Multi-position liquidation price calculation

Multi-position liquidation price calculation:

• otTMM: Sum of maintenance margin of other contracts: $\text { otTMM }=\sum_{k=1}^{n} a b s\left(S_{k} * P_{k} * M_{k}\right)-\operatorname{abs}\left(S_{i} * P_{i} * M_{i}\right)$
• otUPNL: Total unrealized gain or loss of other contracts:

  $\text { otUPNL }=\sum_{k=1}^{n} S_{k} *\left(P_{k}-P_{k}^{o}\right)-S_{i} *\left(P_{i}-P_{i}^{o}\right)$

The liquidation price of the $I^{\text {th}}\;$contract ($P_i^l$):

$$P_{i}^{l}=\frac{S_{i} * P_{i}^{o}-\mathrm{BI}+\text { otTMM - otUPNL }}{S_{i} *\left(1-\operatorname{dir}_{i} * M_{i}\right)}$$

Variables:

• $BI$: Static account equity
• $P_i:$Current oracle price of the $I^{\text {th}}\;$contract
• $P_i^l:$The liquidation price of the $I^{\text {th}}\;$contract
• $P_i^0$: The opening price of the $I^{\text {th}}\;$contract
• $S_i:$ The number of open positions of the $I^{\text {th}}\;$contract, positive for long positions and negative for short positions.
• $dir_i:$The opening direction of the $I^{\text {th}}\;$contract, +1 for long positions and -1 for short positions
• $M_i:$Maintenance margin rate of the $I^{\text {th}}\;$contract

Example:

Initial account balance $BI = 1000$, ETH-USDC maintenance margin rate$M_i = 3$%, the opening price of ETH-USD $P_i^0 = 3000$, Current oracle price $P_i$= 2900; BTC-USDC maintenance margin rate$M_i = 3$%, the opening price of BTC-USDC $P_n^0 = 40000$, Current oracle price $P_n$= 38000. A trader buys $S_i$= 1.5 ETH-USDC, sells $S_n$= -0.1 BTC-USDC.

The liquidation price of the ETH contracts is:

otTMM = $abs(-0.1*38000*3\%)=144$

otUPNL =$-0.1*(38000-40000)=200$

$$P_{i}^{l}=\frac{1.5 *3000 -1000+114-200}{1.5 *(1-3\%)}=2346.39$$

The liquidation price of the BTC contracts is:

otTMM = $abs(1.5*2900*3\%)=130.5$

otUPNL =$1.5*(2900-3000)=-150$

$$P_{n}^{l}=\frac{ -0.1*40000 -1000+130.5+150}{-0.1 *(1+3\%)}=45820.388$$