Calls and Puts | Oracle Papers

Contract Specification

An Oracle option is a European-style contingent claim whose payoff is a deterministic function of the underlying settlement price at a fixed expiry. The underlying is an SPL token, all prices are quoted in SOL, and all contracts are cash-settled in SOL; no transfer of the underlying occurs at settlement.

Each contract is fully specified by the tuple $(\omega, K, T, M)$ :

Side $\omega$ : $+1$ for a call, $-1$ for a put.
Strike $K$ : the reference price in SOL against which moneyness is measured.
Expiry $T$ : the settlement time. Time to expiry in years is denoted $\tau = T - t$ .
Maximum payout $M$ : the supremum of the payoff function in SOL.

The premium is the upfront consideration paid at issuance and constitutes the holder's maximum loss.

Worked Example: Call

Let spot be $S = 0.0010$ SOL. Consider a call with $K = 0.0012$ SOL and $M = 2$ SOL.

For $S_T = 0.0018$ SOL, the return relative to strike is $(S_T - K)/K = 0.5$ , so $\Pi_C = 2 \cdot 0.5 = 1$ SOL.
For $S_T = 0.0030$ SOL, the uncapped value $2 \cdot 1.5 = 3$ SOL exceeds $M$ , so $\Pi_C = M = 2$ SOL.
For $S_T \le K$ , $\Pi_C = 0$ and the holder's loss equals the premium.

Worked Example: Put

Let spot be $S = 0.0010$ SOL. Consider a put with $K = 0.0009$ SOL and $M = 1$ SOL.

For $S_T = 0.00045$ SOL, $(K - S_T)/K = 0.5$ , so $\Pi_P = 1 \cdot 0.5 = 0.5$ SOL, partially offsetting the mark-to-market loss on a spot position.
For $S_T \ge K$ , $\Pi_P = 0$ .

Formal Payoff Definitions

Let $S_T$ denote the settlement price of the underlying in SOL. The vanilla European payoffs per unit of underlying are

\Pi^{\text{van}}_{C}(S_T) = (S_T - K)^{+}, \tag{1}

\Pi^{\text{van}}_{P}(S_T) = (K - S_T)^{+}, \tag{2}

where $(x)^{+} = \max(0, x)$ . Under a risk-neutral measure $\mathbb{Q}$ with zero rate, the vanilla prices satisfy put-call parity,

C - P = S - K. \tag{3}

Oracle contracts are not denominated in units of the underlying. The payoff is instead a fraction of $M$ proportional to the return of $S_T$ relative to $K$ , truncated at $M$ :

\Pi_{C}(S_T) = \min\!\left(M,\; M \cdot \frac{(S_T - K)^{+}}{K}\right), \qquad \Pi_{P}(S_T) = \min\!\left(M,\; M \cdot \frac{(K - S_T)^{+}}{K}\right). \tag{4}

Since $\sup_{S_T} \Pi(S_T) = M$ , the liability of each contract is bounded ex ante, which is the property that admits full collateralization. Valuation is derived in Capped Payoffs and Collateral.