Positive-Part Formulas for Calls and Puts

Introduction

At expiration, an option’s payoff depends on how the underlying asset’s price compares with the strike price. A call payoff cannot be negative for its holder: if exercising would not produce a positive amount, the holder can let the option expire. A put has the corresponding feature in the opposite direction. The positive-part function records this “zero if negative, otherwise keep the amount” rule in one expression.

This tutorial defines that function and uses it to express call and put payoffs at expiration. The formulas describe the payoff for one unit of the underlying, unless a contract multiplier is specified. They do not include the premium paid or received, so they are not profit formulas.

The Positive-Part Function

The notation \(x^+\) is useful whenever an amount is available only if it is positive. For example, if \(x=-4\), then \(x^+=\max(-4,0)=0\); if \(x=4\), then \(x^+=\max(4,0)=4\). It does not mean the absolute value: \(|-4|=4\), whereas \((-4)^+=0\).

For an option payoff, the input to the positive-part function is a difference between a market price and a contractual price. The maximum operation is applied to that difference and zero—not to the two prices separately.

Call and Put Payoffs at Expiration

Let \(S_T\) denote the underlying asset’s price at expiration time \(T\), and let \(K\) denote the option’s strike price. Both are measured in the same currency per unit of the underlying. The formulas below give the expiration payoff to the holder of one option on one unit of the underlying. If a contract covers more than one unit, apply its multiplier after finding the per-unit payoff.

The order of the subtraction matters. A call compares the terminal price with the strike as \(S_T-K\); a put reverses that comparison as \(K-S_T\). The positive-part operation then prevents either holder’s expiration payoff from falling below zero.

These are payoffs, not option prices or profits. They give the amount associated with the option at expiration, before accounting for any premium paid or received earlier. They also do not require an interest rate or a probability model: once \(S_T\) and \(K\) are specified, the payoff is determined directly.

Why the Formulas Hold

At expiration, the call’s exercise amount per unit is the terminal asset price less the strike. If that difference is positive, the payoff is \(S_T-K\); if it is zero or negative, the holder’s expiration payoff is zero. The two cases combine as follows:

For a put, the corresponding difference is the strike less the terminal asset price. A positive difference gives the put’s payoff; a zero or negative difference gives a payoff of zero:

These case definitions and maximum expressions agree in every region, including the boundary \(S_T=K\), where both payoffs equal zero. This justification concerns the expiration payoff only; it does not value an option before expiration.

Reading the Formulas Across the Strike

The strike divides possible terminal prices into three cases. Below the strike, the put has a positive payoff and the call has zero payoff. At the strike, both have zero payoff. Above the strike, the call has a positive payoff and the put has zero payoff. This is a direct consequence of the signs of \(S_T-K\) and \(K-S_T\); it does not require a separate pricing model.

The table is a useful check on any substitution into the formulas. For instance, a proposed positive call payoff when \(S_T<K\) signals either a subtraction error or an error applying the maximum with zero.

From Per-Unit Payoff to Contract Amount

The positive-part formulas first produce an amount per unit of the underlying. A contract multiplier converts that amount to the payoff for one contract. If there are multiple contracts, multiply again by the number of contracts. Keep these steps separate so the strike and terminal price are compared in their quoted units before applying the multiplier.

Common Mistakes and Exam Checks

• Forgetting the zero floor. If \(S_T-K<0\), the call payoff is not a negative amount; it is \((S_T-K)^+=0\). Likewise, a put payoff cannot be negative.
• Reversing the subtraction. Use \(S_T-K\) for a call and \(K-S_T\) for a put. A quick check is to ask which option should have a positive payoff for the stated price comparison.
• Replacing the positive part with an absolute value. The expression \(|S_T-K|\) is generally not either option’s payoff, because it is positive on both sides of the strike.
• Confusing payoff with profit. The formulas here contain only \(S_T\) and \(K\). They do not subtract an earlier premium or include financing of that premium.
• Applying a multiplier to the wrong quantity. First calculate the price difference and take its positive part in the underlying’s quoted units; then convert using the contract multiplier.

A reliable calculation therefore follows the same order each time: identify whether the option is a call or a put, write the correct difference, take the maximum with zero, and only then apply any contract multiplier or contract count. This procedure is aligned with the SOA Exam FM (Financial Mathematics) syllabus.