Mint and redeem of LP tokens

the calculation

Parameters:

Symbol	Meaning
lp	Each LP token’s net asset value in US dollars. $lp=\frac {LP} {TotalSupply}$
LP	total LP tokens’ net asset value, in USD, equal to LP’s cash after instantly closing all user positions without fee
s	net position of each currency in USD for all users. = (loi-soi)
pr	a risk management parameter for each currency. pr $\in (0,1)$
$p_{oracle}$	Oracle price of each currency
$p_{exec}$	execution price of each currency

When calculating the mint and redeem process, Symmetry's algorithm looks at its impact on the overall welfare of the LP holders. Specifically:

• When a liquidity provider mints lp tokens, it reduces the risk (or does not change the risk) other LPs take (from acting as the counterparty of traders). Therefore, $lp_{mint} = lp_{oracle}$

• When a liquidity provider redeems lp tokens, it adds risks (or does not change the risks) other LPs take. Therefore, $lp_{redeem} \leq lp_{oracle}$

Let's go through the calculation with examples. Note: the following examples do NOT include gas fees.

Assumptions:

$LP$ = $100,000,000, $TotalSupply$=1,000,000

BTC-USDC swap: s=$0, $\lambda$ = 0.05, pr=0.8, $p_{oracle}$= $25,000

ETH-USDC swap: s=$-10,000,000, $\lambda$ = 0.05, pr=0.75, $p_{oracle}$= $2000

Mint

If someone wants to mint 250,000 lp, how much USDC would he need (assuming the USDC's price remains at $1)?

$lp_{oracle}= \frac {100,000,000} {1,000,000} = \$100$

$250,000 \times lp_{oracle} = \$25,000,000$=

Redeem

If someone wants to redeem 250,000 lp, how much USDC could he get (assuming the USDC's price remains at $1)?

The maximum liquidity available for redemption is $LP - \sum |s| = \$90,000,000$.

$lp_{oracle}= \frac {100,000,000} {1,000,000} = \$100$

Every lp can be viewed as holding 0 BTC-USDC position and $\frac {-10,000,000} {1,000,000} =\$-10$ ETH-USDC position. Thus, 250,000 $lp$ would hold 0 BTC-USDC position and $-2,500,000 ETH-USDC position.

To close $-2,500,000 ETH-USDC position:

$LP = 100,000,000 - 250,000 \times 100 = \$75,000,000$,

$s = -10,000,000 +2,500,000 = \$-7,500,000$

$T=\$-2,500,000$

$p_{mid} = p_{oracle} \cdot (1 + \frac {\lambda \cdot s} {pr \cdot LP})=\$1,986.67$

$p'_{mid} = p_{oracle} \cdot (1 + \frac {\lambda \cdot (s+T)} {pr \cdot LP}) = \$1,982.22$

$p_{exec} = \$1,984.44$

The calculation assumes 0 trading fees and 0 gas fees.

$slippage = 1,984.44-2,000= \$-15.56$

$PnL= \frac {2,500,000} {2,000} \times (1,984.44-2,000)= \$-19,444.44$

Adding in the 0.1% redemption fee:

$250,000 \times lp_{redeem}= （250,000 \times 100 - 19,444.44)\times(1-0.1\%) = 24,955,575$, which is less than 25,000,000.

Further elaborations on trading prices are explained here.