This post is about a theoretical discovery made by researchers like Dave White, Martin Tassy, Charlie Noyes, and Dan Robinson from Team Paradigm. Ever since Uniswap has released on Ethereum, there have been questions from different angles showing concerns about how it will turn out in the long run. To ensure Uniswap addresses all those, these common questions were openly discussed and researched for the last couple of years. In December 2020, the above team released a blog post explaining the logic and math that answers these questions. Let’s discuss then.

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Uniswap

To quote their website, “Uniswap is an automated liquidity protocol powered by a constant product formula and implemented in a system of non-upgradeable smart contracts on the Ethereum blockchain. It obviates the need for trusted intermediaries, prioritizing decentralization, censorship resistance, and security.” Liquidity Providers (LPs) are the ones who contribute their assets to such liquidity pools developed by the protocol. This month, Uniswap released their V3 to mainnet by incorporating 2 major features: Concentrated Liquidity and Multiple-Fee Tiers.

Questions

Though the monthly trading volume on Uniswap has crossed $30B, there still exists some ambiguity about the fundamental nature of these kinds of protocols. Those have been stripped down to 3 simpler questions as given below:

  1. What is the expected return of a Uniswap LP?

  2. What is the optimal fee to maximize LP wealth?

  3. Can a Uniswap LP’s optimal growth rate exceed a buy-and-hold portfolio’s?

Parrondo’s Paradox & Solutions

To put it in simple terms: Parrondo’s Paradox is a game-theory paradox designed to prove that a combination of losing strategies played simultaneously will become a winning strategy. It is named after the creator, Juan Parrondo, a Spanish physicist who discovered it in 1996. For More, watch this lecture by F. Alberto Grunbaum.

Impermanent Loss is something inevitable in the case of protocols like Uniswap. If you don’t know what’s it, please watch this video explanation made by Finematics:

Hope you’ll refer the maths explained by paradigm from their post itself. Here, I’ll look into the interpretation and how it pans out to be a Parrondo’s paradox.

The Geometric Brownian Motions model is subject to volatility drag due to the fact that it compounds growth. That means rebalancing allows us to partially counteract the volatility drag on the underlying asset.

To quote Dave, “If the average return even without volatility drag is zero or negative, no amount of rebalancing will help us, and we are better off just holding cash, although the rebalanced portfolio will still do better than just HODLing the asset itself. And if the average return without volatility drag is positive:

  • If volatility drag costs the asset more than 200% of its average log return, rebalancing on Uniswap won’t be able to eliminate enough of the drag to make it worthwhile, and you are better off just holding cash.

  • If volatility drag costs the asset less than 66% of its average log return, offsetting drag by rebalancing on Uniswap will not be worth the cost, and you are better off simply holding the asset.

  • Within that range, being a Uniswap LP will eventually make you rich, and richer than you could become by holding any unrebalanced portfolio consisting of cash and the asset. This includes both some assets that will eventually dwindle into nothing and some assets that will go parabolic.”

It’s evident that the impermanent loss is not going anywhere, rather we are turning our losing strategy into a winning one - in other words, we are becoming exponentially wealthy by losing money in every breath.

References

  1. Open Problems: Optimal Uniswap Fees by Charlie Noyes
  2. Uniswap’s Financial Alchemy by Dave White
  3. Liquidity Provider Wealth by Charlie Noyes
  4. Launch of Uniswap V3