macrocephalopod

(Actually it's a little more complicated since k increases every time there is a swap, due to liquidity fees being subtracted from the incoming amount and added to the pool - I'll get to that)

The important thing is that you can express the reserves in terms of k, p as x = √(kp) and y = √(k/p), and whenever there is a price move the new reserves are x' = √(k'p') and y' = √(k'/p')

The total value of the pool is x + y * p = 2√(kp) and after a price move this has changed to x' +y' * p' = 2√(k'p') so the P&L is just 2√(k'p') - 2√(kp)

(Clearly you only have an ownership claim on a % of all the assets in the pool, but since everyone in the pool gets the same results, we may as well model the whole pool as a single liquidity provider)

Now most treatments I see compare this to an alternative where you just held assets X and Y outside of the liquidity pool. But I'm a market neutral kinda guy so let's assume you hedge the linear price exposure with a short futures position, which generates P&L of -y * (p' - p)

Call the price return r, so that p' = p(1+r), then your total P&L is 2√(k'p(1+r)) - 2√(kp) - √(kp) * r which, if you expand out the square root in powers of r, is 2√(kp) * [√(k'/k) * (1 + r/2 - r^2/8 + ...) - 1 - r/2] (the linear term cancels as expected)

Now remember that the pool constant k grows over time. The growth is proportional to the turnover of the pool and to the fees charged so we can write k' = k * (1 + t * f) where t is the turnover as a fraction of pool assets, and f is the liquidity fee (0.3% in Uniswap v2)

Using that in the expression for the P&L, expanding in powers of tf and simplifying, we get P&L = √(kp) * [tf - r^2/4] = x * (tf - r^2/4)