# OLL

## Liquidation Mechanism in Collateralized Debt Positions

### Liquidation Condition

A liquidation event is triggered if and only if the Health Factor falls below 1:

$$\text{Health Factor} < 1$$

This can be expressed mathematically as:

$$\frac{\text{Liquidation Threshold} \times \text{Collateral Value}}{\text{Borrowed Amount}} < 1$$

where:

* Borrowed Amount > 0
* Liquidation Threshold ∈ (0,1)

### Collateral Composition

In our model, the collateral is composed of a long asset (e.g., BTC or ETH) and a put option. The collateral value is thus defined as:

$$\text{Collateral Value} = P + \max(X - P, 0)$$

where:

* P is the current price of the long asset
* X is the strike price of the put option

### Liquidation Scenarios

We can distinguish two scenarios based on the relationship between X and P:

#### Scenario 1: X > P

When the strike price exceeds the current asset price:

$$\text{Collateral Value} = P + (X - P) = X$$

Liquidation occurs if and only if:

$$X < \frac{\text{Borrowed Amount}}{\text{Liquidation Threshold}}$$

#### Scenario 2: X ≤ P

When the strike price is at or below the current asset price:

$$\text{Collateral Value} = P$$

Liquidation occurs if and only if:

$$P < \frac{\text{Borrowed Amount}}{\text{Liquidation Threshold}}$$

### Conclusion

Given that X ≤ P in Scenario 2, we can conclude that in both scenarios, liquidation occurs when:

$$X < \frac{\text{Borrowed Amount}}{\text{Liquidation Threshold}}$$

This unified condition provides a concise criterion for liquidation events in our collateralized debt model.


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