Option Basics - Calls, Puts, and Payoffs › Option Payoffs at Expiration

The payoff of an option is the value received upon exercise, ignoring the initial premium paid. Payoffs are the starting point for understanding option value because they define what the option delivers at expiration. By analyzing payoffs first, we establish the terminal conditions that any pricing model must satisfy. The premium you pay today must be justified by the expected value of these future payoffs, adjusted for time and risk.

At expiration, a call option's payoff depends on whether the underlying price $S_T$ exceeds the strike price $K$. The relationship is binary in nature: either the option is worth exercising or it isn't. This creates the characteristic hockey stick payoff shape that defines call options.

$$\text{Call Payoff} = \max(S_T - K, 0) = (S_T - K)^+$$

where:

• $S_T$: price of the underlying asset at expiration
• $K$: strike price
• $(x)^+$: notation for $\max(x, 0)$

The notation $(x)^+$ means "the positive part of $x$," equivalent to $\max(x, 0)$. This mathematical shorthand appears frequently in derivatives literature and captures the essential asymmetry of options in a compact form.

The logic is straightforward and follows directly from your rational self-interest:

• If $S_T > K$: Exercise the call, buy at $K$, and own stock worth $S_T$. Payoff = $S_T - K$. You pay the strike price to acquire stock worth more than that amount, pocketing the difference as profit.
• If $S_T \leq K$: Don't exercise. You wouldn't pay $K$ for stock worth less than $K$. Payoff = 0. The option expires worthless, but you have lost nothing beyond the premium already paid.

Notice that the payoff function is piecewise linear. Below the strike, it's flat at zero. Above the strike, it increases dollar-for-dollar with the stock price. This kink at the strike price is what creates the asymmetric risk profile that makes options so useful for constructing targeted exposures.

A put option's payoff is the mirror image of the call, reflecting its role as the right to sell rather than buy:

$$\text{Put Payoff} = \max(K - S_T, 0) = (K - S_T)^+$$

where:

• $S_T$: price of the underlying asset at expiration
• $K$: strike price

The reasoning follows the same logical structure as the call but in reverse:

• If $S_T < K$: Exercise the put, sell at $K$ what's worth only $S_T$. Payoff = $K - S_T$. You deliver stock worth less than the strike price and receive the full strike price in return.
• If $S_T \geq K$: Don't exercise. You wouldn't sell at $K$ what you could sell for $S_T$ or more. Payoff = 0. There's no point taking the strike price when the market offers more.

The put's payoff function is also piecewise linear, but the kink works in the opposite direction. Above the strike, the payoff is flat at zero. Below the strike, the payoff increases as the stock price falls, providing protection against declining prices.

Payoff differs from profit in an important way. Profit accounts for the initial premium paid or received. While payoff tells you what the option delivers at expiration, profit tells you whether the overall transaction made money. An option can have a positive payoff but still result in a loss if the payoff doesn't exceed the premium paid.

For a long call and long put respectively:

$$\begin{aligned} \text{Profit}_{\text{long call}} &= \max(S_T - K, 0) - C \\ \text{Profit}_{\text{long put}} &= \max(K - S_T, 0) - P \end{aligned}$$

where:

• $S_T$: stock price at expiration
• $K$: strike price
• $C$: call option premium paid
• $P$: put option premium paid

The premium appears as a subtraction because you pay this amount upfront to acquire the option. For a long call to be profitable, the stock must rise above the strike plus the premium paid. This defines the breakeven point, the price at which the trade neither makes nor loses money.

Short positions have the opposite profit profile of long positions:

$$\begin{aligned} \text{Profit}_{\text{short call}} &= C - \max(S_T - K, 0) \\ \text{Profit}_{\text{short put}} &= P - \max(K - S_T, 0) \end{aligned}$$

where:

• $S_T$: stock price at expiration
• $K$: strike price
• $C$: call option premium received
• $P$: put option premium received

The profit formulas for short positions show the premium as an addition because you receive this amount upfront. Options are a zero-sum game: what one party gains, the other loses. You achieve your maximum profit when the option expires worthless, but you face significant exposure when large price moves occur.

Let's visualize these payoff and profit functions: