure out how Curve CEO designed the algorithm.

1 Refer to Uniswap v3

The definitions of price in this article and Uniswap v3 are reciprocals of each

other. Therefore, we have modified the formulas in the Uniswap v3 white paper

to make them consistent with this article. In short, LLAMMA tries to make

everything dynamic in Uniswap v3 to give a more preferable price for both

crvUSD debtors and liquidators.

2 Compare the constant product formulas

Formula (2.2) from Uniswap v3 whitepaper:

Formula (1) in the Curve stablecoin whitepaper :

Here Pcd means Pcurrent down, Pcu means Pcurrent up.

The corresponding relationship is:

The corresponding constant product formula is:

Among them:

3 Liquidity Calculation Formula Correspondence

Formula (6.7) from Uniswap v3 whitepaper:

Because of the reciprocal relationship between their price definitions, it

corresponds to the formula:

One specific application of this formula is:

Do square expansion to get:

From the above formulas, we can easily find that when y0 remains constant,

The closer pcd and pcu are, the greater the corresponding liquidity I.

In another word:

The liquidity cannot be infinite, and the corresponding minimum tick in

Uniswap v3 will limit the size of L.

From this, it can be deduced that in LLAMMA, we also need to define an

indicator to measure the minimum difference between prices, and continue the

analogy between Uniswap v3 and Curve.

4 Correspondence between the minimum price difference

It can be seen from the definition of A, the closer p and p are, that is, the

larger A is, the higher the liquidity concentration:

In Uniswap v3, only ticks with indexes that are divisible by tickSpacing can

be initialized. Thus, tickSpacing determines the minimum price range for LPs

to allocate their liquidty. The smaller the tickSpacing is, the tighter and more

precise the price ranges. In Uniswap v3 different fee tiers determine different

tickSpacing.

However, there is no need for crvUSD LLAMMA to have so many tickSpac

ing. Just make every tickSpacing = 100basepoint since LLAMMA is just for

ETH-crvUSD. Formula (6.1) from Uniswap v3:

In LLAMMA, A=100, Formula (11) from Curve stablecoin whitepaper:

Set n = -i and A=100, we have:

5 Design pcd and pcu

We hope that LLAMMA has the following properties: when the price of ETH

rises, the pool buys ETH. When ETH falls, the pool sells ETH. Given this, we

define pcd and pcu as functions of po and are steeper than linear functions, so

their growth rates will be faster than po. At the same time, it can be seen from

the figure that the two curves pcu and pcd pass through two points (p , p )

and (p , p ) respectively. The pcd and pcu that meet the above requirements

actually have many curves. The general formula is:

where m < n.

Let’s start from the simplest case:

Substitute pcu and pcd into the square expansion of I:

Then f 2 can be calculated as:

It is not difficult to find that f 2 is hard to comprehend and calculate under this

assumption. What if pcd and pcu are cubic functions of po:

Substitute pcu and pcd into the square expansion of I:

Recalculate f 2 :

It can be seen that when pcd and pcu are cubic functions of po, the whole

mathematical form is much simpler. The square root term is eliminated, and

the calculation is much more convenient. If a higher order is taken, the price

of AMM and po will differ greatly, and thus the cost of buying ETH (when the

price rises) will be much higher, which leads to a greater loss of liquidation. In

summary, it is a better choice to define pcd and pcu as cubic functions of po.

6 Derivation of other parameters

On the basis of assuming that pcd and pcu are cubic functions about po, taking

the special value po = p , it is not difficult to obtain that y = y0 and x = 0,

then:

Given the fomula of I, we can calculate f and g:

From this, we finally get the complete constant product formula:

Transform the above equation into a quadratic equation of y0:

Use the quadratic equation of one unknown to solve y0:

If the price moves so slowly that the oracle price po is fully capable to follow

it, given x and y, using the calculation formula of Uniswap v3, it is possible to

calculate how much y of ETH (if the price rises) or x ↓ of USD will eventually

be in the band (if price drops)

References

 Adams, Hayden, et al. ”Uniswap v3 core.” Tech. rep., Uniswap, Tech. Rep,

2021 from https://uniswap.org/whitepaper-v3.pdf

 Egorov, Michael, and Curve Finance. Curve stablecoin design.

Technical report, Curve Finance, Tech. Rep, 2022 from

https://github.com/curvefi/curve-stablecoin/blob/master/doc/curve

stablecoin.pdf