Paradigm: The Financial Alchemy of Uniswap

Original title: "Paradigm's latest research: Uniswap's financial alchemy"

Original source: Babbitt

Researchers believe that Even though Uniswap Liquidity Providers (LPs) experience losses on every arbitrageur trade, in some cases they can still perform better than simply holding coins due to volatility harvesting, in this case , the fee setting should be as close to zero as possible, not zero, to rebalance as much as possible.

The original authors are researchers Dave White, Martin Tassy, Charlie Noyes and Dan Robinson from the investment institution Paradigm, who are also Uniswap’s seed round investors.

in Under ruthless arbitrage, Uniswap LP becomes rich instead

Question

Oct 14, Charlie Noyes on Twitter Posted a question he and Dan Robinson have been arguing about:

"For any Uniswap asset pair, the best What is the cost? Can this optimal cost beat an unbalanced investment portfolio, remove impermanent losses, and even achieve excess growth?"

Background

An automated market maker (AMM) is a type of decentralized exchange that allows customers to trade between on-chain assets like USDC and ETH . While 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 priced transactions between them based on the size of its reserves, keeping prices in line with the wider market.

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

Setting problem

The asset pool is between stable currency and risk assets with random price changes We also make a particularly cruel assumption that all transactions entering the pool are informed (arbitrage transactions only occur when there is a deviation between the price of the AMM and the normal transaction price).

In other words, the entire pool experiences a loss after each transaction.

Traditional thinking

At first glance, being a Uniswap LP in this case seems is a costly mistake.

Because market makers require that the buying price be lower than the selling price, when the asset price does not move, they directly profit, and they get Buying and selling volumes were roughly balanced. These trades are often called "uninformed" trades because they are not tied to short-term price movements.

On the other hand, a market maker who buys an asset before its price falls, or sells it before it rises, loses money. Therefore, one of the counterparties that market makers fear most are arbitrageurs, who only come to trade when the price changes and leave the market makers behind. Every trade executed by the arbitrageur is a net profit for the arbitrageur and a net loss for the market maker.

Since there are no uninformed transactions in our Uniswap problem setting (actually, we assume that every transaction is an arbitrage transaction), LP Obviously there will be a very large loss.

Challenge

However, Dan and Charlie feel that the story does not end there.

They suspect that being a Uniswap LP still makes sense for some underlying price dynamics.

They posed the question to Steven Shreve, a legend in mathematical finance, and posted it on Twitter, Martin Tassy and I independently asked partial solutions, and then collaborate to extend the full solution to general cases.

Over the next few weeks, the four of us spent some time discussing the results via telegram, looking for mistakes, building our intuition, and these discussions were is the basis of this article.

Solution

If an asset is sufficiently volatile relative to its average return High, then over time, LPs on Uniswap will perform better than HODLers, even when all transactions are arbitrage transactions.

This is due to a phenomenon known as "volatility harvesting": Balanced, they have the potential to outperform any static portfolio. In this context, "rebalancing" refers to making a trade that returns the proportion of the total portfolio value held in each asset back to a fixed allocation, such as 50/50.

So when LPs are arbitraged, they basically pay the market a fee to rebalance the portfolio for them. In this particular mathematical context, you want to do as much of this rebalancing as possible when it is beneficial. This means that liquidity providers should set their fees as low as possible and not zero.

This is good news for Uniswap, as it means that even in the case of arbitrage trading dominance, you can still enjoy low fees, which makes Uniswap Stay competitive as on-chain order books grow and start offering tighter spreads.

That said, it bears repeating that these results apply to a very specific stylized mathematical setting involving assumptions very different from the Black-Scholes option pricing model resemblance. For mathematical convenience, we also assume a different fee structure than that used in Uniswap production.

Comparison

We evaluate different Strategies, these growth rates measure how quickly they compound (or lose) value over the long run.

This quantity is important because a strategy that optimizes it performs better over time than a strategy that is barely certain.

We compare all strategies to an "unbalanced portfolio" where half the value is in stablecoins and the other half is in risky assets, after which Never change again. This is also the community standard for measuring the so-called "impermanent loss" in AMMs.

No matter what happens, the unbalanced portfolio will always hold the same amount of stablecoins, which means that, in the worst case, when the risk asset When losing all of its value, an unbalanced portfolio will consist almost entirely of stablecoins and thus have zero growth rate in the long run.

On the other hand, if the risky asset grows exponentially, it will quickly become the dominant unbalanced portfolio and thus grow at the same rate as the risky asset.

It is worth noting that two portfolios can share the same progressive wealth growth rate while performing very differently at close range. For example, if the growth rate of the risky asset is zero, then the zero-fee Uniswap stake will always be worth less than an unbalanced portfolio, but since neither is expected to compound growth or losses over time, the two The growth rate of their wealth will be zero.

Volatility Drag

Volatility Resistance in the 50% Loss/75% Gain Process

To understand these results, one must first understand volatility The concept of volatility drag. Assume that our risky asset either falls in price by 50% or rises by 75% every year, with equal probability of both happening.

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

Unfortunately, in the real world, our profits won't materialize. If we buy and hold this security, we end up losing everything.

This is because, over time, the compounding of wealth can bring catastrophic losses.

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

As time goes by, the law of large numbers will guarantee us a gain of  -15% per annum, and we will inevitably go bankrupt.

Wait, what happened?

If you've been trained to analyze gambling from the perspective of expected value, chances are the previous section will look very strange, and may even be downright incorrect.

Actually, a little over a week ago, we had a complete, closed-form mathematical solution to this problem, which I had no idea before intuition What does up mean.

At its root: expected value is a theoretical quantity that measures what would happen if we replicated a given game simultaneously in an infinite number of parallel universes.

But this is not the case in reality, we only have one chance for each game, and the effect of the game we play is not instantaneous, but over time Composite.

We can look at it from another angle to help reconcile the math. As we repeated the (-50%/+75%) game, reinvesting the money each time, only a handful of paths were correct, yielding astronomical returns.

Over time, these paths make up a smaller and smaller percentage of all possible paths, and we really see what one of them achieves The chances also shrink to zero.

The value of rebalancing

Facing volatility headwinds, even with positive expectations In some cases, part of the funds should also be reserved. That way, you have less to lose when things go wrong, which increases your compound wealth in the long run.

All of this leads to some fairly familiar concepts as far as trading goes. When the price rises, sometimes part of the position is closed to lock in profits in case the price falls again. When prices are falling, sometimes it makes sense to buy the bottom in order to secure expected future returns at a favorable price.

In some cases, like this one, the best strategy is to constantly rebalance your portfolio so that you always have A fixed ratio of wealth investment, for example, half stable currency, half risk assets. This is not always the optimal balance, and in general you want the more risky assets you have in your portfolio, the higher its return relative to its volatility, but we defer further exploration to future work.

The benefits of rebalancing long-term wealth growth can be huge and could mean the difference between exponential wealth growth and bankruptcy. Even in the context of our setup, it is true that every rebalancing transaction is priced unfavorably and causes instantaneous losses.

Resources

It is likely that you are not satisfied with these explanations and would like to know more many.

You can first review the Kelly formula, which is a theoretically optimal betting strategy based on these principles. The Wealth Formula by @wpoundstone is a highly regarded and easy-to-read book on the history and meaning of the Kelly formula.

On the other hand, for an in-depth study of the mathematics of wealth growth, I highly recommend @ole_b_peters' lecture notes on Ergodic Economics or his publication in Nature article.

If you choose to do your own research, you must be careful, this is a little-known field, in the course of my own research, I found a lot The sources are all wrong, which set my understanding back hours or days.

In particular, if you see someone calling for mean reversion or a log utility function, I suggest you don't stop there and move on. The key results in this area do not require any particular payoff distribution or utility function to be assumed.

Fee Alchemy

In this setup, when is it beneficial to be an LP that rebalances as often as possible to facilitate rebalancing at minimal cost?

(Fees should be as low as possible, not zero, so that rebalancing is triggered by increasingly small price changes. Picking up pennies in the middle".)

However, when the fee is exactly zero, all benefits of rebalancing disappear, and in most cases, the situation of LP Worse than if they just held an unbalanced portfolio.

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

Uniswap uses a "constant product" invariant, which means that without fees, every trade must keep the product of reserve balances constant . We express this as Rα Rβ=C, although readers 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 increase 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 as reserves increase. However, unlike the arithmetic mean, the geometric mean shrinks as reserves become unbalanced, even if their arithmetic mean remains constant.

In the absence of 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 momentum for wealth to grow.

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

For the exact math of how this is calculated, see section 3.1 of Martin and my proof paper.

Mathematics

Having said all this, we can now accurately answer Charlie's original problem statement questions raised in .

To reiterate, they focus on the wealth growth rate G of Uniswap style AMMs, where the percentage fee rate 1-γ affects stablecoins and volatile assets (in geometric Brown Movement volatility, with parameters μdrift and σ volatility) between markets.

LP wealth growth rate

Optimal fees and excess returns

if and only if, As an LP, holding half of the stable currency and half of the risk assets is more profitable than simply holding the currency:

< img alt="Paradigm Research: How to Use Rebalancing Strategy to Profit in Uniswap?" src="https://img.chainnews.com/material/images/9a42732ec88cbdb50291ae6c7ec8d91c.jpg-article">

In this case, LPs should set their fees as low as possible but not zero, and they will achieve a wealth growth rate that gradually approaches< /p>

Explanation

Since geometric Brownian motion simulates compound growth, they are also subject to volatility drag, which is expressed mathematically as

< br>

The growth rate of wealth is:

This gives us a perspective to observe the results : Rebalancing allows us to partially offset the volatility resistance of the underlying assets. If there is no volatility headwind and the average return is zero or negative, then the amount of rebalancing will not help either. While a rebalanced portfolio will still do better than just holding the asset itself, we are better off just holding stablecoins.

On the other hand, if there is no volatility resistance the average return is positive:

If Volatility headwinds make assets lose more than 200% of their average log return, then rebalancing on Uniswap won't remove enough headwinds, and you're better off holding stablecoins.

If the volatility headwinds make the asset lose less than 66% of its average log return, then rebalancing on Uniswap is not worth it and you are better off simply holding on to the asset.

On this spectrum, being a Uniswap LP will end up making you richer, in fact, richer than holding any unbalanced portfolio of stablecoins and volatile assets. This includes both some assets that will eventually become worthless and some that will exhibit parabolic growth.

Proof

A preprint paper on the full proof can be found here. It works by modeling the dynamics for a discrete random walk and then taking behavioral constraints when shrinking the step size to zero.

You can also look at my original proof for the zero log-drift case and do some problem simulations here.

How much trust should we place in these findings?

In my personal opinion, this is very believable.

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

Simulated vs. predicted growth rates of wealth

Nevertheless, this is still a very confusing area, and over the past few My understanding of it has changed many times during the week. If you do find a bug, please feel free to contact us.

Future Work

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

For example, many of our assumptions could be modified or extended:

How to translate these results For the multi-asset case, or when can LPs choose something other than 50/50 like Balancer to rebalance?

What happens when we no longer allow infinite transactions per unit of time?

What about when we introduce transaction costs, which can even vary to reflect priority gas auction dynamics?

There are also some empirical questions:

Can we provide for the securities transactions in today's market Are these parameters estimated?

How many actively traded tokens could benefit from the rebalancing strategy we describe?

Can we determine what percentage of Uniswap LP returns are realized in reality due to volatility gains?

Finally, and perhaps the most interesting. How can we use what we learn here to improve existing protocols, create a new one, or grow the DeFi ecosystem as a whole?

What's the problem? idea? Potential application? We want to hear your voice.

@charlienoyes, @danrobinson, @_Dave__White, @MartinTassy

Thanks Vitalik Buterin, Matt Huang, Georgios Konstantopoulos, and Alex Evans contributed to this article.

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