A buy and hold portfolio has a linear relationship with its value since the value of the portfolio increases or decreases
with price linearly. The liquidity provider has a near linear relationship with price moves around the initial price only. When the price
drops to zero the portfolio value drops sharply to zero, since the liquidity provider is buying more and more of the asset losing its
value until the price is exactly zero and all of the other liquidity asset has dropped to zero. When the price increases the liquidity
provider loses more and more of the upside gains since he is selling more and more of the appreciating asset as the price increases.
This is the reason why the liquidity provider is losing money in both directions. He is buying the asset that is dropping in value and
selling the asset that is rising in value. Ideally the liquidity provider wants the asset prices to hover around his initial price, so one can
therefore think of the liquidity provider being ‘short volatility’ and ‘short convexity’. His position looks similar but is not identical to
someone being short a call and put option (a straddle). The risk profile here is nonlinear.
Figure 5: Risk Profile of a Uniswap v2 liquidity provider and a Buy'n'Hold
The impermanent loss of Uniswap v3 can be calculated similarly to v2, first one needs to solve Eq. (5) and Eq. (6) for x and y,
similarly to Eq. (1),Eq. (2) and Eq. (4), which amounts to finding the roots of a quadratic equation in x and y.
�` + √� a�
1
=�789:
+ =�345b ∙ � + �� ∙ a
=�345
=�789:
− 1b = 0
and
�` + √�a 1
=�789:
+ =�345
� b ∙ � +
�
� ∙ a
=�345
=�789:
− 1b = 0
Which can be solved using the ordinary p-q formula. Substituting this in the formula for the difference in the market to market move
of a 2-asset market making portfolio compared to a fixed quantity portfolio yields the impermanent loss of a Uniswap v3 liquidity
provider. We note from these quadratic equations that when �345 is zero and �789: is infinite we return the same equations as given
Eq. (4) in.
UNISWAP: Impermanent Loss and Risk Profile of a Liquidity Provider
Aigner, Andreas A. 7
Figure 6: Impermanent Loss of various Uniswap v3 liquidity positions and a Uniswap v2 liquidity position
Examples for various price ranges are shown in Figure 6 and compared to the impermanent loss of a Uniswap v2 liquidity provider. It
is obvious that when the fixed range of the liquidity provider approaches the semi-infinite domain of the v2 liquidity provider the
impermanent loss functions become similar. For smaller ranges the impermanent loss gets more symmetric and decreases around the
initial value having steeper losses than an ordinary Uniswap v2 position with a semi-infinite domain will have.
For example, when the price moves by 20% the impermanent loss of a v2 liquidity position will be −0.56% and −0.46% while a
fixed range of 25% and 125% of initial will have an impermanent loss of −4.75% and −3.8%. Some more examples are collected in
Table 1.
Table 1: Impermanent Loss of various Liquidity Positions
%Move/Range -20% Initial 20%
[0%, inf) -0.56% 0 -0.46%
[0%, 200%] -0.86% 0 -0.70%
[25%, 175%] -1.5% 0 -1.22%
[50%, 150%] -2.34% 0 -1.91%
[75%, 125%] -4.75% 0 -3.8%
Consider a fixed range of 80% and 120%, the risk profile of such a liquidity position will look like Figure 7. Since the liquidity
outside of this range is composed entirely of one or the other asset the value of the portfolio will decline linearly below the lower
bound, since the pool consists of 100% of the declining asset, and remain constant above the upper bound, since the pool consists of
100% of the asset that is not appreciating. Compare this to a buy and hold portfolio of the same initial asset distribution, which
increases/decreases linearly with price. Note that when the price drops to zero for one asset your buy and hold portfolio will still be
worth 50% whereas the market making portfolio will be have lost 100% of its value since it was long 100% of the declining asset
already since it breached the lower bound.
UNISWAP: Impermanent Loss and Risk Profile of a Liquidity Provider
Aigner, Andreas A. 8
Figure 7: Risk Profile of an 80%-120% liquidity position.
Figure 8: Impermanent loss of an 80%-120% liquidity position.
Figure 8 shows the corresponding impermanent loss of a [80%, 120%] range liquidity provider. We note here that a liquidity provider
portfolio risk profile is always at a disadvantage to a buy and hold portfolio. Some websites interactive calculators erroneously show a
risk profile that indicate that you will lose less than a buy and hold portfolio. This is not the case. In essence as exhibited by the PNL
profile, Figure 7 and Figure 8, it is similar to being synthetically short variance (short Gamma) on the underlying, earning Theta
(yield). Extreme spot price moves will manifest this short gamma; in further papers we will discuss periodic delta hedging strategies
and short dated puts to ameliorate and subdue the risk characteristics endemic in the form of liquidity positions discussed in this paper.
IV. CONCLUSION
Current investors can purchase crypto using centralized exchanges (CEX), which are companies in the conventional sense that provide
a platform or app to deposit fiat currencies into or offer the option to purchase using credit cards. They provide an on-ramp facility for
investors to exchange fiat currencies into crypto. Many of these platforms offer interest rates on crypto (and fiat) deposits and further
investment products such as locked staking.
In contrast to centralized exchanges there are decentralized exchanges (DEX) which have no central entity managing the deposits.
Instead the interactions between participants are handled by smart contracts on the Ethereum chain. Assets get exchanged using a
smart contract using a deterministic market making function that has a set price for every amount of token that gets offered or bid.
This function of liquidity represents a distribution of bid offers similar to order book type markets provided by centralized exchanges.
Instead of having a dynamic order book with market depth the depth of bid-offers is fixed, as long as the amount of liquidity
UNISWAP: Impermanent Loss and Risk Profile of a Liquidity Provider
Aigner, Andreas A. 9
underlying doesn’t change. The liquidity in such a liquidity pool is not provided by a centralized entity, as would be the case on a
CEX, but instead by other individual market participants who get compensated by trade commissions in return for the risk they take
on. The risk that a liquidity provider takes on is essentially of two kinds. When setting up a position initially an amount of each asset
is provided and usually in a ratio of about 50% each. A liquidity provider therefore has, without any other interaction, a linear risk of
the price of one of the assets changing versus the other asset. This is normally called Delta risk, since Delta in the option market is the
rate of change of one asset with the change in the other asset. Usually the other asset in such case is the base currency such as the US
Dollar. When you have a liquidity position you can have two assets which are both different to your US Dollar, so in practice you
have Delta risk on both currencies already.
The second risk that a Liquidity Provider has is the change in his position due to other market participants interacting with his liquidity
pool. Every time part of his position gets bought or sold, the price of the asset changes. And at either one of the extreme ends of the
prices, albeit 0 and infinity or a fixed range (�345 and �789:) he will have swapped one of the assets into the other assets completely.
The difference between the value of the portfolio of two assets with and without these transactions is called impermanent loss. It can
also be called unrealized loss in this situation, because if the price of the asset reverses to the initial price a liquidity provider would
end up with exactly the same position as what he set out with, having zero loss, but would have earned commissions along the whole
price swing. The Delta risk above is also an unrealized loss, since the Liquidity provider ends up with zero loss or gain when the price
drops back to the same initial price.
To summarize both the Delta as well as the impermanent loss are unrealized as long as the liquidity provider does not withdraw his
liquidity from the pool. At the point where the liquidity provider withdraws his funds from the pool, is when his loss or gain due to the
impermanent loss gets realized. His loss or gain due to the Delta risk is only realized once he sells out of his position or swaps one of
the assets for the other.
Here we have described the underlying mechanics of a market making function in Section II, and have derived the impermanent loss
function for Uniswap v2 and v3, in Section III. We have provided an improved impermanent loss formula for the commonly quoted
Equation for v2 and online calculators for v3. We have charted the risk profile of positions in v2 and v3 and compared various
different ranges of liquidity, showing that v2 is approached in the limit of the range going to infinity.
Uniswap is one of the most liquid decentralized exchanges. There are other exchanges that offer similar market making products that
will work according to different market making functions. There is still ongoing research into these which can be found in our
references below.
What other risks exist for a liquidity provider? One of the key properties of a liquidity provider is that he is willing to own all of either
one of the assets at either his lower price limit or zero (in the case of v2). This means that when one of the assets has huge price
swings or is compromised, as for example in the recent case of Mark Cuban (ironfinance, Titan-DAI). The liquidity provider would
have sold all of his reliable assets to purchase the compromised asset. A liquidity provider cannot be certain that he will own any or
part of either asset at the time of redemption. If a liquidity provider is happy to own either one of the assets at its low and sell out of it
at a higher level, he will get compensated for this through the commissions he will earn when participants are trading on his liquidity
pool.
Since market making functions are using individual liquidity curves there are also various arbitrage opportunities. Some with external
order book type exchanges as well as internally between a trio of currency pairs that end up quoting different cross exchange rates for
the exchange rate in the first pair [72]. Provided the arbitrage opportunity is greater than the transaction costs required it will be
utilized. The work involved in taking advantage of these opportunities is extensive, especially if its across different exchanges. There
is however the advantage of using atomic transactions, meaning one smart contract transaction that will exercise a round trip in the
arbitrage. Building such arbitrage tools as a regular person will be too difficult, but suffice to say it is most certainly done already to
some extent. In fact decentralized exchanges such as Uniswap can represent a ‘fair’ price of assets since it will have been arbed
already, these are called price oracles and Uniswap is regarded as such as well [44] .
As of time of writing the biggest liquidity pool on Uniswap (ETH-USDC) is earning an average of 1.5% weekly [90, 100, 101], which
amounts to 78% annually at the current trading volumes. A liquidity provider who provides an equal amount of liquidity mapped
along the price curve can expect to earn the same or similar return.
There are other risks involved with crypto with have to do with smart contract risk, fraudulent DEXs, regulatory clampdowns, ISP
provider censorship and other exterior risks [102-106]. There are discussed widely in the literature here and here. Useful websites for
monitoring the Defi space and security issues are vfat.tools, RugDoctor and rekt.news.
APPENDIX I: DERIVATION OF THE IMPERMANENT LOSS FUNCTION V2
We start off with the change in the Value of a portfolio of a liquidity provider versus the change of portfolio of a fixed-asset portfolio.
UNISWAP: Impermanent Loss and Risk Profile of a Liquidity Provider
Aigner, Andreas A. 10
�G − �N − L�e8Efg − �NM
Notice how we can cancel the �N term and write
�< + �G ∙ �< − (� + �G ∙ �)
Substituting Equations 1,2 in Equation x gives
=� ∙ �G + �G ∙ h�
�G
− i=� ∙ �N + �G ∙ h�
�N
j
When you expand this you get
√� ∙ L=�G − =�NM + √��G ∙ ih 1
�G
− h 1
�N
j
The initial value of the portfolio is
�N = � + �N ∙ � = =� ∙ �N + �N ∙ h�
�N
= 2=� ∙ �N
Dividing this by �N we get
√� ∙ L=�G − =�NM
2=� ∙ �N
+
√��G
2=� ∙ �N
∙ ih 1
�G
− h 1
�N
j
Cancelling out the terms gives
L=�G − =�NM
2=�N
+
�G
2=�N
∙ a
=�N − =�G
=�N ∙ =�G
b = 1
2ih�G
�N
− 1j +
1
2 h
�G
�N
∙ i1 − h
�G
�N
j
When we introduce the ratio of prices
� = �G
�N
This can simply be written as
� = 1
2 L√� − 1M +
1
2√�L1 − √�M
Which becomes
� = √� − 1
2 ∙ (� + 1)
Comparing this to the commonly quoted impermanent loss function, where the change in value as relative to the final portfolio value
is calculated instead of the initial value, you would divide by �e8Efg instead of �N and get [97, 107-109]
UNISWAP: Impermanent Loss and Risk Profile of a Liquidity Provider
Aigner, Andreas A. 11
� = 2√�
1 + � − 1
APPENDIX II: HOW TO BE A LIQUIDITY PROVIDER
We want to explain the steps involved and the fees involved in initiating a liquidity position. We choose the most liquid and biggest
liquidity pool on Uniswap which is ETH vs USDC. USDC is a stablecoin which is backed by US Dollar 1-1. The company which
issues these stablecoins is backed by investors such as Coinbase.
1. First, we transfer by wire transfer or SEPA say 2000 USD to a centralized exchange for free.
2. Then we buy 1000$ worth of ETH and incur a fee of say 0.00061814 ETH ($1.19 with ETHUSD = 1930$).
3. We transfer this ETH to a wallet accepted by Uniswap and incur a fee of 6.98$ (0.0036ETH). We use Metamask.
4. We purchase 1000$ worth of USDC for a fee of about $1.19.
5. We also transfer this amount of USDC to the accepted wallet and incur a fee of 8$
6. Connecting your wallet to Uniswap we find the most liquid pool using the charts. One of the options is to provide liquidity.
Since all the liquidity pools have to use a wrapped ETH (WETH9) to provide liquidity we have to convert the ETH we have
in our wallet to WETH9. We do this using Uniswap which incurs a 1.17$ fee.
7. Now we can start with setting up the liquidity position. We choose to allocate all of the wrapped ETH (WETH9) and it will
return us the required USDC we need to allocate. If the balance of ETH versus USDC is not exactly 50:50 then you will need
to transfer for USDC from as described above. Alternatively start off with buying more of the assets or reducing the amount
of ETH to deposit.
8. In our example we end up having a balance of 58% versus 42% roughly.
9. Then one confirms the wrapped ETH and USDC position which costs 1.41$ and 1.87$ in fees.
10. After waiting for these two transactions to go through you can confirm the liquidity position as a whole, which costs 13.26$
to mint.
11. After paying this last fee after a couple of minutes have passed one will see the position listed under ‘your positions’ and can
UNISWAP: Impermanent Loss and Risk Profile of a Liquidity Provider
Aigner, Andreas A. 12
12. View the Ethereum Address of this position on the Ethereum chain using Etherscan. One can also view the running fees
collected for the liquidity pool. Which in this case is already around 0.54$ in fees after about 1-2hrs online.
In the whole process we have spent 35,07 USD or in this example 1.25% of the total value of the portfolio, which is less than the
theoretical return in commissions in one week (1.5%).

How does Uniswap work

Liquidity providers deposit funds into Uniswap and receive trading fees from traders (supply-side fees).

Who governs Uniswap

UNI tokenholders own and govern the protocol. They do not currently take a cut of the total trading fees paid by traders (revenue). Uniswap's investors include Paradigm, a16z, USV, etc.

Crypto Winter Trends & Analysis

The first occurred after December, 2017, following a steep rise in the price of Bitcoin and Ethereum over that year. The second occurred after Q1 2022, following broad declines across all asset classes.

Crypto winters are characterized by a seemingly long period of lethargic sentiment towards cryptocurrencies. In the 2017-18 winter, the price remained somewhat lackadaisical until the stimulus of Covid-19 occurred in Q1 2020. This stimulus boosted incomes and thus financial experimentation, particularly among retail users who delved with digital finance in both crypto and stock markets.

The winding down of that stimulus in 2022, following interest rate hikes by the fed, led to a significant decline of -67% in the price of Bitcoin, putting it closer to the 2017 level of roughly $20,000 from the peak at $69,000.

Silicon Valley Bank, Zero Knowledge L2s, and wallet hacks helped define the narrative in the first quarter or 2023. Here are some insights on defining moments in the first 3 months!

Uniswap Sets Record Trading Vol. After Silicon Valley Bank Collapse

On March 11th, one day after the collapse of Silicon Valley Bank and takeover by the FDIC, trading volume of Uniswap’s decentralized exchange hit ATHs.

• $25B - highest trading volume day in history on March 11th, 2023
• $77M - total trading fees earned by liquidity providers (LPs) in March
• 35% - the total volume of Uniswap volumes happening on L2s

The elevated trading volumes on Uniswap was largely driven by traders selling USDC and buying other stablecoins like USDT (Tether) and DAI because ~$3 billion worth of assets that backed USD Coin were held in Silicon Valley Bank.

The total value locked in defi isis on a steady decline, and as of September 1, 2023, it’s teetering at roughly $37.59 billion. Rewinding to February 9, 2021, this was the last time the TVL stood at such a level, right before it surged to its record peak.

By May 11, 2021, it swelled to an impressive $121 billion, and by November 8, 2021, it soared to a staggering $178 billion. Although there was a brief resurgence to $161 billion on April 2, 2022, the TVL has been in a consistent descent since the all-time high.

As of the first of September this year, LIDO reigns supreme as defi’s leading protocol by TVL, boasting a $14.06 billion TVL. Hot on its heels are Makerdao, Aave, Justlend, and Uniswap, completing the top-tier quintet of defi protocols based on TVL size. Makerdao stands at $5.05 billion, Aave at $4.49 billion, Justlend at approximately $3.33 billion, and Uniswap close behind with about $3.27 billion this weekend.

Uniswap Performance in the Turmoil

beyond the scaling of its core business, Uniswap has also adopted a horizontal growth approach, launching a wallet and wading into the aggregation market with UniswapX, which leverages off-chain orders, allowing the free market to compete to fill bids.

Uniswap’s early mover advantage perfectly showcases the ability of the market incumbent to compound its gains. And in the bear market, as forks and clones die off, the advantages of being a market leader have magnified in intensity.

Despite possessing no unique defensibility, Uniswap reigns supreme in the DEX landscape. Trading volumes trended upwards for the first half of 2023 and have remained steady since June. This year, Uniswap’s Spot Trading Volume has exceeded Coinbase’s. A perfect microcosm of the trend toward decentralized service providers – easily observable when comparing DEX to CEX spot trading volume.

DEX Activity Based on Network Effect :

The role of network effects in public chain competition is very obvious. It is represented in the ecological flywheel that enhances the number of users, the number of developers, and capital size reinforce each other. Numerous “Ethereum killer” public chains were all falling under this powerful rule.

Network effects also exist in multilateral user markets such as trading and lending platforms. On Uniswap, the larger the trading volume, the more market makers will be attracted to make the market to provide liquidity. On the other hand, the improvement in market-making depth will in turn increase the user stickiness of traders, so as to achieve mutual reinforcement.

However, due to the open source nature of blockchain projects and the rise of liquidity mining, new projects can use Fork project code + mining subsidies to attract users and market makers of existing projects. SushiSwap uses this approach to catch up with Uniswap, which was previously considered to have strong network effect.

A large number of new Dex on BSC also challenge PancakeSwap in a similar way. However, it turns out that the success rate of this approach is getting lower and lower. Uniswap and Pancake are still the Dex with the highest number of active users on Ethereum and BSC respectively. The moat built by network effects helps them resist a certain degree of attack.

TVL Fluctuations in DeFi: The Total Value Locked (TVL) in DeFi is a key metric reflecting the state of the ecosystem. It has experienced fluctuations, reaching a recent low of $37.59 billion in September 2023, a level last seen in February 2021.

Uniswap's Competitive Ranking: In the DeFi TVL hierarchy, Uniswap stands strong, closely following the leading protocols like LIDO, Makerdao, and Aave. Uniswap's TVL hovers around $3.27 billion, solidifying its position.

Horizontal Growth Strategy: Uniswap's strategy extends beyond its core business, with the introduction of a wallet and the foray into the aggregation market through UniswapX. This approach demonstrates its adaptability.

Network Effects in DEX: Uniswap's dominance in the decentralized exchange (DEX) landscape is a testament to the power of network effects. As trading volumes rise, more market makers are drawn to provide liquidity, enhancing user experiences.

Challenges from Competitors: While network effects provide a moat, Uniswap faces challenges from competitors like SushiSwap and new DEXs on different blockchains. Despite the rise of forks and clones, Uniswap maintains its position as a leader.

In conclusion, Uniswap’s commitment to decentralization, governance by tokenholders, and adaptability have allowed it to weather the crypto winter and remain a prominent DeFi protocol. Its enduring popularity and ability to navigate challenging market conditions highlight the resilience of decentralized finance in the broader blockchain landscape.

Introduction

Are Uniswap v3 liquidity providers (LP) profitable ?

Recent research shows that at least in the most popular Uniswap v3 pool (USDC/WETH 0.05%), LPs have been suffering large losses. Is Uniswap¹ liquidity provisioning an irrational behavior that is expected to disappear with time? Not necessarily.

To show why, I will make the following arguments:

1. The published LP profit-and-loss analyses are sometimes misinterpreted to apply more broadly than they actually do. Additionally, the LVR idea itself provides a way how to identify +EV (positive expected value) pools.

2. It is not realistic to select a single, one-size-fits-all LP profitability metric. LPs have different goals when depositing to Uniswap. Currently, the most widely used metric is “EV in USD terms compared to HODL”. However, it is not always the right one to measure what “success” means for an LP; for example, range orders, just-in-time liquidity, and potentially also hedging, can all lead to incorrect conclusions when this metric is used.

¹ — I’ll skip “v3” further in this article.

Background

Feel free to skip this if you’re familiar with the context and existing discussions.

A decentralized exchange (DEX) can only be attractive to traders if its price impact and fees are low enough to be competitive with centralized exchanges or other alternatives. In order to have a low price impact, a DEX requires deep liquidity to be present near the current price. Low fees are only possible if liquidity providers (LP) price this liquidity cheaply enough.

These actors have conflicting interests:

• Traders: want fees to be small and liquidity highly concentrated relative to volatility
• LPs: want fees to be big and liquidity concentration low relative to volatility

Given these pressures, is realistic to expect that LPs deposit large amounts of non-incentivized liquidity? Or is the current state a historical aberration that will be corrected when the markets become more rational? Are liquidity providers expected to be profitable, on the average?

These are not new questions: doubts on Uniswap LP profitability have been showing up since at least January 2019 (for Uniswap v1 at the time). But at the moment, mostly thanks several recent analyses by several researchers that look at impermanent/divergence loss (IL), markouts and Loss versus Rebalancing (LVR)², as well as increased conceptual awareness of the order flow toxicity, a consensus seems to be forming that Uniswap LP is highly unprofitable.

Perhaps at this point the community’s opinion risks shifting too much in this direction. Certainly there is a lot of room for improvements in DEX design, but they are either yet to be tested in the real world, or come with their own limitations (e.g. small number of tradable assets in GMX).

² —Further in this article I don’t always distinguish between LVR and IL analyses. Under certain assumptions, they result in the same expected value of PnL .

In quantitative finance, we assume that market trends are unpredictable, and asset price behavior follows a Geometric Brownian Motion. We simulate the future random walk based on historical returns and volatility, which is also known as a price random walk.

So how do we describe a random walk? The variation in asset prices follows a stochastic differential equation, which is a historical expected return μ and a historical volatility σ:

Here, St represents the price of the asset at time t, dSt represents the change in the asset within an infinitesimal time interval, and Wt~N(0,1) follows a normal distribution with a mean of 0 and a standard deviation of 1, representing the random component in the asset price movement.

The random walk is an ideal framework for liquidity providers to assess asset price trends. As opponents of trend bettors, LPs must assume that the asset will not deviate from a specific trend and focus only on the fluctuations within a certain range. Their profits come from this range, and they welcome bettors to participate. Let’s look at the following graph to understand the concept of a random walk and price intuition.

Next, let’s solve the above geometric Brownian motion equation accurately to obtain the analytical expression of the asset at time t. Let’s assume the current price of ETH is 1800, and we simulate the asset’s price movement for the next 90 days without deviating from a specific trend, using μ=0.1 and σ=0.2 as parameters.

The graph above shows a sample path of asset price variation that follows a geometric Brownian motion. We expect that, given the parameters of expected returns and volatility, after simulating hundreds of paths, the prices will converge to a specific range at a given future time t. We performed 200 random walk simulations, and based on the

terminal prices, approximately 70% of the paths eventually returned to the range between 1610.87 and 1980.13. You may have already guessed the connection between this range and the LP’s market-making range, but let’s first consolidate some intuitions about LP strategies before we quickly link them to practical applications.

Let’s review the thinking process of a robust LP strategy to help with further discussions in the next sections:

1. LP opposes trend bettors.
2. Assume that the asset will not deviate from a specific trend.
3. Use a stochastic model to assess the asset’s price movement.
4. Evaluate LP market-making strategies.

The parameters in the Brownian motion, such as expected returns (μ), historical volatility (σ), and time (t), directly affect the LP’s market-making strategy. To assist decision-making, we abstract these parameters. According to the formula, for assets with high returns, the opportunity cost for LPs is too high, so it is better to hold these assets directly rather than invest in the liquidity pool.

Similarly, high volatility assets increase the LP’s risk because there is a higher probability that the asset’s price movement will deviate from the average path, increasing the possibility of LP’s exposure to one-sided risk.

Lastly, the mathematical meaning of time in the Brownian motion suggests that as time goes on, the asset price is more likely to return to the profitable range. Therefore, it is advisable to avoid frequent adjustments of the market-making range.

Based on these intuitions, LPs should try to avoid participating in high-risk liquidity pools that may bring high returns, such as meme-meme or meme-USDC pools. Although these pools may offer high annual percentage yields (APY) in the short term due to abundant liquidity, in the long run, the price movements of these assets often deviate from a robust LP strategy.

Furthermore, why do many experts recommend deploying LPs in the BTC-ETH trading pool? Despite BTC and ETH individually offering higher returns, their return ratio in the liquidity pool remains relatively stable. This allows high returns to increase liquidity and balance the deviation from the LP range. The stability of this return ratio also validates our theoretical assumption that even if asset prices temporarily exceed the normal range, over time, they are likely to return to the LP’s profitable range because BTC and ETH tend to rise together in the long term. The delta of BTC-ETH hedges the potential losses from high volatility and impermanent loss (IL).

Qualitative screening for LP trading pools:

• Asset pairs
• Trading volume
• Liquidity

Quantitative evaluation of LP trading strategies:

• Asset returns
• Asset volatility
• Holding period for asset pairs

I must emphasize that the purpose of the above analysis is to provide investors with a concise and intuitive thinking model and method framework. You have complete freedom to apply various assumptions and mathematical tools. Mathematics and tools are only aids to help investors abstract quality investment intuitions. However, the key is to gain a deep understanding of the market-making range through these tools, enabling you to make the best investment decisions.

Misinterpretation Risks

What can, and what cannot be said about LP profitability using the existing analyses Let us focus on the WETH/USDC pairs

Other analyses focus on ETH/stablecoin or large-asset pairs. Mid-cap asset pairs have been much less studied, and long-tail asset pairs almost not at all: the only paper that I know is about Uniswap v2, and analyses three “exotic” v2 token pairs (the paper from ETH Zurich). This focus is understandable because a large part of Uniswap liquidity belongs to the larger pairs. However, generalizations between different-sized pools should be done cautiously, because small-cap token dynamics are very different from ETH/USDC dynamics.

The LPs are still expected to have all of their principal back if the price reverts to the starting point. The problem is in the assumption that the price is going to revert at some point. This assumption is not true in general for volatile assets. Mean-reverting price action for volatile assets is an exception, not the norm. And if the LP believes that it is true for their particular pair of assets, then why not trade these assets instead of LP’ing? (Buy low, sell high.) The LVR approach to predict expected profitability is useful in particular when the LP cannot predict the price evolution of the assets.

Furthermore, the performance of the LP positions does, obviously, depend on the price action in the analysis period. It is really surprising that WETH/WBTC pairs have been performing better than WETH/USDC? In my opinion, not at all, because the WETH/WBTC price dynamics since Uniswap v3 launch date have been nearly ideal for LPs: range-bound and with almost no expected IL. Meanwhile, ETH is down nearly 50% since June 2021, when significant amounts of liquidity was first added to the WETH/USDC pools.

Which pools are profitable (against HODL)?

The LVR idea describes how arbitragers extract value from LP. Arbitrage order flow is considered “toxic”, as it forces the LPs to buy or sell against the market. The other flow, presumably mostly retail driven, is considered “uniformed”. If the uniformed flow dominates over the toxic flow, the pool does not suffer from the LVR problem.

The mechanics of LVR require the presence of another, more liquid market. However, for many assets, this is not the case. Initial DEX Offerings (IDO), airdrops, the latest and hottest meme coins, and various other long-tail assets may be traded predominantly on DEXes, including Uniswap. If liquidity on Uniswap is deeper than the liquidity elsewhere, the DEX price determines the “market” price, not the other way around. As a result, the toxic flow is smaller than the uninformed flow.”

LP’ing these assets are certainly not risk-free, on the contrary. However, they are likely to avoid the problem of LVR, and thus beat the HODL strategy. The main issue for LP here is the need to guess whether HODLing these assets is likely to give a positive EV in dollar or ETH terms; but at least they are relatively safe from LVR.

Range orders as LP positions

In Uniswap v3, creating and closing LP positions can be a form of trading. Range orders can be used as either buy-limit or take-profit orders. The fact that the order earns fees is a nice side effect, but typically not critical for traders. From an analytical perspective, it’s important to note that traders are expected to close their LP position as soon as the order is filled. Furthermore, the LP position is expected to be highly concentrated, and during its lifetime, the price action is expected to be directional, not mean-reverting. Both of these features imply high IL.

Let’s show the math on an example. Let’s consider a range order that buys WETH. The order is 1% wide and set to begin exactly at the current price P. This implies that the average buy price of WETH is equal to sqrt(0.99 P * 1.0 P) = 0.994987 P. At the moment when the order is filled, the price of ETH is just 0.99 P, and the value of the position is just to be 99.4987% of the original position’s value. Hence, the IL of the position is going to be slightly above 0.5%. Unless this is the 1% fee-tier pool, the fees are likely to be lower than that; the position is in loss.

Mind that this is an idealized example; in the real life, due to Uniswap v3 tick math limitations, the loss is likely to be higher:

• its usually not possible to open an LP position perfectly close to the current price;
• 1% wide LP positions may not be possible in the higher fee tiers;
• opening and closing the position is not instantaneous.

All this means that range orders generate significant negative EV from the IL perspective even if they are successful.

Just-in-time (JIT) liquidity

JIT liquidity is a form of MEV, so the mechanics and intentions of its actors are typically not openly discussed. One can guess, though, that JIT LPs would use some form of hedging. Or maybe JIT is combined with arbitrage trading. For instance, consider a JIT LP that has ETH and USDC both in a wallet and in a CEX. When the LP observes a large ETH -> USDC swap in the mempool, they deploy a LP position on Uniswap, and at the same time submit an ETH buy order on the CEX. After both trades, the LP has the same total amount of ETH and USDC as before. Assuming that the following holds, the LP are also profitable:

fees_DEX + price_impact_DEX > fees_CEX + price_impact_CEX.

According to the analysis as much as 37% of all JIT LP positions were not profitable, and the average JIT LP loses to the basic HODL strategy. Many other JIT LP positions have profits that are very low, likely below the “risk free” DeFi rates of e.g. staking ETH or supplying stablecoins in Aave. However, I’d argue that these results are likely to underestimate the profitability of JIT LP, due to two methodological problems described further: the lack of transparency on other arms of the trade, and potential issues with computing the closing price.

Two potential problems are present when analyzing the Uniswap JIT data.

The first problem is that it’s impossible to say what exactly happens from the LP’s point of view and thus compute their PnL accurately: only one part of the trade is recorded in the blockchain as a Uniswap swap, there may be other part(s) on other CEX and DEX. The second problem is related to the closing price, used to compute the JIT LP’s divergence loss. If the price after the swap is used to compute the IL, then higher price impact implies higher expected divergence loss. But clearly, price impact is a good thing for the LP, so using this closing price to compute IL is totally wrong! The analysis partially avoids this problem by using the average price “in the N blocks after the swap”. However, it still might not be the right one to use: Uniswap pools are not arbitraged back instantly due to the fee payment causing a “liquidity gap” next to the pool’s price. A market price from multiple other exchanges, not directly affected by the swap, should be used instead.

On the other hand, perhaps there is a real problem with JIT LPs, because they may be more exposed to toxic order flow compared with other types of LPs. Research shows that most of swaps are actually profitable for LPs; only swaps above a certain size have negative EV.

This suggests a new kind of strategy: inverse-JIT. An LP could monitor the mempool, and remove their liquidity whenever a swap over the size threshold appears in the mempool. When the swap is executed, the LP adds their liquidity back, likely in a different range. In L2 chains, the gas fees could be low enough for this to work.

Hedging for sideways markets

“How to hedge an Uniswap position to make it delta-neutral” is something of a holy grail topic in the LP community. Many have attempted this, few if any of those I have talked to have succeeded.

The payoff function of a LP position is concave. LP’ing with borrowed assets does not change this. As this demo shows, hedging via borrowing still leaves the exposed to unlimited downside risk, while limiting the max payoff.

Since Uniswap LP payoff function is similar to the payoff function of option sellers, another natural way to hedge would be to buy options. Buying both put and call option for the same asset results in a long straddle position. It has a highly concave payoff function; when combined with a partially hedged LP position, as shown in the figure, it can result in a delta-neutral (flat) total payoff. To put it in another way, Uniswap LPs are selling volatility, straddles are buying volatility, added together they may cancel out.

In any case, Uniswap positions can effectively function as hedges, generating good returns in case the market goes sideways.

As a philosophical aside: providing liquidity on Uniswap has been compared with selling options³. While it’s a debatable topic, it’s interesting to consider that buying options can be used either as a hedge (insurance), or for leverage (gambling).

Buying insurance and buying lottery ticks are two very different actions, but they do have one thing in common — negative expected value⁴. With insurance, one sells large, unpredictable loss and buys a small, predictable loss. With lottery tickets, one incurs some small fixed costs, and buys a large, unpredictable win. Most people would agree that at least one of them (buying insurance) is a rational action. Hence, not all finance-related actions need have a positive EV in monetary terms in order to appear rational.

Often, reducing risks and having a peace of mind and are often more important in crypto than maximizing one’s perceived EV. The real discussion here should about exactly how expensive it is to hedge through Uniswap LP, and how to select the right parameters for one’s goals: pool’s fee tier, position’s range, duration etc.