Lesson 17 - Slippage and Impermanent Loss

Slippage Formula

$SlippageAmount=Actual{y}_{out}-Expected{y}_{out}$
$Actual{y}_{out}=\mathrm{\Delta }y$
$(y-\mathrm{\Delta }y)\cdot (x+\mathrm{\Delta }x)=x\cdot y$
$y\cdot x+y\mathrm{\Delta }x-\mathrm{\Delta }y\cdot x-\mathrm{\Delta }y\mathrm{\Delta }x=x\cdot y$
$-\mathrm{\Delta }y(x+\mathrm{\Delta }x)+y(x+\mathrm{\Delta }x)=x\cdot y$
$-\mathrm{\Delta }y(x+\mathrm{\Delta }x)=x\cdot y-y(x+\mathrm{\Delta }x)$
$-\mathrm{\Delta }y(x+\mathrm{\Delta }x)=-y\mathrm{\Delta }x$
$\mathrm{\Delta }y=\frac{y\mathrm{\Delta }x}{x+\mathrm{\Delta }x}=Actual{y}_{out}$

$Expected{y}_{out}=p_x{x}_{in}$
$\frac{y}{x}\cdot x_{in}$

$SlippageAmount=\frac{y\cdot x_{in}}{x+x_{in}}-\frac{y}{x}\cdot x_{in}$

Formula for Impermanent Loss

Impermanent loss needs to show the different in V for pair in the liquidity pool versus held in the wallet, where it adjusts based on market price.

Value = (Price Token X)(Amount Token X) + Amount Token Y

This is where Y is the token in which value is expressed, and Price Token X is with respect to price Y.

Impermanent Loss represents the difference in the value of what is in the Pool vs what is in the Wallet

$V_{pool}-V_{hold}=p_{now}\cdot x_{now}+y_{now}-(p_{now}\cdot x_{old}+y_{old})$

The impermanent loss occurs when an arbitrageur trades an (x,y) pair at the price of the AMM pool and is able to take profits. The profit taking comes by, say, trading out a token at a lower price relative to, say, USDC, than the external markets.

This is why $x_{now}$ and $y_{now}$ are different from $x_{old}$ and $y_{old}$