Alexis Direr, a researcher at the University of Orleans in France, published a paper summarizing the mathematical foundations of Uniswap and other exchanges based on automated market makers.
Automated market maker (AMM) is a term that refers to a decentralized exchange represented by Uniswap, which will be widely welcomed in 2020.
In short, these exchanges have abandoned traditional order books and instead rely on liquidity pools controlled by mathematical formulas. Even for the least liquid tokens, traders can always trade with the liquidity pool, but every transaction will affect the price of the assets they trade-this phenomenon is called “slippage”.
The mathematical formula defines how the price changes with the size of a particular order. For example, this formula might say that you can get $3,500 if you exchange 10 Ethereum (ETH) for Dai (DAI), but you can only get $3,400 if you trade 10 Ethereum. This means that in the former case, the price of 1 Ethereum is $350, but in the latter case it is only $340. This formula is often called the “joint curve” because the various possible combinations describe a specific price curve. Uniswap is hyperbolic, although other AMMs may have more complex shapes to optimize different scenarios.
AMM relies on liquidity providers-individuals and entities that invest their funds in liquidity pools to facilitate transactions and reduce slippage. In return, the liquidity provider receives transaction fees paid by the user.
Although this sounds like a good deal, the liquidity provider needs to bear the “impermanence” loss. When prices fluctuate sharply in one direction, liquidity providers may eventually receive less funds than they initially invested. Compared with the 50:50 portfolio of related assets, this portfolio has significantly underperformed and the price deviation is large.
The researchers explained that this phenomenon is caused by the existence of arbitrage traders. External market prices do not follow the joint curve, so continuous action must be taken to keep Uniswap prices in balance with other markets. However, when arbitrageurs re-adjust the liquidity pool to the correct value, they will trade according to the “suboptimal exchange rate” defined by the joint curve. This also allows arbitrageurs to obtain value from liquidity providers.
This loss is often referred to as “impermanence” loss, and liquidity providers can even compare it with a 50:50 portfolio. Leaving aside the situation where prices reach a new equilibrium permanently, Direr raises a question:
“The fact that the two strategies produce the same results at first seems disturbing. In the liquidity pool strategy, the liquidity pool generates two arbitrage costs. In the holding strategy, investors completely avoid arbitrage costs, but In the end get the same final wealth. How is this possible?”
The researcher’s answer is that the way benchmarks are usually conducted is misleading. Uniswap constantly rebalances the liquidity pool as it increases or decreases, so that liquidity providers can have fewer rising assets and relatively more falling assets.
Liquidity providers effectively use the profit and cost averaging method in both methods. When the price of an asset rises, they lock in some profits and gradually buy more when the price falls.
Similar to the working principle of this averaging method, even if the price returns to the initial figure, a 50:50 portfolio that is constantly rebalancing will make a profit. In contrast, the value of the liquidity pool just remains at its original level.
Therefore, “Impermanence Loss” seems to be a misleading term. Loss is always permanent, but in an optimistic situation, it only reduces the gains that can be gained by the same strategy.
Bancor V2 and Mooniswap have adopted techniques to reduce such losses. Bancor V2 uses an oracle to obtain real market prices and balance the liquidity pool accordingly, while Mooniswap delays price updates to minimize the profits of arbitrage traders.
