Paradigm Research: How to use rebalancing strategies to profit from Uniswap?


Researchers believe that even if Uniswap liquidity providers (LPs) experience losses in every arbitrage transaction, in some cases, due to volatility gains, they can still perform better than simply holding currency. In this case, the commission setting should be as close as possible to zero, not zero, in order to rebalance as much as possible.

The authors of the original text are researchers Dave White, Martin Tassy, ​​Charlie Noyes, and Dan Robinson from investment agency Paradigm, who are also seed round investors of Uniswap.

Under ruthless arbitrage, Uniswap LP became rich instead


On October 14, Charlie Noyes posted on Twitter a question he had been arguing with Dan Robinson:

“For any Uniswap asset pair, what is the optimal fee? Can this optimal fee beat an unbalanced investment portfolio and achieve the elimination of impermanent losses and even excess growth?”


Automated market maker (AMM) is a type of decentralized exchange that allows customers to trade between on-chain assets like USDC and ETH. Uniswap is the most popular AMM on Ethereum. Like most asset management systems, Uniswap facilitates transactions between specific asset pairs by holding reserves of two assets. It determines the transaction price between them based on the size of its reserves, so that the price is consistent with the broader market.

Anyone who is willing to join a certain pair of asset pools, we call them liquidity providers, or LPs for short, these people will contribute assets to two reserve assets at the same time, and they have to bear part of the transaction risk in exchange for part of the fee return .

Set up problem

The asset pool provides liquidity between stablecoins and risky assets with random price changes. We also made a particularly cruel assumption that all transactions entering the pool are informed (arbitrage transactions are only between AMM prices and normal It only occurs when there is a deviation between transaction prices).

In other words, the entire pool will experience losses after each transaction .

Traditional thinking

At first glance, in this situation, becoming a Uniswap LP may seem like a costly mistake.

Since market makers require that the buying price be lower than the selling price, when the asset price does not move, they directly profit, and their buying and selling volumes are roughly balanced. These transactions are often referred to as “unwitting” transactions because they have nothing to do with short-term price movements.

On the other hand, market makers will lose money if they buy assets before prices fall, or sell before prices rise. Therefore, one of the most feared counterparties of market makers is arbitrageurs, who will only trade when prices change and leave market makers behind. Every transaction executed by the arbitrageur is a pure profit for him, and a pure loss for the market maker.

Since there are no unknowing transactions in our Uniswap problem setting (in fact, we assume that every transaction is an arbitrage transaction), therefore, the LP will obviously experience very large losses.


However, Dan and Charlie believe that the story does not end here.

They suspect that it still makes sense to become a Uniswap LP for certain potential price dynamics.

They threw this problem to Steven Shreve, a legend in the field of mathematical finance, and then announced the problem on Twitter. Martin Tassy and I independently proposed a partial solution, and then cooperated to extend the complete solution to general situations.

In the next few weeks, the four of us spent some time discussing the results via telegram, looking for errors, and building our intuition, and these discussions are the basis of this article.


If the volatility of an asset is sufficiently high relative to its average rate of return, then over time, the LP on Uniswap will perform better than the HODLer, even if all transactions are arbitrage transactions Case.

This is due to a phenomenon called “volatility harvesting”: under certain conditions, by periodically rebalancing two assets, they may outperform any static portfolio. In this case, “rebalancing” refers to trading so that the proportion of the total portfolio value held in each asset is returned to a fixed allocation, such as 50/50.

Therefore, when LPs are arbitraged, they will basically pay a fee to the market to rebalance their investment portfolio. In this particular mathematical environment, when this rebalancing is beneficial, you should do as much as possible. This means that liquidity providers should set their fees as low as possible and not zero.

This is good news for Uniswap, because it means that even when arbitrage trading dominates, you can still enjoy low fees, which allows Uniswap to remain competitive when on-chain orders continue to increase and begin to provide smaller spreads .

In other words, it is worth repeating that these results are applicable to very specific stylized mathematical settings, and the assumptions involved are very similar to the Black-Scholes option pricing model. For mathematical convenience, we also assumed a different cost structure than the one used in Uniswap production.

Control standard

We evaluate different strategies by comparing their progressive wealth growth rates, which measure how quickly they compound (or lose) value over the long term.

This number is important because over time, strategies that optimize it perform better than strategies that are almost uncertain.

We compare all strategies with the “unbalanced portfolio”. Half of the value of the latter is stable currency and the other half is risky assets, which will never change after that. This is also the community standard for measuring the so-called “impermanence loss” in AMM.

No matter what happens, the unbalanced portfolio will always hold the same amount of stablecoins, which means that in the worst case, when the risky assets lose their full value, the unbalanced portfolio will be almost entirely covered by stablecoins. Composition, so the growth rate will be zero in the long run.

On the other hand, if risky assets grow exponentially, it will soon become the dominant unbalanced portfolio, so its growth rate is the same as that of risky assets.

It is worth noting that the two investment portfolios can share the same progressive wealth growth rate, but have very different performance at close range. For example, if the growth rate of risky assets is zero, then the value of the shares of Uniswap with zero fees will always be lower than the unbalanced portfolio, but since it is expected that over time, neither of them will compound growth or loss. The growth rate of the wealth of both will be zero.

Volatility Drag

50% loss/75% volatility resistance in the process of gains

To understand these results, we must first understand the concept of volatility drag. Assume that the price of our risky asset either falls by 50% or rises by 75% each year, and the probability of the two occurrences is equal.

In any year, if we invest 100 USD in this asset, our expected value is (50+175)/2=112.5 USD. If we simply buy and hold, the expected value of our portfolio will increase by 12.5% ​​year by year, which seems to be a good deal.

Unfortunately, in the real world, our profits will not be realized. If we buy and hold such securities, we will eventually lose everything.

This is because the recombination of wealth will bring catastrophic losses over time.

If we lose 50% in one year, and then grow by 75% in the next year, then we end up with 87.5% of the investment (50% * 175%).

As time goes by, the law of large numbers will guarantee our income every year -15%, and we will inevitably go bankrupt.

Wait, what happened?

If you are trained to analyze gambling from the perspective of expected value, the previous section may look very strange or even completely incorrect.

In fact, more than a week ago, we had a complete, closed-form mathematical solution to this problem. Before that, I had no idea what it meant intuitively.

The root is that the expected value is a theoretical quantity, which measures what will happen if we replicate a given game in countless parallel universes at the same time.

But the reality is not like this. We only have one chance for each game, and the effect of the game we play is not instant, but compounded over time.

We can look at it from another angle to help reconcile mathematics. As we continue to repeat the game (-50%/ +75%), we reinvest our funds every time, and only a few paths are correct, resulting in astronomical returns.

As time goes by, these paths account for smaller and smaller proportions of all possible paths, and we really see that the chance of realizing one of these paths will shrink to zero.

Rebalancing value

In the face of volatility resistance, even when the expected value is positive, some funds should be reserved. In this way, when things go wrong, your losses will be reduced, which will increase your compound wealth in the long run.

As far as trading is concerned, all of this leads to some fairly familiar concepts. When the price rises, sometimes part of the position is closed to lock in profits, in case the price falls again. When prices fall, sometimes it makes sense to buy bottoms in order to obtain the expected future returns at a favorable price.

In some cases, such as this time, the best strategy is to constantly rebalance your investment portfolio so that you always have a fixed proportion of wealth investment in each position, for example, half stable currency and half risky assets. This is not always the best balance. Generally speaking, the more risky assets you want in your portfolio, the higher the volatility of the return rate relative to it, but we will postpone further exploration to future work.

The benefits of rebalancing long-term wealth growth can be substantial and can mean the difference between exponential wealth growth and bankruptcy. Even in the context of our setting, the price of each rebalancing transaction is very unfavorable and causes instantaneous losses. This is also true.


Chances are you are not satisfied with these explanations and want to learn more.

You can first review the Kelly formula, which is a theoretically optimal betting strategy based on these principles. @wpoundstone’s “Fortune Formula” is a highly regarded and easy-to-read book about the history and meaning of Kelly’s formula.

On the other hand, for in-depth research on the mathematics of wealth growth, I highly recommend @ole_b_peters’s lecture notes on ergodic economics or his article published in the journal Nature.

If you choose to research on your own, you must be careful. This is a little-known field. During my own research, I found errors in many sources, which made my understanding go backwards by hours or days. day.

In particular, if you see someone calling for mean regression or logarithmic utility functions, I suggest you don’t stop and move on. The key results in this area do not require any specific return distribution or utility function.

Cost alchemy

Under this setting, when is it beneficial to become an LP? Should LP rebalance as often as possible to promote rebalancing at the least cost?

(Charges should be as low as possible, not zero, in order to trigger rebalancing through increasingly small price changes. Dan Robinson calls it “picking up pennies in a quantum bubble.”)

However, when the cost is completely zero, all the benefits of rebalancing will disappear, and in most cases, the situation of LPs is worse than if they simply hold an unbalanced portfolio.

Understanding this seemingly abnormal phenomenon helps reveal the rest of the problem.

Uniswap uses a “constant product” invariant, which means that every transaction must keep the product of the reserve balance constant without charging fees. We express this as R α R β = C, although readers who are already familiar with Uniswap may be more accustomed to writing it as x*y = k.

However, it turns out that in order to achieve rebalancing, the number of this product C must be increased in order to provide us with excess wealth growth.

Why is C so important? We say that √C is the geometric mean of our reserve balances Ra and Rb. Like the arithmetic mean, the geometric mean increases with the increase in reserves. However, unlike the arithmetic mean, the geometric mean shrinks with the imbalance of reserves, even if their arithmetic mean remains the same.

Without charging any fees, C is constant, so transactions always result in larger reserves or more balanced reserves. The two never exist at the same time, so there is no motivation for wealth growth.

However, in the real world Uniswap, or in the environment we set up, non-zero fees guarantee that C will increase for every transaction. Over time, this means that the reserves are not only growing, but also balanced, which provides the benefits discussed above.

To understand the exact mathematics of how this is calculated, see section 3.1 of Martin and my proof paper.


Having said that, we can now accurately answer the questions raised in Charlie’s initial problem statement.

To reiterate, they are concerned about the wealth growth rate G of Uniswap style AMM, in which the percentage rate 1-γ affects the difference between stable coins and volatile assets (fluctuates in geometric Brownian motion with parameters μ drift and σ volatility) market.

LP wealth growth rate

Optimal fees and excess returns

If and only under the following circumstances, as an LP holding half of the stable currency and half of the risky assets, the return is more than simply holding the currency:

In this case, LPs should set their fees as low as possible but not zero, and they will achieve a wealth growth rate approaching gradually


Since geometric Brownian motion simulates compound growth, they are also affected by volatility resistance, which is expressed mathematically as

The growth rate of wealth is:

This gives us a perspective on the results: rebalancing allows us to partially offset the volatility resistance of the underlying asset. If there is no volatility resistance, and the average return is zero or negative, then the amount of rebalancing will not help. Although the rebalanced portfolio will still do better than just holding the assets themselves, it is better to only hold stablecoins.

On the other hand, if the average return without volatility resistance is positive:

  1. If the volatility resistance makes the asset loss more than 200% of its average log return, then rebalancing on Uniswap will not eliminate enough resistance, then you’d better hold stablecoins.
  2. If volatility resistance makes the asset loss less than 66% of its average log return, then rebalancing on Uniswap is not worthwhile, and you’d better simply hold the asset.
  3. Within this range, becoming a Uniswap LP will eventually make you rich, in fact, richer than any unbalanced portfolio of stablecoins and volatile assets you hold. This includes assets that will eventually become worthless, as well as assets that will exhibit parabolic growth.


You can find a preprinted paper with full proof here. Its working principle is to establish a dynamic model for discrete random walks, and then adopt behavior restrictions when the step size is reduced to zero.

You can also check my original proof of the zero logarithmic drift, and perform some problem simulations here.

How much should we trust in the results of these studies?

In my personal opinion, this is very credible.

We have two independent proof methods, and they will produce the same result when the domains overlap. We also have some simulations to verify our predictions:

Simulated and predicted wealth growth rate

Nevertheless, this is still a very confusing area, and my understanding of it has changed many times in the past few weeks. If you do find an error, please feel free to contact us.

future job

Although we hope you agree that these research results are theoretically interesting (or crazy), there is still a lot of work to be done to determine their relevance to the real world.

For example, many of our assumptions can be modified or expanded:

  1. How to turn these results into a multi-asset case, or when can LP choose a ratio other than 50/50 like Balancer to rebalance?
  2. What happens when we no longer allow unlimited transactions per unit of time?
  3. What happens when we introduce transaction costs that can even be changed to reflect the dynamics of priority gas auctions?

There are also some empirical questions:

  1. Can we estimate these parameters for securities transactions on the market today?
  2. How many actively traded tokens can benefit from the rebalancing strategy we describe?
  3. Can we determine what percentage of Uniswap LP returns are realized in reality due to volatility returns?

Finally, perhaps the most interesting is. How can we use the knowledge learned here to improve existing protocols, create a new protocol, or develop the DeFi ecosystem as a whole?

What’s the problem? idea? Potential applications? We want to hear your voice.

@ charlienoyes, @danrobinson, @_Dave__White , @MartinTassy

Thanks to Vitalik Buterin, Matt Huang, Georgios Konstantopoulos, and Alex Evans for their comments on this article.

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