USH Redemption

as of 02/01/2025

The Redemption Mechanism is the most important feature of the Isolated Pools. It provides a quick and simple process that anyone can execute when USH depegs from the dollar. Essentially, it allows users to perform a liquidation-like operation on healthy accounts by repaying loans and seizing collateral. Both redeemers and borrowers can benefit financially in this process:

• For redeemers: They buy USH from the market at a price below its peg and seize collateral valued at the peg price.

  • Example: Alice buys 100 USH for $99 on the market. She redeems it through the protocol, repaying 100 USH and seizing $100 worth of Bob’s collateral. Alice earns $1.

• For borrowers: Borrowers don’t lose money as long as USH returns to its peg since their borrow balance and collateral decrease proportionally. Revenue from the redemption can also be shared between redeemers and borrowers.

  • Example: If Alice redeems 100 USH at an equivalent value of $99.5, both Alice and Bob earn $0.5 each.

This explanation only introduces the surface of the Redemption Mechanism. Let’s now explore the details:

1. Depeg Detection: A Depeg Strategy smart contract monitors the market and, if USH depegs, opens a redemption window where anyone can redeem.

2. Execution Process: When a redeemer repays an amount $R$ of USH, the protocol identifies healthy but risky accounts and distributes the redemptions across them, prioritizing the riskiest accounts.

   • The protocol uses a Red-Black Tree data structure to efficiently find and prioritize these accounts.

   • The total amount of USH repaid equals the following summation:

$$R = \sum_{a = 1}^{N_a} \Delta B_a.$$

We want to find the $N_a$ riskiest but healthy accounts and redeem them in a way that makes all of them have the same resulting health. An account’s health is directly related to the account’s borrow limit used $L$, given by the following relationship:

$$L = \frac{B_a\cdot p_{\text{USH}}}{\text{LTV} \cdot C_a \cdot p_{\text{EGLD}}}$$

However, all accounts are affected by the same price quotient and collateral factor, so we can abstract all that out and just use the following rate:

$$s_a^{\star} = \frac{B_a}{C_a}.$$

In redemption, we want to find the accounts with the higher $s_a^{\star}$ that are still healthy and:

• repay a portion of their borrows $\Delta B_a$ while seizing collateral,
• make all of them have the same resulting rate $s_{a,f}^{\star}$.

An account being redeemed by an amount $\Delta B_a$ will see the following change in its rate:

$$s_{a,f}^{\star} = \frac{B_a - \Delta B_a}{C_a - \Delta C_a}.$$

The amount being repaid $\Delta B_a$ and the collateral being seized $\Delta C_a$ are related by the following relationship:

$$\Delta C_a \cdot p_{\text{EGLD}} = \Delta B_a \cdot F_D \cdot p_{\text{USH}},$$

where $F_D$ is the Depeg factor that allows sharing the revenue between redeemer and borrowers. Then, we have:

$$\Delta C_a = \Delta B_a \cdot F_D \cdot \frac{p_{\text{USH}}}{ p_{\text{EGLD}}}.$$

For simplicity, we can define the redemption price as:

$$p_{\text{redeem}} = F_D \cdot \frac{p_{\text{USH}}}{ p_{\text{EGLD}}},$$

such that:

$$\Delta C_a = \Delta B_a \cdot p_{\text{redeem}}.$$

Then, the final rate is given by:

$$s_{a,f}^{\star} = \frac{B_a - \Delta B_a}{C_a - \Delta B_a \cdot p_{\text{redeem}}}.$$

With a bit of algebra, we can rearrange the previous equation and get:

$$\Delta B_a = \frac{B_a - s_{a,f}^{\star} \cdot C_a}{1 - s_{a,f}^{\star} \cdot p_{\text{redeem}}}$$

Finally, since we know that all accounts will end up with the same health (or borrow limit used or rate), we can use the equation for $R$ and get:

$$s_{f}^{\star} = \frac{\sum_a B_a - R}{\sum_a C_a - R \cdot p_{\text{redeem}}}$$

With previous equations, we first obtain $s_{a,f}^{\star} = s_{f}^{\star}$ for all accounts and then compute $\Delta B_a$ for each account.