Original title: "The Double Parachute Model: a mathematical model for using debt-backed stable coins as collaterals》
Author: Yaron Velner
Original compilation: ChinaDeFi
Original compilation: ChinaDeFi
Debt-backed stablecoins like DAI, LUSD, sUSD, and FRAX are sources of passive income in DeFi (like Curve LP or Yearn Vaults). Users can benefit from highly leveraged positions in such assets. If leveraged against other stablecoins like USDC, the user's liquidation risk is considered minimal (as long as the collateralized stablecoin remains pegged).
As such, the lending market benefits from providing high leverage to such users, but could be at risk of bad debts if the collateral stablecoin loses its peg. Such bad debts can be mitigated by setting an appropriate liquidation threshold (aka LTV) that will enable the platform to properly liquidate collateral when it is unpegged. But at the same time, it also limits the leverage available to users.
In this paper, we propose a mathematical model to reason about liquidation thresholds for stable collateral assets. Our proposed framework assumes that a mathematical model already exists 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 Gearbox's leverage protocol, focusing on the scenario of borrowing USDC against stablecoin collateral.
double parachute model
The LUSD stablecoin is quite elegant as it is backed by a single collateral (i.e. ETH) and it has a built-in mechanism where users’ bad debts are 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 due to permanent price decoupling, which ignores temporary decoupling due to illiquidity. In such a setup, the price of LUSD is only affected by the percentage of ETH it is backing, so we can treat a user position with LUSD collateral and USDC debt as de facto a position where the collateral is ETH (the debt is still USDC ).
Both Liquity (the protocol that runs LUSD) and the lending marketplace (Gearbox in our case) will try to prevent bad debt from accumulating.
As shown in the chart below, as the price of ETH drops, the first mitigation line will be activated and Liquity will attempt to prevent bad debts from accumulating in the LUSD system. When Liquity's attempt fails, and the price of ETH continues to drop, then the bad debts of the LUSD system will reduce the price of LUSD itself, at which point the Gearbox system will step in and try to reduce bad debts on its own platform.
In the double parachute analogy, the first parachute is Liquity, whose strength is determined by the ETH backing it currently has. The second parachute is the Gearbox, whose strength comes from the configured liquidation threshold, the lower the threshold the stronger the protection. In particular, the second parachute can degenerate and be set at 100% (minus applicable liquidation penalties and known oracle bias) when the ETH-to-LUSD backing ratio is high enough.
formal framework
Formally, we view 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 value at risk/bad amount, which can be done in any standard stress testing environment. Gearbox's liquidation threshold is then set to compensate bad debts in the LUSD system. For example, if the expected value at risk 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, LUSD is expected to have a VaR of 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 relative to the corresponding DEX liquidity (usually Curve Finance's liquidity).
Most of these stablecoins are not subject to risk-free arbitrage even when above or below the peg. However, one might expect the price to bounce back to 1.
By examining the short-term timeframe of 1 hour, we observe that the trading volume of these assets is quite one-sided.
As the chart below shows, when decomposing the Curve Finance volume for FRAX transactions into 1-hour windows, (volume weighted) on average over 90% of the volume is unilateral.
That said, FRAX remains perfectly pegged due to the large amount of Curve liquidity (over $0.5B) owned almost entirely by the FRAX protocol itself.
This is not the case for LUSD, which has fewer side trades per hour but suffers from an almost permanent upward decoupling.
Finally, sUSD is the most balanced in terms of one-sided transactions, but is still basically one-sided.
Therefore, we also consider the stability of DEX liquidity relative to USDC, and assume that the reverse organic volume will not mitigate cascading liquidations. That said, as long as the stablecoin is solvent, the volatility of the asset will remain low and, as a result, relatively few liquidations are expected.
formal framework
To be on the safe side, we simulated a situation where all of Gearbox's stable collateral is liquidated within a day, without any price recovery after each liquidation.
Classification
original assets
For raw assets such as sUSD and LUSD, we simulate according to the double parachute model and the price fluctuation model, and set the liquidation threshold to the minimum of these two proposals.
Curve LP Token
Curve LP tokens such as LUSD/3crv LP tokens are special because their price is higher than the USDC price ($1). This is due to their technical limitations in terms of price forecasting.
On the other hand, depositing assets into the Curve system introduces additional smart contract risk. This risk can be mitigated by charging users higher fees. In any case, the Curve smart contract has been rigorously tested and is considered low risk.
Algorithmic Stablecoins
Algorithmic Stablecoins
In this case, we can apply DPM with FXS as collateral asset. However, since FRAX has an agreement and liquidity compared to USDC, this liquidity will also be considered as FXS liquidity.
