Interest Rates and Utilization

Everything is 100% collateralized. When the liquidity pool sells you a call option, it locks in one SOL for every call option. It is impossible for the liquidity pool to default. Olive only sells covered calls and cash-secured puts, so the liquidity pool always locks enough collateral in the custody account to make sure there is zero risk of default.

Fixed rates are used for options and expiry futures. Once you buy an option or expiry future, you lock in that fixed rate. You can only sell back to the liquidity pool in that fixed rate.

Variable rates are used for perpetual futures. The interest-rate payments accrue and change hourly depending on demand to borrow.

Variable Rates and Utilization

The ratio of locked assets to total assets is called the utilization ratio U. The variable interest-rate function is based on the utilization of the pool. The function has two slopes and a kink at optimal utilization.

$$r_\text{variable}(U)= \begin{cases} r^{\min}+\left(r^{\text{kink}}-r^{\min}\right)\dfrac{U}{U_{\text{optimal}}}, & 0\le U\le U_{\text{optimal}},\\ r^{\text{kink}}+\left(r^{\max}-r^{\text{kink}}\right)\dfrac{U-U_{\text{optimal}}}{1-U_{\text{optimal}}}, & U_{\text{optimal}}<U\le 1. \end{cases}$$

Fixed Rates and 2D Utilization

For the fixed rate, Olive uses a negative exponential model with normalized time variables:

$$r_\text{fixed}(t,U,U_{2d}) = r_\text{variable}(U) + U_{2d} \beta \left(1 - e^{-\tau_y t_y}\right)$$

Where:

• r_variable(U) is the current variable interest rate
• beta is the scaling factor
• U_2d is the current 2D utilization of the pool
• tau is the average expiry of all fixed-rate assets
• t is the quoted position time to expiry
• tau_y = tau / (365 days)
• t_y = t / (365 days)

This normalization keeps the exponent dimensionless while preserving the intended pool economics:

• larger average pool expiry tau increases the fixed-rate premium
• larger quoted expiry t increases the fixed-rate premium

Liquidity taken represents the total amount of liquidity reserved by financial instruments that expire at a certain date. The 2D utilization is calculated by:

$$U_{2d} = \frac{\text{Liquidity Taken}}{\text{TVL } \cdot 365 \text{ days}}$$

The average expiry tau is:

$$\tau \;=\; \frac{\sum_{i=1}^{n} \mathrm{OI}_i \,(T_i - t_0)}{\sum_{i=1}^{n} \mathrm{OI}_i}$$

Where:

• OI_i is the open interest of the expiry instrument
• T_i is the expiry date of the instrument
• t_0 is the current time