Interpreting the Double Drop Mountain Model: Why is it so low risk?

原文：《The Double Parachute Model: a mathematical model for using debt-backed stable coins as collaterals》
Compiled by: ChinaDeFi
Debt-backed stablecoins such as DAI, LUSD, sUSD, and FRAX are sources of passive yield in DeFi (e.g. Curve LP or Yearn Vaults). Users can benefit from highly leveraged positions in such assets. If leverage is applied to other stablecoins such as USDC, the user's liquidation risk is considered minimal (as long as the collateralized stablecoin remains pegged).
Therefore, the lending market benefits from providing high leverage to such users, but may face the risk of bad debt if the collateral stablecoin loses its peg. Such bad debt can be mitigated by setting an appropriate liquidation threshold (also known as LTV), which will enable the platform to properly liquidate the collateral when it is unpegged. But at the same time, it will also limit the leverage that users can use.
In this paper, we propose a mathematical model to reason about liquidation thresholds for stable collateral assets. Our proposed framework assumes that there is already a mathematical model to explain liquidation thresholds for volatile assets. Therefore, this new framework can be applied to any existing stress testing environment.
This paper proposes a model for setting up the Gearbox Leverage Protocol and focuses on the scenario of borrowing USDC with stablecoin collateral.
Double parachute model
The LUSD stablecoin is quite elegant because it is backed by a single collateral (i.e. ETH) and it has a built-in mechanism whereby a user’s bad debt is socialized across all borrowers. Therefore, we use LUSD to demonstrate our framework, but similar principles apply when analyzing DAI and sUSD.
The Double Parachute Model (DPM) is designed to simulate bad debts caused by permanent price decoupling, ignoring temporary decoupling due to insufficient liquidity. In such a setting, the price of LUSD is only affected by the percentage of ETH it supports, so we can treat a user position with LUSD collateral and USDC debt as a de facto position with ETH as the collateral asset (the debt is still USDC).
Both Liquity (the protocol that runs LUSD) and the lending market (in our case, Gearbox) will try to prevent the accumulation of bad debt.
As shown in the figure below, as the price of ETH drops, the first mitigation line will be activated, and Liquity will try to prevent bad debts from accumulating in the LUSD system. When Liquity's attempt fails and the price of ETH continues to fall, the bad debts in the LUSD system will reduce the price of LUSD itself, and the Gearbox system will intervene and try to reduce bad debts on its own platform.
In the dual parachute analogy, the first parachute is Liquity, and its strength depends on the ETH backing it currently has. The second parachute is Gearbox, and its strength is derived from the configured liquidation threshold, with lower thresholds providing stronger protection. In particular, when the ratio of ETH backing to LUSD is high enough, the second parachute can degenerate and be set to 100% (minus applicable liquidation penalties and known oracle bias).
Formal Framework
Formally, we treat the LUSD system as a single user with X amount of ETH collateral and Y amount of LUSD debt. We stress test the LUSD system to find the expected risk value/bad debt amount, which can be done in any standard stress testing environment. Gearbox's liquidation threshold is then set to compensate for bad debt in the LUSD system. For example, if the expected risk value in the LUSD system is 15% of the LUSD supply, then Gearbox will set a liquidation threshold of 85%.
We note that under normal circumstances, the risk value of LUSD is expected to be 0%.
Price Fluctuations
Most decentralized stablecoins have no physical mechanism to force them to trade at exactly $1. Instead, they fluctuate around $1, with volatility tied to the corresponding DEX liquidity (usually Curve Finance’s liquidity).
Most of these stablecoins are not subject to risk-free arbitrage even when trading above or below their pegs. However, one might expect the price to rebound to 1.
By examining the shorter term timeframe of 1 hour, we observe that the trading volumes for these assets are quite one-sided.
As the chart below shows, when breaking down Curve Finance volume on FRAX into 1-hour windows, (volume-weighted) on average over 90% of volume is one-sided.
That said, FRAX maintains a perfect peg due to the massive amount of Curve liquidity (over $0.5B) owned almost entirely by the FRAX protocol itself.
This is not the case with LUSD, which has less one-sided trading volume per hour but suffers from an almost permanent upward decoupling.
Finally, sUSD is the most balanced in terms of one-sided trades, but it is still essentially a one-sided trade.
Therefore, we also consider the stability of DEX liquidity relative to USDC and assume that reverse organic volume will not mitigate cascading liquidations. That is, as long as stablecoins are solvent, the volatility of the asset will remain low, and therefore, liquidations are expected to be relatively rare.
Formal Framework
To be safe, we simulated a scenario where all of Gearbox’s stable coin collateral was liquidated in a single day, without any price recovery after each liquidation.
 Classification
Original assets
For primitive assets such as sUSD and LUSD, we simulate based on the double parachute model and the price volatility model, and set the liquidation threshold to the minimum of these two recommendations.
Curve LP Tokens
Curve LP tokens, such as LUSD/3crv LP tokens are special because their price is higher than the USDC price (1 USD). This is due to their technical limitations in price prediction.
Thus, we get special cases among LP tokens like LUSD/3crv, where LUSD is redeemable for $0.99 of ETH as long as it is solvent, and since its imperfect oracle caps the price at $1, we get that the asset is not subject to price volatility and therefore its DEX liquidity is negligible.
On the other hand, depositing assets into the Curve system carries additional smart contract risk. This risk can be mitigated by charging users higher fees. In any case, Curve smart contracts have been rigorously tested and are considered to be low risk.
Algorithmic Stablecoins
The FRAX stablecoin is also partially backed by its FXS governance token, which is minted whenever it loses some of its support.
In this case, we can apply DPM with FXS as collateral asset. However, since FRAX has a protocol and liquidity compared to USDC, this liquidity will also be considered as FXS liquidity.