Capped Payoffs and Collateral

Capped Linear Payoffs

Let $M$ denote the maximum payout in SOL, $K$ the strike, and $S_T$ the settlement price. The payoff functions are

\text{Payoff}_{\text{call}} = \min\!\left(M,\; M \cdot \max\!\left(0, \frac{S_T - K}{K}\right)\right), \tag{1}

\text{Payoff}_{\text{put}} = \min\!\left(M,\; M \cdot \max\!\left(0, \frac{K - S_T}{K}\right)\right). \tag{2}

Each payoff is piecewise linear in $S_T$ , with slope $M/K$ on the interval between the strike and the cap. For the call, the cap binds for $S_T \ge 2K$ . For the put, since $S_T \ge 0$ implies $(K - S_T)/K \le 1$ , the cap is never strictly binding and $\Pi_P = M$ is attained only at $S_T = 0$ .

Static Replication

Under the base model, the capped payoff admits closed-form valuation as a linear combination of vanilla options. Equation (1) decomposes as

\text{Payoff}_{\text{call}} = \frac{M}{K}\Big[(S_T - K)^{+} - (S_T - 2K)^{+}\Big], \tag{3}

a $K$ / $2K$ call spread scaled by $M/K$ . By no-arbitrage its value is $V_{\text{call}} = \tfrac{M}{K}\left[C(K) - C(2K)\right]$ , with each $C(\cdot)$ evaluated by the pricing engine at the corresponding point on the volatility surface. Since the put cap does not bind on $S_T \gt 0$ , the put payoff reduces to $\tfrac{M}{K}(K - S_T)^{+}$ with value $V_{\text{put}} = \tfrac{M}{K}P(K)$ .

Admission Constraint

Let $L_{\text{pool}}$ denote total pool collateral in SOL and $M_i$ the maximum payout of open contract $i$ . A contract with maximum payout $M_{\text{new}}$ is admitted only if aggregate maximum payout, including the new contract, does not exceed pool collateral:

\sum_{i \in \text{open}} M_i + M_{\text{new}} \le L_{\text{pool}}. \tag{4}

Defining free collateral $L_{\text{free}} = L_{\text{pool}} - \sum_{i \in \text{open}} M_i$ , a per-contract concentration limit additionally applies:

M_{\text{new}} \le 0.25 \cdot L_{\text{free}}. \tag{5}

Constraint (5) bounds the fraction of residual capacity any single contract may consume, preserving capacity for other participants and limiting pool concentration at a single strike and expiry. For $L_{\text{free}} \ge 0$ , (5) implies (4).

The admission predicate is:

free      = pool_collateral - sum(open.max_payout)
admit     = (new.max_payout <= free) and (new.max_payout <= 0.25 * free)
on admit  : lock new.max_payout; credit premium

Full Collateralization Invariant

Since $M$ is locked at issuance and released only upon settlement or early close, the invariant $L_{\text{pool}} \ge \sum_{i \in \text{open}} M_i$ holds at every state. The pool is therefore solvent against the simultaneous maximum payoff of all open contracts. Three properties follow:

Zero counterparty default. The pool cannot fail to honor a contract it has written.
No socialized loss. In-the-money holders are never subject to pro-rata haircuts arising from losses elsewhere in the pool.
No margining. Holders post only the premium; no variation margin is required after issuance.

The trade-off is capital efficiency: collateral is reserved against the worst case rather than an expected or stressed loss, and pool size bounds the open interest each market can support.