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Liquidation Penalty

Rationale for Liquidation Penalty values
Goals: - liquidation is lucrative for liquidator, - liquidation improves health of the borrower;
Given
health=1bcu/bchealth = 1 - bcu/bc\\
for
  • deposited asset A worth of d with collateral factor
    cfAcf_{A}
    ,
  • borrowed asset B worth of b with liquidation threshold
    ltBlt_{B}
Liquidation starts when
h0=1(b/ltB)(dcfA)=1bdcfAltB=0,d/b=1/(cfAltB)h_0 = 1 - \frac{(b/lt_B)}{(d*cf_A)} = 1 - \frac{b}{d*cf_A*lt_B} = 0,\\ \Downarrow\\ d/b = 1/(cf_A*lt_B)
Upon liquidation of portion
α\alpha
of B with penalty
pAp_A
, health must increase:
h1=1(b(1α))cfAltB(dαb(1+pB))>h0=0b(1α)<cfAltB(dαb(1+pB))(1α)<cfAltB((d/b)α(1+pB))1α<1α(1+pB)cfAltBα>α(1+pB)cfAltB(1+pB)cfAltB<1(1+pB)<1cfAltBpB<1cfAltB1h_1 = 1 - \frac{(b(1-\alpha))}{cf_A*lt_B*(d-\alpha * b (1 + p_B))} > h_0 = 0\\ \Downarrow\\ b(1-\alpha) < cf_A*lt_B*(d-\alpha * b (1 + p_B))\\ \Downarrow\\ (1-\alpha) < cf_A*lt_B*((d/b)-\alpha * (1 + p_B))\\ 1 - \alpha < 1- \alpha * (1 + p_B) * cf_A*lt_B \\ \alpha > \alpha * (1 + p_B) * cf_A*lt_B\\ (1 + p_B) * cf_A*lt_B < 1\\ (1 + p_B) < \frac{1}{cf_A*lt_B}\\ \Downarrow\\ p_B <\frac{1}{cf_A*lt_B} - 1
is the condition for liquidation to increase health. Given
max(cf)=0.9max(cf)=0.9
p<10.9lt1p <\frac{1}{0.9*lt} - 1
is then a safe condition for liquidation penalty for any asset.