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 marktomarket 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.
$\begin{array}{l}IL=Value(initial\text{}LP\text{}position\text{}at\text{}current\text{}price)Value(LP\text{}at\text{}current\text{}price)\\ IL=\left\{US{D}_{0}ET{H}_{0}\cdot {p}_{1}\right\}\left\{US{D}_{1}+ET{H}_{1}\cdot {p}_{1}\right\}\\ IL=\left\{US{D}_{1}US{D}_{0}\right\}{p}_{1}\left\{ET{H}_{1}ET{H}_{0}\right\}\end{array}$
This simplifies to a net token change valued at the ending price
Equation (1)  net LP token change: $IL=\Delta USD+{p}_{1}\cdot \Delta ETH$
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.
$\begin{array}{l}\Delta ETH=liq\cdot \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)\\ \Delta USD=liq\cdot \left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)\end{array}$
Plugging this in and doing algebra, we get a squareroot function that, multiplied by liquidity, generates the second metric for the LP's convexity cost:
$\begin{array}{l}IL=liq\cdot \left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)+{p}_{1}\cdot liq\cdot \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)\\ IL=liq\cdot \left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}+\sqrt{{p}_{1}}\raisebox{1ex}{${p}_{1}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)\\ IL=liq\cdot \left(2\sqrt{{p}_{1}}\sqrt{{p}_{0}}\raisebox{1ex}{${p}_{1}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)\\ IL=liq\cdot \frac{\left(2\sqrt{{p}_{0}{p}_{1}}{p}_{0}{p}_{1}\right)}{\sqrt{{p}_{0}}}\end{array}$
equation (2)  square of square roots: $IL=liq\cdot \frac{{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}^{2}}{\sqrt{{p}_{0}}}$
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 prefees.
The arbitrage profit is how much the arb is traded times the price changes.
$ArbitragePnL=\Delta ETH\cdot \left({p}_{1}fillPrice\right)$
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.
$\begin{array}{l}fillPrice=\frac{\Delta USD}{\Delta ETH}=\frac{liq\cdot \left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}{liq\cdot \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)}=\frac{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}\cdot \frac{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)}\\ fillPrice=\frac{{p}_{1}+{p}_{0}2\sqrt{{p}_{1}\cdot {p}_{0}}}{2\raisebox{1ex}{$\sqrt{{p}_{0}}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.\raisebox{1ex}{$\sqrt{{p}_{1}}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.}\\ fillPrice=\frac{\sqrt{{p}_{1}\cdot {p}_{0}}\cdot \left(\raisebox{1ex}{$\sqrt{{p}_{1}}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.+\raisebox{1ex}{$\sqrt{{p}_{0}}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.2\right)}{\left(2\raisebox{1ex}{$\sqrt{{p}_{0}}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.\raisebox{1ex}{$\sqrt{{p}_{1}}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)}\\ fillPrice=\sqrt{{p}_{1}\cdot {p}_{0}}\end{array}$
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.
$$\begin{array}{l}ArbitragePnL=liq\cdot \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{1}}$}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\right)\cdot \left({p}_{1}\sqrt{{p}_{1}{p}_{0}}\right)\\ ArbitragePnL=liq\cdot \left(\sqrt{{p}_{1}}\raisebox{1ex}{${p}_{1}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\sqrt{{p}_{0}}+\sqrt{{p}_{1}}\right)\\ ArbitragePnL=liq\cdot \left(2\sqrt{{p}_{1}}\raisebox{1ex}{${p}_{1}$}\!\left/ \!\raisebox{1ex}{$\sqrt{{p}_{0}}$}\right.\sqrt{{p}_{0}}\right)\\ ArbitragePnL=liq\cdot \frac{\left(2\sqrt{{p}_{1}\cdot {p}_{0}}{p}_{1}{p}_{0}\right)}{\sqrt{{p}_{0}}}\\ ArbitragePnL=liq\cdot \frac{{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}^{2}}{\sqrt{{p}_{0}}}\end{array}$$ 

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 = USD_{0} + ETH_{0}*p_{0}
The delta for an LP position can be confirmed by doing the math:
$\frac{dValu{e}_{LP}}{\partial p}=delta=\Delta =\frac{liq}{\sqrt{p}}=ET{H}_{LP}$
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:
$\begin{array}{l}netLP=LPposition+hedge\\ netL{P}_{0}=\left\{US{D}_{0}+ET{H}_{0}\cdot {p}_{0}\right\}ET{H}_{0}\cdot {p}_{0}=US{D}_{0}\\ netL{P}_{1}=\left\{US{D}_{1}+ET{H}_{1}\cdot {p}_{1}\right\}ET{H}_{0}\cdot {p}_{1}\\ pnl=L{P}_{1}L{P}_{0}=\left\{US{D}_{1}+ET{H}_{1}\cdot {p}_{1}\right\}ET{H}_{0}\cdot {p}_{1}US{D}_{0}\\ pnl=\Delta USD+{p}_{1}\cdot \Delta ETH\end{array}$
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 BlackScholes 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.
$theta=\frac{1}{2}gamma\cdot variance$
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:
$\begin{array}{l}LPvalu{e}_{LP}=2\cdot liquidity\cdot \sqrt{p}\\ \frac{dLPvalu{e}_{LP}}{\partial p}=delta=\Delta =\frac{liq}{\sqrt{p}}=ET{H}_{LP}\\ \frac{{d}^{2}LPvalu{e}_{LP}}{\partial {p}^{2}}=gamma=\Gamma =\frac{liq}{2\cdot {p}^{1.5}}\\ theta=\frac{1}{2}gamma\cdot variance=\frac{1}{2}\cdot \left(\frac{liq}{2\cdot {p}^{1.5}}\right)\cdot \left({p}^{2}\cdot {\sigma}^{2}\right)\end{array}$
Simplifying, we get a function that uses a variance term:
Equation (3) variance over 4: $theta=liq\cdot \frac{\sqrt{p}\cdot {\sigma}^{2}}{4}$
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 deltahedged.
The LvR function is identical to equation 3, though it looks different because the pool's market value is used to normalize it:
$\frac{theta}{LPvalue}=\frac{{\sigma}^{2}}{8}$
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
$LPvalue=2\cdot liq\cdot \sqrt{p}$
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).
$liq\cdot \frac{{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}^{2}}{\sqrt{{p}_{0}}}\approx liq\cdot \frac{\sqrt{{p}_{0}}\cdot {\sigma}^{2}}{4}$
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.
$net\text{}token\text{}change:\Delta US{D}_{0,1}+{p}_{1}\cdot \Delta ET{H}_{0,1}$
$square\text{}of\text{}square\text{}root:liq\cdot \frac{{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}^{2}}{\sqrt{{p}_{0}}}$
$variance\text{}over\text{}4:liq\cdot \frac{\sqrt{{p}_{0}}\cdot {\sigma}_{0,1}^{2}}{4}$
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.
$net\text{}token\text{}change\text{}PnL:\Delta US{D}_{0,1}+{p}_{1}\cdot \Delta ET{H}_{0,1}$
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.
$square\text{}of\text{}square\text{}root\text{}PnL:USDCvolumetraded\cdot feeliq\cdot \frac{{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}^{2}}{\sqrt{{p}_{0}}}$
$variance\text{}over\text{}4\text{}PnL:USDCvolumetraded\cdot feeliq\cdot \frac{\sqrt{{p}_{0}}\cdot {\sigma}_{0,1}^{2}}{4}$
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 restrictedrange 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 ETHUSDC v2 pool on Ethereum Mainnet
1/1/233/5/24
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.
$Impermanent\text{}Loss=liq\cdot \frac{{\left(\sqrt{{p}_{1}}\sqrt{{p}_{0}}\right)}^{2}}{\sqrt{{p}_{0}}}$
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 intraperiod 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.
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