Liquidation Threshold

In most other projects, all users share a unified LTV threshold that determines the liquidation line. In our project, however, to ensure system security, we adopt a more rigorous approach: from a theoretical and mathematical perspective, we derive the optimal liquidation threshold individually for each user.

Consider the underlying price $P_t$ satisfies the following geomatric Brownian motion under the risk neutral probability:

$$\left\{ \begin{aligned} dP_t &= \mu P_t \, dt + \sigma P_t \, dW_t, \\ P_0 &= p_0. \end{aligned} \right.$$

Let the liquidation threshold be denoted by L, the optimal L should satisfies

$$L - \frac{\lambda}{m} \geq \frac{1}{m} \left[ \frac{1 - \eta}{\eta} + \frac{2\eta - 1}{\eta} \cdot \min \left( 1, \; 2\Phi \left( \sqrt{1 - \eta} \left[ \left( \frac{\sigma}{2} - \frac{\mu}{\sigma} \right) \sqrt{\frac{(mL + 1)P_0}{a(m + 1)}} - \frac{1}{\sigma} \sqrt{\frac{a(m + 1)}{(mL + 1)P_0}} \right] \right) \right) \right]$$

where $\lambda$ denotes the proportional reward ratio, a denotes the constant rate of price decrease in Dutch Auction, m=$\frac{N_L}{N_S}$.