Last updated 12 September 2026

Payoff & pricing

Stochastic.Finance options are fully collateralised and fractional: a payoff is a share of the USDC posted against the option, never a claim beyond it. That makes the payoff function — and therefore the price — different from a vanilla option's, and worth understanding precisely before you size a position.

Payoff at expiry

Write \(C\) for the collateral, \(K\) for the strike, \(S_T\) for the settlement price and \(\delta = 0.1\) for the zero shift.

For a call, the long leg receives the fraction

$$\varphi_{call} = \frac{\max(S_T - K, 0) + \delta K}{\max(S_T - K, 0) + K}$$

and for a put,

$$\varphi_{put} = \frac{(1 - \delta)\max(K - S_T, 0) + \delta K}{K}$$

with the payoffs themselves being

$$\text{Payoff}_{long} = C \cdot \varphi \qquad \text{Payoff}_{short} = C \cdot (1 - \varphi)$$

The two legs sum to exactly \(C\), always. Settlement redistributes the collateral; it never creates or destroys value, and the contract can never owe more than it holds.

What the zero shift does

Without \(\delta\), a far out-of-the-money option would settle at exactly zero, and — more consequentially — would trade at near zero on the AMM. That region is hazardous: on a constant-product curve, a tiny absolute price near zero represents an enormous relative move, so an attacker can mint near-worthless options and swap them into a pool to extract a disproportionate amount of USDC. Liquidity providers absorb the loss.

Setting \(\delta = 0.1\) removes the region entirely:

So the instrument is bounded on both sides. An option that expires worthless returns a tenth of its collateral to its buyer; one that expires deep in the money approaches all of it, rather than growing without limit as a vanilla call would.

Worked values, per unit of collateral

For a call:

Next

• Dynamic swap fee — what the model price is compared against
• Exercise & settlement — the payoff formulas applied in practice
• Core concepts — the intuition, without the algebra