Open Option

Option Pricing

The liquidity pool sells options using Black-Scholes pricing.

$$V(S, K, r, \sigma, T)= \begin{cases} S\,N(d_1)-K e^{-rT}\,N(d_2), & \text{Call option},\\ K e^{-rT}\,N(-d_2)-S\,N(-d_1), & \text{Put option}. \end{cases}$$ $$d_1=\frac{\ln\!\left(\frac{S_0}{K}\right)+\left(r+\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}, \qquad d_2=d_1-\sigma\sqrt{T}.$$

Where:

• S is the oracle price
• sigma is the volatility
• K is the strike price
• T is the time to expiry
• r is the interest rate

$$r_{\text{option}}= \begin{cases} r_\text{TOKEN-fixed}, & \text{Call option},\\ -r_\text{USDC-fixed} & \text{Put option}. \end{cases}$$

The formula above gives you the price of a single option. You can buy any number of options in a single order as long as the minimum order size is above $10.

For opening an option position you can submit your order via either a limit or market order. You can choose to pay for the option in either currency of the pool.

Option Restrictions

Strike prices are restricted to the following bounds:

$$K_L(S,n,r, \sigma, T)=S \div \exp\!\left(r_\text{USDC-fixed}T + n\,\sigma\sqrt{T}\right)$$ $$K_U(S, m, r,\sigma, T)=S \times \exp\!\left(r_\text{TOKEN-fixed}T + m\,\sigma\sqrt{T}\right)$$

Where:

• m and n are the number of standard deviations for the upper and lower bounds
• S is the oracle price
• sigma is the volatility
• K_L is the lower bound and K_U is the upper bound of the strike price
• T is the time to expiry
• r is the interest rate

Olive starts with n = 1 and m = 1. Expiry time cannot exceed one year and must exceed one day, so T must stay within 1 day and 365 days.

Locking Collateral

Everything is 100% collateralized because the liquidity pool only sells covered calls and cash-secured puts.

For every call option the liquidity pool locks in one token as collateral.

For each put contract the pool locks K USDC, where K is the strike price, to make a cash-secured put.