Why is curve’s slippage larger than uniswap V3?

In the previous article, we compared the slippage difference between curve and uniswap V3 through data. We can find that under comparable conditions, the slippage of uniswap V3 is smaller than that of curve, but some people want to know why. So today we will start from the technical principles of curve and uniswap to analyze why the slippage of the curve is larger than that of uniswap V3. As we all know, curve adopts the mixed curve equation of constant product and constant sum. As for the constant sum market making curve, the exchange price has no slippage with limited liquidity, but as for the constant product market making curve, the exchange price exists slippage with unlimited liquidity. To achieve lower slippage of exchange for AMM, curve combines the above two curves and proposes a hybrid market making curve:

A(P0x+y)+xy=k

Wherein P0 is the balanced price

y=(k-P0Ax)/(X+A)

Derivation of the above formula can get the price

https://preview.redd.it/hsxxddiv86591.png?width=320&format=png&auto=webp&s=ceb8a363b02f6a1079f1d2b42d8b76240ea16e9f

As parameter A approaches infinity

https://preview.redd.it/fnb6slax86591.png?width=451&format=png&auto=webp&s=29181ee4112d21b6543638b4ae5cbfeeeb080a42

Therefore, the larger A , the closer the exchange price to P0, that is, the smaller the slippage,

According to the market making curve of uniswap V3,

https://preview.redd.it/d4a1vo4z86591.png?width=439&format=png&auto=webp&s=3f6f5e824215efbaea3108f8736f3f5e88790f7d

Derivation of the above formula we can get the price

https://preview.redd.it/y5did51196591.png?width=747&format=png&auto=webp&s=70abb431f203018d3db72e5ba3929014e54218f5

m is the magnification factor of x

as parameter m approaches infinity

https://preview.redd.it/k5o2ahu296591.png?width=577&format=png&auto=webp&s=f75296ae40f41e8b54aa78bdaab5ee2d98f5ace6

It can be seen from the above that the exchange price of curve is strongly related to A, while the exchange price of uniswap V3 is strongly related to m, and we know that the amplification factor A in curve is determined internally by the protocol and cannot be changed by external users, while the amplification factor m of uniswap V3 is completely determined by the price range set by the external liquidity provider. In other words, the amplification factor m can theoretically be infinitely. However the amplification factor A of curve cannot be infinitely in order to ensure infinite liquidity. So the slippage of the curve must be larger than that of uniswap V3.

https://preview.redd.it/xjhhtqw496591.png?width=718&format=png&auto=webp&s=dc876af56f0640425bcdc295b8a8685b5b8ba951

https://x3finance.medium.com/why-is-curves-slippage-larger-than-uniswap-v3-e9dd735c88ba

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