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Signed vanilla option payoff and simple profit

For sign q = +1 long or −1 short and units U, simple profit is qU(h−π), with per-unit holder payoff h and premium π.

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Define the contract payoff

A call holder payoff per unit is max(S−K,0); a put holder payoff is max(K−S,0). Here S is the nonnegative terminal price and K the positive strike. The contract payoff remains a nonnegative holder amount.

Apply the side and quantity

Signed settlement is qUh. Initiation premium cash flow is −qUπ, where π is the nonnegative per-unit premium. Simple profit is their sum.

Πsimple=qU(h−π),q∈{+1,−1}\Pi_{simple}=qU(h-\pi),\quad q\in\{+1,-1\}

h is the holder payoff; premium is not financed to expiry.

Solve the positive-premium profit crossing

For a call the strike/premium boundary is K+π. For a put it is K−π if nonnegative. If π exceeds K, a put holder cannot recover that premium from any nonnegative terminal price under the simple model.

Handle zero premium carefully

When π = 0, simple profit equals signed payoff. A call has zero profit throughout S≤K and a put throughout S≥K, so K is a boundary of a zero region rather than a unique isolated breakeven point.

Further references