Fee (theta)

Points:

• Interest Rate Model like Aave was applied to Perpetual futures as Lending

• Protocol Fee was applied to the Lending fee

• Rebalance Fee is required to maintain the protocol

Utilization

The interest rate algorithm is calibrated to manage liquidity risk and optimize utilization. The borrow interest rates are derived from the Utilisation Rate $U = \frac{TotalBorrowAmount}{TotalSupplyAmount}$

Interest Rate Model

Liquidity risk materializes when utilization is high, and this becomes more problematic as $U$ gets closer to 100%. To tailor the model to this constraint, the interest rate curve is split in two parts around an optimal utilizationn rate $U_{optimal}$. Before $U_{optimal}$the slope is small, after it begins rising sharply.

The interest rate$R$ follows the model:

$$R=Base+\frac{U}{U_{optimal}}Slope1\ (U<U_{optimal})$$

$$R=Base+Slope1+\frac{U-U_{optimal}}{1-U_{optimal}}Slope2\ (U≥U_{optimal})$$

	Base	U_optimal	Slope1	Slope2
USDC	0.004	0.9	0.04	1.4
ETH	0.004	0.85	0.04	1.4
variance of √ETH	0.022	0.45	0.14	1.6

Fee for USDC and ETH

$$TotalBorrowInterest=TotalBorrowAmount*R*\frac{TimeElapsed}{1 year}$$

$$TotalSupplyInterest = TotalBorrowInterest*(\frac{1-TokenReserve}{1})$$

where $TokenReserve = 0.08$ is set as a parameter of the protocol fee.

Fee for 2√ETH

$$TotalBorrowPremium=TotalBorrowAmount*\frac{\sqrt{x} \sigma^2}{4}*\frac{TimeElapsed}{1 year}$$

$$TotalSupplyPremium=TotalUniswapFee+TotalBorrowPremium*(\frac{1-PremiumReserve}{1})$$

where $PremiumReserve = 0.04$ is set as a parameter of the protocol fee and $\sigma$ is calculated with Interest Rate Model.