⌛Staking & Impermanent Loss

One of the main issues with liquidity providing on an LP is impermanent loss. IL is the loss incurred by a market making position vs. keeping the initially allocated amounts fixed. Another approach of IL could be referred as unrealized profit or loss (uPNL): this a more accurate way to call such profit or loss, as it only becomes realized if one chooses to sell out of a position which has changed in value.

Given an initial price $P_0$, the value of $P_0$ of a 2-asset portfolio is initially given by:

$V = V_0 + P_0 \cdot y$

On the other hand, a new value $V_1$ after the first transactions $P_1$ is given by:

$V_1 = x' + P_1 \cdot y'$

Given a new value and new prices, the equation will still be the following:

$x+ P_1 \cdot y$

The IL or uIL is the delta between the portfolio change of the market making portfolio and the change in value of a portfolio of assets with fixed quantities. This is the loss on top of a mark to market move of an equivalent fixed-quantity portfolio:

$x' + P_1 \cdot y' - (x + P_0 \cdot y) - (x + P_1 \cdot y - (x + P_0 \cdot y))$

Which simplifies to:

$x' - x + P_1 \cdot y' - P_0 \cdot y - (P_1 - P_0) \cdot y$

Also, we will call $R$ the ratio between the current and previous price given a fixed-period of time $t$:

$R = \frac{P_1}{P_0}$

Hence, unrealized impermanent loss $\epsilon$ will be given by the following equation:

$\epsilon = \frac{uPNL}{V_0} = \sqrt{R} - \frac{1}{2} \cdot (R+1)$