Monday, April 08, 2024

Spurious High Frequency Autocorrelation

 A curious aspect of high-frequency data is that most come from centralized limit order books (CLOBs) where the bid-ask spread makes the data look negatively autocorrelated as trades are randomly made at the bid and the ask.

The returns driving this pattern are well below transaction costs, so they do not generate an arbitrage opportunity. However, one might be tempted to use the high-frequency data to estimate variance for pricing options or convexity costs (aka impermanent loss, loss versus rebalancing). This is a problem because the 1-minute Gemini returns generate a variance estimate 40% higher than one derived from daily data.

Variance grows linearly over time; volatility grows with the square root of time. Thus for a standard stochastic process, the variance should be the same when divided by the frequency. If the return horizon is measured in minutes, the variance of the 5-minute return should be half of the variance of the 10-minute return, etc. Variance(ret(M minutes))/M should be constant for all M. It’s helpful to divide the data by a standard, which in my case is the variance of the 1-day return over this sample (1440 minutes), so we can clearly identify those frequencies where variance is over and under-estimated. If the ratio is above 1.0, this implies mean-reversion at this frequency (negative autocorrelation), and if below 1.0, this implies momentum (positive autocorrelation).

Here is the ratio of the m-minute return variance divided by m for various minutes, normalized by dividing by the 1-day return. It asymptotes to 1.0 at 300 minutes.

I added the ETH-USDC 5 and 30 bp pools, and we can see the effect of stasis created by the fee and lower transaction volume. The one-minute return variance ratios for AMMs are well below 1.0, implying momentum—positive autocorrelation—at that frequency. Again, the effect driving this is well below 5 basis points, so it’s not a pattern one can make money off.

A common and easy way to calculate variance is to grab the latest trade and update an exponentially weighted moving average. This would generate a variance estimate at an even higher frequency than once per minute. The perils of this are clear when many important data are linear in variance, such as the expected convexity cost, and 40% is big.

As a refresher, I show how the common concept for impermanent loss equaling a function linear in variance because it’s not obvious. We start with the original IL definition

This is measured as

The following AMM formulas can substitute liquidity and prices into the above equation.

plugging these in, we can derive the following formula.

This ‘difference in the square roots squared’ function calculates a realized IL over a period (from 0 to 1). If one estimated this using daily data over several months, it would equal the expected IL via the law of large numbers.

One can estimate the expected IL using the expected variance and gamma of the position. The key nonlinear function on the AMM LP position is the square root of price, which has negative convexity. This implies a simple variance adjustment to the current square root to get the expected future square root.

Note that in the above, the variance is for the time period from 0 to t, so if it’s an annualized variance applied to daily data, you would divide it by 365. Returning to the above IL formulated as a function of square roots, we can substitute for E[sqrt(p1)] to see how this equals the ‘LVR’ equation with the variance.

Using (p1/p0 - 1)2 for the variance will generate the same formula as the difference in square roots squared if you use the same prices. We can’t measure expected returns, but average returns equal expected returns over time, and that’s all we have. One chooses some frequency to estimate the IL, regardless of the method. Just be careful not to use returns under a 300-minute horizon, as these will bias your estimate. 

Thursday, March 14, 2024

Moallemi's Auction-Managed AMM

A recent paper by Columbia professor Ciamac Moallemi and three Uniswap affiliates (Adams, Reynolds, and Robinson) presents a mechanism for recapturing the convexity costs. It builds upon Moallemi's previous work on automated market makers (AMMs) and arbitrage profit, published a year ago, which Moallemi presented at a16z crypto last summer. In that talk, he mentioned auctions as a way to reduce adverse selection costs for liquidity providers (LP).

AMMs' big problem is that LPs generally lose money in their popular capital-efficient (v3) pools. An LP's net profitability consists of revenue in the form of fee income and convexity.

LP profits = volume*fee – convexity costs

The LPs present traders with an option, and like all options, these are costly for the sellers. LP convexity costs have three mathematically equivalent formulations, and I posted three equivalent ways on Monday. Many only consider LP fee revenue as if convexity costs are just an academic hypothetical relative to an ideal. However, unlike the hedge fund sniping expense, option expenses are real, as reflected in standard option premiums above intrinsic value.

More relevant to this post, the convexity cost in an AMM is determined by two variables: the variance of the asset, which is exogenous to the AMM, and the liquidity in the AMM pool, which is endogenous. Both have a direct, linear impact on convexity costs.

AMM convexity costs = liquidity * sqrt(p0) * variance / 4

The expected value of this formula is unaffected by the granularity of the data used to generate the variance, highlighting that this cost has nothing to do with the frequency of blocks, trading, or hedging. The convexity cost is the inverse of the gross arbitrage profit (noted in my Monday post), which is the profit to the arbitrageur collective pre-fees

The largest AMM on blockchains is the ETH-USDC 5 basis point v3 pool, which processes hundreds of millions of USDC daily. This year, the 5 bp pool is actually making money because its liquidity is down about 60%, which has a first-order effect on convexity costs. For example, last year, their liquidity was usually around 30MM, which implies a $100k trade would generate slippage of only 0.01%; currently, liquidity is around 10MM, so the slippage would be 0.04%. I don't think traders mind, as this is still far better liquidity than anywhere else. This LP profitability could be temporary, but this pool has not had this low level of liquidity for three months since 2021, suggesting LPs have wizened up. With only $150MM in capital in the pool, this annualizes to an excellent 10% annualized return. 

Average Daily Stats for ETH-USDC 5bp Uniswap v3 Pool

Unfortunately, this year's ETH-USDC 5bp pool performance is not typical, and regardless, addressing LP convexity costs is important. The larger capital efficient restricted range (v3) pool LPs lose money. Since January 1, 2023, the ETH-USDC 5bp pool has lost an average of $6k a day net, including $142k in fee revenue and $148k in convexity costs. If one could reduce these costs by a mere 10%, the pool LPs would be profitable.

The auction mechanism is presented formally per Moallemi's previous work, but the gist is straightforward. Arbitrageurs bid for the right to be the 'pool manager' who gets all fee revenue and sets the pool fee in a future period (e.g., a day). As they get all the fees paid, the pool manager will effectively trade for free because the fees come back to him as revenue, while other arbitrageurs would pay the X bp fee. This gives the pool manager a competitive advantage in arbitraging the pool, as those paying the fee will find any mispricing less than the fee unprofitable, while the pool manager could capture it. If the pool manager pays the expected fees and just half of the arbitrage revenue, virtually all v3 pool LPs would be profitable. This is because gross arbitrage revenue—i.e., excluding fees—is the flip side of their convexity costs.

Chicago professor Eric Brudish has been promoting frequent batch auctions as an alternative to CLOBs for the past decade (see here and here), highlighting the tax generated by hedge fund sniping, where speedy high-frequency traders scoop up resting limit orders before the hapless market makers can remove them. Milionis, Moamelli, Roughgarden, and Zhang's Loss vs. Rebalancing paper alludes to hedge fund sniping in the context of AMMs, but that's far-fetched. AMMs do not have race conditions analogous to traders picking off stale limit orders. Even within the snail's pace of blockchains, you do not see LPs trying to do the inverse of just-in-time liquidity to get out of the way of trades (removing liquidity before a big trade and then adding it back in after the trade). The sniping form of adverse selection is sexy, focusing on unsympathetic high-frequency traders portrayed by best-selling author and SBF fanboy Michael Lewis. Still, it's, at best, a curiosity. A generous estimation of its effect is 0.4 basis points on average CLOB stock market spread of 3.5 basis points—a trivial effect—and irrelevant in AMMs. The bottom line is that arbitrage profits generate standard adverse selection costs for all market makers. With AMMs, this cost has an explicit formula.

Nonetheless, the sniping-motivated auction literature explains why Professor Moamelli had an auction solution ready. Academics like auctions because, in theory, they do not have any deadweight losses and are efficient given simplifying assumptions. In practice, however, auction complexity generates problems, which is why most asset markets use CLOBs, which have reduced stock trading costs by 90% from 1990 through 2010.

Uncertainties in the Bidder's Expected Arbitrage Profit

Expected arbitrage profits are a linear function of variance, the square of volatility. The average daily variance using minute-downsampled data is 15 basis points, but the distribution is flat with a long tail, so we should expect bidders to assume something well below the mean.

Expected arbitrage profit is also a linear function of liquidity, which is also variable, especially in the popular v3 pools [they acknowledge their mathematical model only applies to v2 pools, but clearly, the application would be to v3 pools]. An intriguing example occurred on March 6, where the ETH-USDC 30 basis point pool liquidity was a whopping 30 times higher than its standard value over a 5% price interval, from $3,612 to $3,812. This was due to a massive LP position above the current price, where the LP probably thought it would be an excellent place to sell their ETH without any hassle. Interestingly, in this case, the LP lost $2.6MM in convexity costs and captured $2.2MM in fees, for a net loss of about 40 basis points on the $100MM in ETH they effectively sold. This was not a good deal, but not horrible given the size. The bottom line is that the expected variance of arb profit in the face of these liquidity spikes increases the expected arb profit volatility even more, making it more difficult because the LPs can withdraw it at any time.

Huge LP Position Taken Out Last Week



New Gaming Strategies

There is also the issue of the unknown unknowns in this setup. For example, i
n March 2020, a bidder rigged an auction to buy $8.32 million in ETH for zero DAI.  While that attack has been exposed, who knows what strategies this approach generates? For example, one could handicap an arbitrageur by withdrawing all of your liquidity when an adversary wins the pool manager's right. This could be a part of a long game where the punisher is willing to incur a cost to inflict a cost on others, getting them to stop bidding. As withdrawals are standard in v3 pools, it is difficult to see how to lock in liquidity and avoid this.

While the pool manager would have an advantage in arbitrage trading, they would not prevent noise traders from inadvertently taking some of their profit, as people will randomly buy when the AMM price is low. To prevent that, a pool manager might target traders using high latency front ends like Uniswap, and front-run these trades. Conversely, traders could see the current mispricing and anticipate an arbitrage trade, sandwich attacking the arbitrageur. While chains without the slow ETH mainchain could avoid explicit MEV tactics, one can expect motivated traders to see these transactions and get in front of them regardless, as latency for a decentralized blockchain is orders of magnitude greater than anything on centralized exchanges (100s of milliseconds vs 100s of microseconds). Unlike co-location on centralized exchanges, there would be back-room deals for access to the sequencers or RPC nodes, creating a deadweight loss to users.

Even without front-running, there is the question of how much of the total arbitrage profit a zero-cost ab could capture. These uncertainties would incentivize the Bidder to price arbitrage profit well below its expected value.

The paper also acknowledges the potential for an increase in sandwich attacks because the zero fee would make these more attractive. This would be like the front-running attack on lucky noise traders, but instead, it would target any large trade. The authors mention that off-chain auctions or private relays could address this, but this complicates the mechanism even further (how are private relays incented?). A strategy space grows exponentially in the number of moves players can make. Simplicity is the key to security.

Lastly, there is the issue of to what extent arbitrageurs are hedging off the blockchain. Given the mispricing distribution, we can see that the AMM price is generally within the fee for the 5 bp pool. As centralized exchanges charge at least 2.5bps, and there is always slippage (i.e., price impact, spread), the total cost of a hedge is at least 5 bps at the size they would trade (chunks of $50k). The arbs cannot hedge each AMM arbitrage trade with a CEX trade and lock in a profit. They probably hedge every hour or day or when their on-chain position reaches some absolute level. 

This practical nature of hedging implies the arbs will have the opposite risk profile as the LPs, becoming short during price declines and long on the opposite. As this arb would have tens of millions at work, it creates an endogenous risk amplification mechanism because a price decline would make the arb profit and also increase his short position, so he would have the means and motive to push it down further to hit stop-loss orders on centralized exchanges.


If average convexity costs for a pool were 100, the uncertainty about future liquidity and variance and how much of the arb could be captured would lower the didder's price well below that, say, half. The arb would have to pay hedging costs and gas fees, reducing this further. Then there is the loss of the fees the current arbs pay. It's possible that LPs would make less money, just as ObamaCare, in theory, was supposed to reduce medical costs but, in practice, increased them. 

Nevertheless, this is a step in the right direction, addressing a major problem in the status quo. Given that the difference between revenues and costs on these pools is 3%, the bidder could pay a mere 10% of the expected total arbitrage profits and bring LPs back to profitability. The biggest downside is that there are many complexities generated by this approach, implying a lot of work for a temporary solution, as the best-case scenario would capture a small fraction of the arbitrage profits, leaving significant room for improvement. 


Monday, March 11, 2024

LvR, Impermanent Loss, Theta, and Arbitrage Profits

arbprofit

Columbia professor Ciamac Moallemi, Uniswap's Adams, Reynolds, and Dan Robinson released a paper last week on recapturing some of the LP convexity costs by having the arbitrageurs bid for the right to trade at a reduced rate. I wanted to dive into that, but I thought it would be helpful to see how arbitrage profits relate to LP convexity costs, as this is not obvious. Further, people in this space use different terms and formulas for what I am calling the LP convexity costs, and it would be useful to clarify how they differ.

Ultimately, loss vs. rebalancing (LvR), impermanent loss (IL), convexity costs, and theta, represent the same thing. The estimation method for these metrics will use daily data generating random errors that become irrelevant due to the Law of Large Numbers. That is, resetting the IL benchmark each day is like hedging once a day or using daily data to estimate variance.

Impermanent Loss

The OG term for the peculiar LP cost in constant product AMMs is 'impermanent loss.' It was called that because while an LP could have a loss relative to their initial LP position, they would not lose money if the price moved back, implying that loss was not permanent. One could say that about any mark-to-market loss, but whatever, it's a commonly used term, and generally, people know what it means.

The basic idea was to compare the LP's current pool position with the original LP token deposit, though one can compare it to the prior tokens at any time in the past. I will use ETH and USDC as the tokens in my example, but the analysis generalizes to any tokens A and B, making it easier to intuit.

An LP position is necessarily adversely selected, losing ETH when the price rises gaining ETH when its price falls. It is this change in token quantities that drives the IL.

IL=Value(initial LP position at current price)Value(LP at current price) IL= US D 0 ET H 0 p 1 US D 1 +ET H 1 p 1 IL= US D 1 US D 0 p 1 ET H 1 ET H 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGjb Gaamitaiabg2da9iaadAfacaWGHbGaamiBaiaadwhacaWGLbGaaiik aiaadMgacaWGUbGaamyAaiaadshacaWGPbGaamyyaiaadYgacaqGGa GaamitaiaadcfacaqGGaGaamiCaiaad+gacaWGZbGaamyAaiaadsha caWGPbGaam4Baiaad6gacaqGGaGaamyyaiaadshacaqGGaGaam4yai aadwhacaWGYbGaamOCaiaadwgacaWGUbGaamiDaiaabccacaWGWbGa amOCaiaadMgacaWGJbGaamyzaiaacMcacqGHsislcaWGwbGaamyyai aadYgacaWG1bGaamyzaiaacIcacaWGmbGaamiuaiaabccacaWGHbGa amiDaiaabccacaWGJbGaamyDaiaadkhacaWGYbGaamyzaiaad6gaca WG0bGaaeiiaiaadchacaWGYbGaamyAaiaadogacaWGLbGaaiykaaqa aiaadMeacaWGmbGaeyypa0ZaaiWaaeaacaWGvbGaam4uaiaadseada WgaaWcbaGaaGimaaqabaGccqGHsislcaWGfbGaamivaiaadIeadaWg aaWcbaGaaGimaaqabaGccqGHflY1caWGWbWaaSbaaSqaaiaaigdaae qaaaGccaGL7bGaayzFaaGaeyOeI0YaaiWaaeaacaWGvbGaam4uaiaa dseadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGfbGaamivaiaadI eadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWGWbWaaSbaaSqaaiaa igdaaeqaaaGccaGL7bGaayzFaaaabaGaamysaiaadYeacqGH9aqpda GadaqaaiaadwfacaWGtbGaamiramaaBaaaleaacaaIXaaabeaakiab gkHiTiaadwfacaWGtbGaamiramaaBaaaleaacaaIWaaabeaaaOGaay 5Eaiaaw2haaiabgkHiTiaadchadaWgaaWcbaGaaGymaaqabaGcdaGa daqaaiaadweacaWGubGaamisamaaBaaaleaacaaIXaaabeaakiabgk HiTiaadweacaWGubGaamisamaaBaaaleaacaaIWaaabeaaaOGaay5E aiaaw2haaaaaaa@AF65@

This simplifies to a net token change valued at the ending price

Equation (1) - net LP token change: IL=ΔUSD+ p 1 ΔETH MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaadY eacqGH9aqpcqqHuoarcaWGvbGaam4uaiaadseacqGHRaWkcaWGWbWa aSbaaSqaaiaaigdaaeqaaOGaeyyXICTaeuiLdqKaamyraiaadsfaca WGibaaaa@4564@

The AMM generates simple formulas for ΔETH and ΔUSD as a function of liquidity and price changes. Note these are the pool token changes, excluding fees.

ΔETH=liq 1 p 1 1 p 0 ΔUSD=liq p 1 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHuo arcaWGfbGaamivaiaadIeacqGH9aqpcaWGSbGaamyAaiaadghacqGH flY1daqadaqaamaaliaabaGaaGymaaqaamaakaaabaGaamiCamaaBa aaleaacaaIXaaabeaaaeqaaaaakiabgkHiTmaaliaabaGaaGymaaqa amaakaaabaGaamiCamaaBaaaleaacaaIWaaabeaaaeqaaaaaaOGaay jkaiaawMcaaaqaaiabfs5aejaadwfacaWGtbGaamiraiabg2da9iaa dYgacaWGPbGaamyCaiabgwSixpaabmaabaWaaOaaaeaacaWGWbWaaS baaSqaaiaaigdaaeqaaaqabaGccqGHsisldaGcaaqaaiaadchadaWg aaWcbaGaaGimaaqabaaabeaaaOGaayjkaiaawMcaaaaaaa@585A@

Plugging this in and doing algebra, we get a square-root function that, multiplied by liquidity, generates the second metric for the LP's convexity cost:

IL=liq p 1 p 0 + p 1 liq 1 p 1 1 p 0 IL=liq p 1 p 0 + p 1 p 1 p 0 IL=liq 2 p 1 p 0 p 1 p 0 IL=liq 2 p 0 p 1 p 0 p 1 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGjb Gaamitaiabg2da9iaadYgacaWGPbGaamyCaiabgwSixpaabmaabaWa aOaaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaqabaGccqGHsislda GcaaqaaiaadchadaWgaaWcbaGaaGimaaqabaaabeaaaOGaayjkaiaa wMcaaiabgUcaRiaadchadaWgaaWcbaGaaGymaaqabaGccqGHflY1ca WGSbGaamyAaiaadghacqGHflY1daqadaqaamaaliaabaGaaGymaaqa amaakaaabaGaamiCamaaBaaaleaacaaIXaaabeaaaeqaaaaakiabgk HiTmaaliaabaGaaGymaaqaamaakaaabaGaamiCamaaBaaaleaacaaI WaaabeaaaeqaaaaaaOGaayjkaiaawMcaaaqaaiaadMeacaWGmbGaey ypa0JaamiBaiaadMgacaWGXbGaeyyXIC9aaeWaaeaadaGcaaqaaiaa dchadaWgaaWcbaGaaGymaaqabaaabeaakiabgkHiTmaakaaabaGaam iCamaaBaaaleaacaaIWaaabeaaaeqaaOGaey4kaSYaaOaaaeaacaWG WbWaaSbaaSqaaiaaigdaaeqaaaqabaGccqGHsisldaWccaqaaiaadc hadaWgaaWcbaGaaGymaaqabaaakeaadaGcaaqaaiaadchadaWgaaWc baGaaGimaaqabaaabeaaaaaakiaawIcacaGLPaaaaeaacaWGjbGaam itaiabg2da9iaadYgacaWGPbGaamyCaiabgwSixpaabmaabaGaaGOm amaakaaabaGaamiCamaaBaaaleaacaaIXaaabeaaaeqaaOGaeyOeI0 YaaOaaaeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaaqabaGccqGHsisl daWccaqaaiaadchadaWgaaWcbaGaaGymaaqabaaakeaadaGcaaqaai aadchadaWgaaWcbaGaaGimaaqabaaabeaaaaaakiaawIcacaGLPaaa aeaacaWGjbGaamitaiabg2da9iaadYgacaWGPbGaamyCaiabgwSixp aalaaabaWaaeWaaeaacaaIYaWaaOaaaeaacaWGWbWaaSbaaSqaaiaa icdaaeqaaOGaamiCamaaBaaaleaacaaIXaaabeaaaeqaaOGaeyOeI0 IaamiCamaaBaaaleaacaaIWaaabeaakiabgkHiTiaadchadaWgaaWc baGaaGymaaqabaaakiaawIcacaGLPaaaaeaadaGcaaqaaiaadchada WgaaWcbaGaaGimaaqabaaabeaaaaaaaaa@9566@

equation (2) - square of square roots: IL=liq p 1 p 0 2 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaadY eacqGH9aqpcqGHsislcaWGSbGaamyAaiaadghacqGHflY1daWcaaqa amaabmaabaWaaOaaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaqaba GccqGHsisldaGcaaqaaiaadchadaWgaaWcbaGaaGimaaqabaaabeaa aOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaamaakaaaba GaamiCamaaBaaaleaacaaIWaaabeaaaeqaaaaaaaa@47F6@

Arbitrage Profit

As the LP's IL is based on net tokens coming in, someone took the opposite side; if the LPs net +1 ETH and 4000 USDC, traders net +1 ETH and 4000 USDC. As the LPs always lose money, the trader collective always makes money pre-fees.

The arbitrage profit is how much the arb is traded times the price changes.

ArbitragePnL=ΔETH p 1 fillPrice MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadk hacaWGIbGaamyAaiaadshacaWGYbGaamyyaiaadEgacaWGLbGaamiu aiaad6gacaWGmbGaeyypa0JaeuiLdqKaamyraiaadsfacaWGibGaey yXIC9aaeWaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Ia amOzaiaadMgacaWGSbGaamiBaiaadcfacaWGYbGaamyAaiaadogaca WGLbaacaGLOaGaayzkaaaaaa@5496@

We have the ΔETH function noted above, so we need the AMM fill price. This is the geometric mean of the starting and ending price.

fillPrice= ΔUSD ΔETH = liq p 1 p 0 liq 1 p 1 1 p 0 = p 1 p 0 p 1 p 0 p 1 p 0 1 p 1 1 p 0 fillPrice= p 1 + p 0 2 p 1 p 0 2 p 0 p 1 p 1 p 0 fillPrice= p 1 p 0 p 1 p 0 + p 0 p 1 2 2 p 0 p 1 p 1 p 0 fillPrice= p 1 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb GaamyAaiaadYgacaWGSbGaamiuaiaadkhacaWGPbGaam4yaiaadwga cqGH9aqpdaWcaaqaaiabfs5aejaadwfacaWGtbGaamiraaqaaiabfs 5aejaadweacaWGubGaamisaaaacqGH9aqpdaWcaaqaaiaadYgacaWG PbGaamyCaiabgwSixpaabmaabaWaaOaaaeaacaWGWbWaaSbaaSqaai aaigdaaeqaaaqabaGccqGHsisldaGcaaqaaiaadchadaWgaaWcbaGa aGimaaqabaaabeaaaOGaayjkaiaawMcaaaqaaiaadYgacaWGPbGaam yCaiabgwSixpaabmaabaWaaSGaaeaacaaIXaaabaWaaOaaaeaacaWG WbWaaSbaaSqaaiaaigdaaeqaaaqabaaaaOGaeyOeI0YaaSGaaeaaca aIXaaabaWaaOaaaeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaaqabaaa aaGccaGLOaGaayzkaaaaaiabg2da9maalaaabaWaaeWaaeaadaGcaa qaaiaadchadaWgaaWcbaGaaGymaaqabaaabeaakiabgkHiTmaakaaa baGaamiCamaaBaaaleaacaaIWaaabeaaaeqaaaGccaGLOaGaayzkaa aabaWaaeWaaeaadaGcaaqaaiaadchadaWgaaWcbaGaaGymaaqabaaa beaakiabgkHiTmaakaaabaGaamiCamaaBaaaleaacaaIWaaabeaaae qaaaGccaGLOaGaayzkaaaaaiabgwSixpaalaaabaWaaeWaaeaadaGc aaqaaiaadchadaWgaaWcbaGaaGymaaqabaaabeaakiabgkHiTmaaka aabaGaamiCamaaBaaaleaacaaIWaaabeaaaeqaaaGccaGLOaGaayzk aaaabaWaaeWaaeaadaWccaqaaiaaigdaaeaadaGcaaqaaiaadchada WgaaWcbaGaaGymaaqabaaabeaaaaGccqGHsisldaWccaqaaiaaigda aeaadaGcaaqaaiaadchadaWgaaWcbaGaaGimaaqabaaabeaaaaaaki aawIcacaGLPaaaaaaabaGaamOzaiaadMgacaWGSbGaamiBaiaadcfa caWGYbGaamyAaiaadogacaWGLbGaeyypa0ZaaSaaaeaacaWGWbWaaS baaSqaaiaaigdaaeqaaOGaey4kaSIaamiCamaaBaaaleaacaaIWaaa beaakiabgkHiTiaaikdadaGcaaqaaiaadchadaWgaaWcbaGaaGymaa qabaGccqGHflY1caWGWbWaaSbaaSqaaiaaicdaaeqaaaqabaaakeaa caaIYaGaeyOeI0YaaSGaaeaadaGcaaqaaiaadchadaWgaaWcbaGaaG imaaqabaaabeaaaOqaamaakaaabaGaamiCamaaBaaaleaacaaIXaaa beaaaeqaaaaakiabgkHiTmaaliaabaWaaOaaaeaacaWGWbWaaSbaaS qaaiaaigdaaeqaaaqabaaakeaadaGcaaqaaiaadchadaWgaaWcbaGa aGimaaqabaaabeaaaaaaaaGcbaGaamOzaiaadMgacaWGSbGaamiBai aadcfacaWGYbGaamyAaiaadogacaWGLbGaeyypa0ZaaSaaaeaadaGc aaqaaiaadchadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWGWbWaaS baaSqaaiaaicdaaeqaaaqabaGccqGHflY1daqadaqaamaaliaabaWa aOaaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaqabaaakeaadaGcaa qaaiaadchadaWgaaWcbaGaaGimaaqabaaabeaaaaGccqGHRaWkdaWc caqaamaakaaabaGaamiCamaaBaaaleaacaaIWaaabeaaaeqaaaGcba WaaOaaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaqabaaaaOGaeyOe I0IaaGOmaaGaayjkaiaawMcaaaqaamaabmaabaGaaGOmaiabgkHiTm aaliaabaWaaOaaaeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaaqabaaa keaadaGcaaqaaiaadchadaWgaaWcbaGaaGymaaqabaaabeaaaaGccq GHsisldaWccaqaamaakaaabaGaamiCamaaBaaaleaacaaIXaaabeaa aeqaaaGcbaWaaOaaaeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaaqaba aaaaGccaGLOaGaayzkaaaaaaqaaiaadAgacaWGPbGaamiBaiaadYga caWGqbGaamOCaiaadMgacaWGJbGaamyzaiabg2da9maakaaabaGaam iCamaaBaaaleaacaaIXaaabeaakiabgwSixlaadchadaWgaaWcbaGa aGimaaqabaaabeaaaaaa@D914@

As the fill price is positive, I ignored the sign in the above algebra. Substituting the fill price and ETH delta formula into the original equation generates the negative of equation(2), the square root function above.

ArbitragePnL=liq 1 p 1 1 p 0 p 1 p 1 p 0 ArbitragePnL=liq p 1 p 1 p 0 p 0 + p 1 ArbitragePnL=liq 2 p 1 p 1 p 0 p 0 ArbitragePnL=liq 2 p 1 p 0 p 1 p 0 p 0 ArbitragePnL=liq p 1 p 0 2 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGbb GaamOCaiaadkgacaWGPbGaamiDaiaadkhacaWGHbGaam4zaiaadwga caWGqbGaamOBaiaadYeacqGH9aqpcaWGSbGaamyAaiaadghacqGHfl Y1daqadaqaamaaliaabaGaaGymaaqaamaakaaabaGaamiCamaaBaaa leaacaaIXaaabeaaaeqaaaaakiabgkHiTmaaliaabaGaaGymaaqaam aakaaabaGaamiCamaaBaaaleaacaaIWaaabeaaaeqaaaaaaOGaayjk aiaawMcaaiabgwSixpaabmaabaGaamiCamaaBaaaleaacaaIXaaabe aakiabgkHiTmaakaaabaGaamiCamaaBaaaleaacaaIXaaabeaakiaa dchadaWgaaWcbaGaaGimaaqabaaabeaaaOGaayjkaiaawMcaaaqaai aadgeacaWGYbGaamOyaiaadMgacaWG0bGaamOCaiaadggacaWGNbGa amyzaiaadcfacaWGUbGaamitaiabg2da9iaadYgacaWGPbGaamyCai abgwSixpaabmaabaWaaOaaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqa aaqabaGccqGHsisldaWccaqaaiaadchadaWgaaWcbaGaaGymaaqaba aakeaadaGcaaqaaiaadchadaWgaaWcbaGaaGimaaqabaaabeaaaaGc cqGHsisldaGcaaqaaiaadchadaWgaaWcbaGaaGimaaqabaaabeaaki abgUcaRmaakaaabaGaamiCamaaBaaaleaacaaIXaaabeaaaeqaaaGc caGLOaGaayzkaaaabaGaamyqaiaadkhacaWGIbGaamyAaiaadshaca WGYbGaamyyaiaadEgacaWGLbGaamiuaiaad6gacaWGmbGaeyypa0Ja amiBaiaadMgacaWGXbGaeyyXIC9aaeWaaeaacaaIYaWaaOaaaeaaca WGWbWaaSbaaSqaaiaaigdaaeqaaaqabaGccqGHsisldaWccaqaaiaa dchadaWgaaWcbaGaaGymaaqabaaakeaadaGcaaqaaiaadchadaWgaa WcbaGaaGimaaqabaaabeaaaaGccqGHsisldaGcaaqaaiaadchadaWg aaWcbaGaaGimaaqabaaabeaaaOGaayjkaiaawMcaaaqaaiaadgeaca WGYbGaamOyaiaadMgacaWG0bGaamOCaiaadggacaWGNbGaamyzaiaa dcfacaWGUbGaamitaiabg2da9iaadYgacaWGPbGaamyCaiabgwSixp aalaaabaWaaeWaaeaacaaIYaWaaOaaaeaacaWGWbWaaSbaaSqaaiaa igdaaeqaaOGaeyyXICTaamiCamaaBaaaleaacaaIWaaabeaaaeqaaO GaeyOeI0IaamiCamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadcha daWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaaaeaadaGcaaqaai aadchadaWgaaWcbaGaaGimaaqabaaabeaaaaaakeaacaWGbbGaamOC aiaadkgacaWGPbGaamiDaiaadkhacaWGHbGaam4zaiaadwgacaWGqb GaamOBaiaadYeacqGH9aqpcaWGSbGaamyAaiaadghacqGHflY1daWc aaqaamaabmaabaWaaOaaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaa qabaGccqGHsisldaGcaaqaaiaadchadaWgaaWcbaGaaGimaaqabaaa beaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaamaaka aabaGaamiCamaaBaaaleaacaaIWaaabeaaaeqaaaaaaaaa@D0D5@

 

Remember this arbitrage pnl number excludes fees, which are an expense to the arbitrageur. Looking at a net price movement over an interval will also include incidentally lucky noise traders instead of true arbitrageurs. However, the critical point is that we can see the fundamental symmetry of arbitrage revenue and an LP's convexity expense.

Delta Hedged LP PnL

The LP's position has a delta, or price risk, that can be hedged. We can apply this hedge to any frequency we want, where the complexity of handling more data offsets the benefit of more frequent hedging. Removing the delta generates a much more efficient estimate of an LP's pnl.

The initial LP position is just {ETH, USDC}, so it is long whatever amount of ETH they deposited. The LP can just eyeball his ETH position to determine his hedge amount.

Position = USD0 + ETH0*p0

The delta for an LP position can be confirmed by doing the math:

dValu e LP p =delta=Δ= liq p =ET H LP MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaamOvaiaadggacaWGSbGaamyDaiaadwgadaWgaaWcbaGaamit aiaadcfaaeqaaaGcbaGaeyOaIyRaamiCaaaacqGH9aqpcaWGKbGaam yzaiaadYgacaWG0bGaamyyaiabg2da9iabfs5aejabg2da9maalaaa baGaamiBaiaadMgacaWGXbaabaWaaOaaaeaacaWGWbaaleqaaaaaki abg2da9iaadweacaWGubGaamisamaaBaaaleaacaWGmbGaamiuaaqa baaaaa@521E@

If we look at the change in the hedged LP position over time and assume we are looking at the change in ETH and USD without fees, we see it goes back to eq(1), the value of the net token change:

netLP=LPposition+hedge netL P 0 = US D 0 +ET H 0 p 0 ET H 0 p 0 =US D 0 netL P 1 = US D 1 +ET H 1 p 1 ET H 0 p 1 pnl=L P 1 L P 0 = US D 1 +ET H 1 p 1 ET H 0 p 1 US D 0 pnl=ΔUSD+ p 1 ΔETH MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGUb GaamyzaiaadshacaWGmbGaamiuaiabg2da9iaadYeacaWGqbGaamiC aiaad+gacaWGZbGaamyAaiaadshacaWGPbGaam4Baiaad6gacqGHRa WkcaWGObGaamyzaiaadsgacaWGNbGaamyzaaqaaiaad6gacaWGLbGa amiDaiaadYeacaWGqbWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0Zaai WaaeaacaWGvbGaam4uaiaadseadaWgaaWcbaGaaGimaaqabaGccqGH RaWkcaWGfbGaamivaiaadIeadaWgaaWcbaGaaGimaaqabaGccqGHfl Y1caWGWbWaaSbaaSqaaiaaicdaaeqaaaGccaGL7bGaayzFaaGaeyOe I0IaamyraiaadsfacaWGibWaaSbaaSqaaiaaicdaaeqaaOGaeyyXIC TaamiCamaaBaaaleaacaaIWaaabeaakiabg2da9iaadwfacaWGtbGa amiramaaBaaaleaacaaIWaaabeaaaOqaaiaad6gacaWGLbGaamiDai aadYeacaWGqbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaiWaaeaa caWGvbGaam4uaiaadseadaWgaaWcbaGaaGymaaqabaGccqGHRaWkca WGfbGaamivaiaadIeadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWG WbWaaSbaaSqaaiaaigdaaeqaaaGccaGL7bGaayzFaaGaeyOeI0Iaam yraiaadsfacaWGibWaaSbaaSqaaiaaicdaaeqaaOGaeyyXICTaamiC amaaBaaaleaacaaIXaaabeaaaOqaaiaadchacaWGUbGaamiBaiabg2 da9iaadYeacaWGqbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaamit aiaadcfadaWgaaWcbaGaaGimaaqabaGccqGH9aqpdaGadaqaaiaadw facaWGtbGaamiramaaBaaaleaacaaIXaaabeaakiabgUcaRiaadwea caWGubGaamisamaaBaaaleaacaaIXaaabeaakiabgwSixlaadchada WgaaWcbaGaaGymaaqabaaakiaawUhacaGL9baacqGHsislcaWGfbGa amivaiaadIeadaWgaaWcbaGaaGimaaqabaGccqGHflY1caWGWbWaaS baaSqaaiaaigdaaeqaaOGaeyOeI0IaamyvaiaadofacaWGebWaaSba aSqaaiaaicdaaeqaaaGcbaGaamiCaiaad6gacaWGSbGaeyypa0Jaeu iLdqKaamyvaiaadofacaWGebGaey4kaSIaamiCamaaBaaaleaacaaI XaaabeaakiabgwSixlabfs5aejaadweacaWGubGaamisaaaaaa@BF35@

Given this result, we can use the logic above to show that this is equivalent to the square of square roots formula. Thus, comparing an initial token position to an initial token position is the same as the net pnl for an LP position hedged at some earlier token position. It is a different way to think about impermanent loss.

Theta

Finally, the Black-Scholes approach uses theta, which equals the expected loss from convexity in equilibrium (profits = 0). In their proof, they create a riskless portfolio of an option and a synthetically replicated inverse of that option via a dynamic hedging strategy that targets the delta. Negative convexity generates an adverse delta change in the option, whether the price goes up or down, which generates losses for the hedged seller. If one rehedges frequently, the losses will be smaller, but there will be more of them, and on average, they generate the same expected return whether one rehedges daily or every second. A convex position's theta equals the position's gamma, or second derivative, times the underlying asset's variance, divided by 2.

theta= 1 2 gammavariance MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadI gacaWGLbGaamiDaiaadggacqGH9aqpcqGHsisldaWcaaqaaiaaigda aeaacaaIYaaaaiaadEgacaWGHbGaamyBaiaad2gacaWGHbGaeyyXIC TaamODaiaadggacaWGYbGaamyAaiaadggacaWGUbGaam4yaiaadwga aaa@4C76@

Here, the variance is scaled to whatever one's frequency is. For example, with daily data, one would take the annual variance divided by 365. Variance is linear in time, which is why people trade variance swaps and not volatility swaps. Fortunately, calculating gamma for AMMs from its primitive is simple, so it leads to the following:

LPvalu e LP =2liquidity p dLPvalu e LP p =delta=Δ= liq p =ET H LP d 2 LPvalu e LP p 2 =gamma=Γ= liq 2 p 1.5 theta= 1 2 gammavariance= 1 2 liq 2 p 1.5 p 2 σ 2 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGmb GaamiuaiaadAhacaWGHbGaamiBaiaadwhacaWGLbWaaSbaaSqaaiaa dYeacaWGqbaabeaakiabg2da9iaaikdacqGHflY1caWGSbGaamyAai aadghacaWG1bGaamyAaiaadsgacaWGPbGaamiDaiaadMhacqGHflY1 daGcaaqaaiaadchaaSqabaaakeaadaWcaaqaaiaadsgacaWGmbGaam iuaiaadAhacaWGHbGaamiBaiaadwhacaWGLbWaaSbaaSqaaiaadYea caWGqbaabeaaaOqaaiabgkGi2kaadchaaaGaeyypa0Jaamizaiaadw gacaWGSbGaamiDaiaadggacqGH9aqpcqqHuoarcqGH9aqpdaWcaaqa aiaadYgacaWGPbGaamyCaaqaamaakaaabaGaamiCaaWcbeaaaaGccq GH9aqpcaWGfbGaamivaiaadIeadaWgaaWcbaGaamitaiaadcfaaeqa aaGcbaWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaamitai aadcfacaWG2bGaamyyaiaadYgacaWG1bGaamyzamaaBaaaleaacaWG mbGaamiuaaqabaaakeaacqGHciITcaWGWbWaaWbaaSqabeaacaaIYa aaaaaakiabg2da9iaadEgacaWGHbGaamyBaiaad2gacaWGHbGaeyyp a0Jaeu4KdCKaeyypa0JaeyOeI0YaaSaaaeaacaWGSbGaamyAaiaadg haaeaacaaIYaGaeyyXICTaamiCamaaCaaaleqabaGaaGymaiaac6ca caaI1aaaaaaaaOqaaiaadshacaWGObGaamyzaiaadshacaWGHbGaey ypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGNbGaamyy aiaad2gacaWGTbGaamyyaiabgwSixlaadAhacaWGHbGaamOCaiaadM gacaWGHbGaamOBaiaadogacaWGLbGaeyypa0ZaaSaaaeaacaaIXaaa baGaaGOmaaaacqGHflY1daqadaqaamaalaaabaGaamiBaiaadMgaca WGXbaabaGaaGOmaiabgwSixlaadchadaahaaWcbeqaaiaaigdacaGG UaGaaGynaaaaaaaakiaawIcacaGLPaaacqGHflY1daqadaqaaiaadc hadaahaaWcbeqaaiaaikdaaaGccqGHflY1cqaHdpWCdaahaaWcbeqa aiaaikdaaaaakiaawIcacaGLPaaaaaaa@BDC0@

Simplifying, we get a function that uses a variance term:

Equation (3) -variance over 4: theta=liq p σ 2 4 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiaadI gacaWGLbGaamiDaiaadggacqGH9aqpcaWGSbGaamyAaiaadghacqGH flY1daWcaaqaamaakaaabaGaamiCaaWcbeaakiabgwSixlabeo8aZn aaCaaaleqabaGaaGOmaaaaaOqaaiaaisdaaaaaaa@47B2@

While AMMs, unlike option markets, do not have theta per se as there is no option premium to decay, it still captures the option cost for any short gamma position. This cost is a function of convexity, not the delta, so it presumes the position is delta-hedged.

The LvR function is identical to equation 3, though it looks different because the pool's market value is used to normalize it:

theta LPvalue = σ 2 8 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WG0bGaamiAaiaadwgacaWG0bGaamyyaaqaaiaadYeacaWGqbGaamOD aiaadggacaWGSbGaamyDaiaadwgaaaGaeyypa0JaeyOeI0YaaSaaae aacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakeaacaaI4aaaaaaa@468C@

This is a handy formula, as volatility per day for many coins is 5%; it generates a cost 3.125 basis points per day, a fun fact. In practice, however, we need to get the liquidity and price data anyway to get the market value we would apply this to anyway. Given the LP market value function

LPvalue=2liq p MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaadc facaWG2bGaamyyaiaadYgacaWG1bGaamyzaiabg2da9iaaikdacqGH flY1caWGSbGaamyAaiaadghacqGHflY1daGcaaqaaiaadchaaSqaba aaaa@468D@

If we multiply variance/8 by the LP value, we get equation (3), the variance function for the LP's convexity cost.

Unfortunately, my math/stats skills are insufficient to prove equations 2 and 3 are the same, but they are (or at least 99% approximations of each other).

liq p 1 p 0 2 p 0 liq p 0 σ 2 4 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaadM gacaWGXbGaeyyXIC9aaSaaaeaadaqadaqaamaakaaabaGaamiCamaa BaaaleaacaaIXaaabeaaaeqaaOGaeyOeI0YaaOaaaeaacaWGWbWaaS baaSqaaiaaicdaaeqaaaqabaaakiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaaakeaadaGcaaqaaiaadchadaWgaaWcbaGaaGimaaqaba aabeaaaaGccqGHijYUcaWGSbGaamyAaiaadghacqGHflY1daWcaaqa amaakaaabaGaamiCamaaBaaaleaacaaIWaaabeaaaeqaaOGaeyyXIC Taeq4Wdm3aaWbaaSqabeaacaaIYaaaaaGcbaGaaGinaaaaaaa@5301@

This gives us three ways to calculate IL. If we use the same starting and ending times, they should be the same. That is, the variance term should use the variance in the [0,1] interval, and the ΔETH and ΔUSDC should also use the [0, 1] interval.

net token change:ΔUS D 0,1 + p 1 ΔET H 0,1 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaadw gacaWG0bGaaeiiaiaadshacaWGVbGaam4AaiaadwgacaWGUbGaaeii aiaadogacaWGObGaamyyaiaad6gacaWGNbGaamyzaiaacQdacqqHuo arcaWGvbGaam4uaiaadseadaWgaaWcbaGaaGimaiaacYcacaaIXaaa beaakiabgUcaRiaadchadaWgaaWcbaGaaGymaaqabaGccqGHflY1cq qHuoarcaWGfbGaamivaiaadIeadaWgaaWcbaGaaGimaiaacYcacaaI Xaaabeaaaaa@5683@

square of square root:liq p 1 p 0 2 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaadg hacaWG1bGaamyyaiaadkhacaWGLbGaaeiiaiaad+gacaWGMbGaaeii aiaadohacaWGXbGaamyDaiaadggacaWGYbGaamyzaiaabccacaWGYb Gaam4Baiaad+gacaWG0bGaaiOoaiaadYgacaWGPbGaamyCaiabgwSi xpaalaaabaWaaeWaaeaadaGcaaqaaiaadchadaWgaaWcbaGaaGymaa qabaaabeaakiabgkHiTmaakaaabaGaamiCamaaBaaaleaacaaIWaaa beaaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGcba WaaOaaaeaacaWGWbWaaSbaaSqaaiaaicdaaeqaaaqabaaaaaaa@5820@

variance over 4:liq p 0 σ 0,1 2 4 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaadg gacaWGYbGaamyAaiaadggacaWGUbGaam4yaiaadwgacaqGGaGaam4B aiaadAhacaWGLbGaamOCaiaabccacaaI0aGaaiOoaiaadYgacaWGPb GaamyCaiabgwSixpaalaaabaWaaOaaaeaacaWGWbWaaSbaaSqaaiaa icdaaeqaaaqabaGccqGHflY1cqaHdpWCdaqhaaWcbaGaaGimaiaacY cacaaIXaaabaGaaGOmaaaaaOqaaiaaisdaaaaaaa@5300@

These are all costs to the LP and need to be subtracted from the LP fees to get the LP's total profitability. However, in practice, the net token change data from Uniswap's 'swap' event logs includes fees, so one does not need the daily volume to generate the net LP pnl; removing the fees would take more work. Thus, we do not need to add fee revenue if we use the swap data to get the LP's net token changes.

net token change PnL:ΔUS D 0,1 + p 1 ΔET H 0,1 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaadw gacaWG0bGaaeiiaiaadshacaWGVbGaam4AaiaadwgacaWGUbGaaeii aiaadogacaWGObGaamyyaiaad6gacaWGNbGaamyzaiaabccacaWGqb GaamOBaiaadYeacaGG6aGaeuiLdqKaamyvaiaadofacaWGebWaaSba aSqaaiaaicdacaGGSaGaaGymaaqabaGccqGHRaWkcaWGWbWaaSbaaS qaaiaaigdaaeqaaOGaeyyXICTaeuiLdqKaamyraiaadsfacaWGibWa aSbaaSqaaiaaicdacaGGSaGaaGymaaqabaaaaa@59BF@

For the other approaches, we can multiply the absolute value of the USDC traded by the fees to get LP revenue. While the fees are paid in whatever token is sold to the pool, using the numeraire is an efficient approximation.

square of square root PnL:USDCvolumetradedfeeliq p 1 p 0 2 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaadg hacaWG1bGaamyyaiaadkhacaWGLbGaaeiiaiaad+gacaWGMbGaaeii aiaadohacaWGXbGaamyDaiaadggacaWGYbGaamyzaiaabccacaWGYb Gaam4Baiaad+gacaWG0bGaaeiiaiaadcfacaWGUbGaamitaiaacQda caWGvbGaam4uaiaadseacaWGdbGaamODaiaad+gacaWGSbGaamyDai aad2gacaWGLbGaamiDaiaadkhacaWGHbGaamizaiaadwgacaWGKbGa eyyXICTaamOzaiaadwgacaWGLbGaeyOeI0IaamiBaiaadMgacaWGXb GaeyyXIC9aaSaaaeaadaqadaqaamaakaaabaGaamiCamaaBaaaleaa caaIXaaabeaaaeqaaOGaeyOeI0YaaOaaaeaacaWGWbWaaSbaaSqaai aaicdaaeqaaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikda aaaakeaadaGcaaqaaiaadchadaWgaaWcbaGaaGimaaqabaaabeaaaa aaaa@6FDD@

variance over 4 PnL:USDCvolumetradedfeeliq p 0 σ 0,1 2 4 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaadg gacaWGYbGaamyAaiaadggacaWGUbGaam4yaiaadwgacaqGGaGaam4B aiaadAhacaWGLbGaamOCaiaabccacaaI0aGaaeiiaiaadcfacaWGUb GaamitaiaacQdacaWGvbGaam4uaiaadseacaWGdbGaamODaiaad+ga caWGSbGaamyDaiaad2gacaWGLbGaamiDaiaadkhacaWGHbGaamizai aadwgacaWGKbGaeyyXICTaamOzaiaadwgacaWGLbGaeyOeI0IaamiB aiaadMgacaWGXbGaeyyXIC9aaSaaaeaadaGcaaqaaiaadchadaWgaa WcbaGaaGimaaqabaaabeaakiabgwSixlabeo8aZnaaDaaaleaacaaI WaGaaiilaiaaigdaaeaacaaIYaaaaaGcbaGaaGinaaaaaaa@6ABD@

The nice thing about the net token change approach is that you can avoid the noise generated by the fact that liquidity is not constant, especially in restricted-range pools, where it can vary quite a bit over the day. Unless you calculate these data using each trade, any liquidity number you use will be approximate and generate noise in one's estimate. Thus, I prefer the net token change approach.

I estimated the daily pnl for a pool and used the day's price return squared for the variance number.

Uniswap ETH-USDC v2 pool on Ethereum Mainnet

1/1/23-3/5/24

Net token change pnl:             $7,231

Square root pnl:                       $7,152

Variance pnl:                           $7,097

As the standard deviation in this period was $12k, these estimates are in complete agreement. The correlation between the square root and variance metrics is 99.9%, while the token change pnl correlation with those two is around 99.6%

Impermanent Loss != LvR ?

For reasons I find irrelevant, Milionis, Moallemi, Roughgarden, and Zhang emphasize their LvR approach (the variance function) as superior to the square root function. First, they note that IL depends on the initial position, as it uses a squared term, so the magnitude is different for given starting prices.

Impermanent Loss=liq p 1 p 0 2 p 0 MathType@MTEF@5@5@+= feaahCart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 gacaWGWbGaamyzaiaadkhacaWGTbGaamyyaiaad6gacaWGLbGaamOB aiaadshacaqGGaGaamitaiaad+gacaWGZbGaam4Caiabg2da9iaadY gacaWGPbGaamyCaiabgwSixpaalaaabaWaaeWaaeaadaGcaaqaaiaa dchadaWgaaWcbaGaaGymaaqabaaabeaakiabgkHiTmaakaaabaGaam iCamaaBaaaleaacaaIWaaabeaaaeqaaaGccaGLOaGaayzkaaWaaWba aSqabeaacaaIYaaaaaGcbaWaaOaaaeaacaWGWbWaaSbaaSqaaiaaic daaeqaaaqabaaaaaaa@53F9@

This is like saying the return on a stock that moved from 100 to 150 is different for someone who bought it at 50 and someone who bought it at 100: one made 100%, the other only 50% over that price increase. In practice, we apply consistent starting prices for the collective of those long and short, or all the LPs, which is intuitive and straightforward. It is like assuming they delta hedge once each period.

Another concern is that the square root function is noisier. Yet, one can apply any of these formulas at whatever frequency you want: by minute, hour, day. Using hourly data is a lot more work with negligible benefit. The errors generated will be normally distributed and diversify away in the same way regardless of one's approach. If you use daily data to generate variance or generate your square of square roots, you lose the information from the intra-period price spike, but when you have hundreds of daily observations, the benefits of greater granularity are trivial.

What you call the LP's convexity cost does not matter. What matters is you use something close to daily data intervals and eliminate the LP's delta risk, as all the above metrics do.