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Babbitt Information (ID: bitcoin8btc)

, by Tarun Chitra, Guillermo Angeris, and Alex Evans, translated by Free and Easy Xi, published with permission.

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(Obviously, it's hard for people to hear the shape of a drum (Marc Kac, 1966), what about CFMM?)

The rise of Uniswap in 2019 is a watershed for DeFi transactions. The simplicity, gas efficiency and performance of Uniswap quickly made it the main place for on-chain transactions. And Curve, launched earlier this year, showed that even small changes to the design of a constant-function market maker (CFMM) can dramatically improve capital efficiency and performance. In particular, Curve pioneered locally smoother curves, which provide lower slippage for transactions between stablecoins. This adjustment allows Curve to capture significant trading volume while routinely outperforming existing centralized exchanges and OTC exchanges. Due to the success of Curve, curvature is increasingly recognized as an integral part of the CFMM design space. However, the exact impact of the choice of curvature on market behavior has not been thoroughly studied.

In this series of articles, we begin to introduce the concept of CFMM curvature shape. We discuss the impact of curvature choice on equilibrium price, stability, liquidity provider (LP) returns, and market microstructure. The opinions of these articles come from our upcoming paper "When does the dog's tail wag?" Curvature and Market Making". We will publish this paper at the same time as the third article in this series.

In the first article, we will provide a definition of curvature and discuss its impact on liquidity and price stability.

The summer of 2020 changed the face of CFMM. Largely due to the impact of yield farming activities, the CFMM market is increasingly becoming the most liquid market for various asset pairs. This requires a new analytical framework to study these markets. We find that curvature provides the missing link for studying CFMM-dominated markets. When CFMM becomes the most liquid trading venue, most other trading venues will adjust according to CFMM's price. The first step in our framework is to understand how places with limited mobility interact with each other.

Two Market Models

Suppose there are two trading venues that can trade a given asset pair. And one of the trading venues is more liquid than the other. So how do we model differences in mobility? A simple exercise is to look at the impact of trades of a fixed size, if trades of the same size result in a larger price change in one market than another, we can roughly say "the former is less liquid". In the case of CFMMs, this simple model is surprisingly descriptive. CFMM implements a specific curve for each asset pair, allowing us to precisely describe the impact of a given trade, which is where the curvature comes from. Informally, curvature describes the absolute change in price reported by a CFMM after a small trade. All else being equal, a CFMM with a higher amount of funds stored will exhibit lower curvature. However, for a given value of stored funds, some CFMMs have lower curvature than others. By comparing Uniswap and Curve, we can see the difference. Starting from the point where the amount of storage is equal, it can be seen from the figure below that Uniswap has a higher curvature than Curve at the point x = y = 5.

Most CFMM models assume a CFMM with limited liquidity and a "reference" market with unlimited liquidity. These models suggest that, under fairly common conditions, CFMM prices will be adjusted by arbitrageurs to reflect prices in the reference market. These models work well in practice because arbitrage problems on Uniswap versus other trading venues are often convex, so arbitrageurs can easily figure out how to adjust reserves to reflect current market prices. This theory underpins the use of CFMMs as price oracles in various on-chain applications (e.g. Uniswap v2 oracles). However, after the CFMM boom of summer 2020, we need a model that better captures the reality of CFMM-driven markets.

To do this, flip the script. Suppose we have a highly liquid (low curvature) CFMM and a less liquid (high curvature) reference market. Reference markets can be based on CFMMs, order books, request-for-quote systems, auctions, or any combination. The choice of the market does not affect the model as long as it has non-zero curvature (finite liquidity). If the prices in the two markets differ, an arbitrageur can profit by making offsetting trades in each market until the prices reported by the two markets match. If the two markets are equally liquid, we expect the resulting arbitrage-free price to be between the pre-trade prices of the two markets. However, if CFMM is more liquid, the final price will be closer to the CFMM quote before the arbitrage. In other words, if the liquidity of the CFMM is significantly higher than that of the reference market, then changes in the reference market price will have less impact on the no-arbitrage price.

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Arbitrage between Uniswap and Balancer

image descriptionEthereumArbitrage between Uniswap and Curve

Curve has much lower curvature than Uniswap when trading asset pairs that are roughly equal in price. This means that even if prices on less liquid venues fluctuate wildly, the final price will not deviate too much from what Curve quotes. Note that this type of arbitrage is extremely common in practice.

Ethereum

The arbitrage bots on Facebook are constantly adjusting prices on Balancer, Uniwap, Curve pools, and order book-based exchanges. In our forthcoming paper, we have determined this effect mathematically. If the CFMM has higher liquidity relative to the reference market, then even large deviations in the reference market price will have minimal impact on the no-arbitrage price. We show that this holds true as long as price jumps are bounded by some (potentially large) constant. This assumption rules out corner cases, such as a complete decoupling of the stablecoin peg. Finally, in footnotes 0 and 1, we outline some technical and mathematical considerations that need to be considered when formally describing curvature.

The Curious Case of sUSD

We have seen that low-curvature CFMMs can "impose their will" on the wider market. It also helps explain another phenomenon: price stability. Starting from March 2020, Synthetix announced that it will incentivize the liquidity of sUSD on Curve to better support the sUSD peg exchange rate. The creation of this sUSD pool on Curve had a near-direct effect on the peg: sUSD began to track the prices of other stablecoins more closely. Below, we show the price of sUSD on Uniswap from late 2019 to September 2020. This sUSD pool was officially launched in mid-March 2020 (restarted shortly after the security incident). From the end of March to the beginning of June 2020, the price of sUSD on Uniswap was well anchored. We expect that arbitrage between Curve and Uniswap contributes to this effect: as long as the price volatility of sUSD around the peg is bounded, arbitrageurs are incentivized to keep Uniswap’s price in line with Curve’s.

Note that sUSD is illiquid in all other markets except Curve, which results in a very large curvature difference between Curve and all other markets.

The price of curvature

footnote:

Low curvature is a trade-off, if the CFMM has zero curvature, the CFMM's quote will not change regardless of volume. Thus, constant sum curves (like mStable) set boundaries for each stablecoin a CFMM can hold, preventing LPs from fully holding the worst performing assets.

• Low curvature CFMMs perform better when assets are highly correlated and mean reverting. In this environment, CFMM is able to attract more volume and fees through lower curvature, while mean reversion moderates the impact of impermanent losses. Stablecoins and stablecoin CFMMs basically follow this principle now, and the same is true for CFMMs for maturing assets such as bonds. In the next post, we will discuss the curvature tradeoff of LP in the case of asymmetric information, mean reversion, and impermanence loss.

• footnote:

• [0] One of the main differences between Curve and Uniswap is that Curve's pricing function is "smoother" in certain regions of the price-volume space and "steeper" in other price regions. The economic intuition for why people prefer this shift in the pricing curve is as follows:

We have two assets whose prices (relative to the other) are mean reverting and low spread (e.g. their prices are usually equal);

Deals that keep these assets close to each other (e.g. "soft" pegs) should be cheap, as they encourage arbitrageurs to implement pegs. This is achieved by flattening the curve, which reduces the slippage and market shocks traders face;

However, traders face higher slippage when assets are “unpegged.” This is actually to compensate liquidity providers for deviating from the peg, and to ensure they don't exit liquidity, freezing the market;

Unlike Uniswap which has a more uniform level of curvature for all prices, Curve adapts to the price process it is expected to trade on (e.g. mean reversion, bounded variance). This example shows that the choice of CFMM pricing function is closely related to the type of assets traded and the incentives needed to keep liquidity providers happy.

[1] Is there a way for us to formalize this other than the nebulous notion of "smoother" or "steperer"? The answer is yes, thanks to Carl Friedrich Gauss. Over the centuries, mathematicians have quantified and categorized geometric intuitions through analysis and algebra. One of the main connections between analysis and algebra comes from the concept of inherent curvature. The intrinsic curvature of a surface is the ratio of the area of ​​a small triangle on the surface to the area of ​​a triangle of the same length on a plane. A key feature of intrinsic curvature is that it does not depend on the orientation or parameterization of the surface. For example, the inherent curvature of a beach ball does not change when it is rotated by any angle in any direction. We can define "intrinsic" properties more generally as: