Payoff and Profit Are Not the Same

Option payoff describes the value generated by exercise at expiration. Profit adjusts that payoff for the premium paid or received. Ignoring the premium can therefore produce a misleading picture of the economics of an option trade.

Long Call

A long call gives the holder the right, but not the obligation, to buy the underlying asset at the strike price. In the formulas below, \(S_T\) represents the price of the underlying asset at expiration, \(K\) represents the strike price, and \(C\) represents the call option premium paid by the buyer.

The payoff of a long call at expiration is:

\[
\text{Payoff}_{\text{Long Call}} = \max(S_T-K,0)
\]

The payoff does not include the premium paid to purchase the option. To calculate the actual profit or loss at expiration, the call premium \(C\) must also be taken into account:

\[
\text{Profit}_{\text{Long Call}} = \max(S_T-K,0)-C
\]

If \(S_T\) is equal to or below \(K\), the option expires worthless and the holder loses the premium paid. Therefore, the maximum loss of a long call is limited to \(C\).

If \(S_T\) rises above \(K\), the option begins to have a positive payoff. Since the underlying price can theoretically continue to rise without an upper limit, the potential profit from a long call is theoretically unlimited.

At expiration, the position reaches breakeven when the underlying price equals the strike price plus the premium paid:

\[
S_T = K + C
\]

For example, if the strike price is $100 and the call premium is $5, the breakeven price at expiration is $105. Above $105 the position generates a profit, while below $105 it generates a loss, with the maximum loss limited to the $5 premium.

Short Call

A short call is created by selling a call option. The seller receives the option premium but assumes the obligation to sell the underlying asset at the strike price if the option is exercised. In the formulas below, \(S_T\) represents the price of the underlying asset at expiration, \(K\) represents the strike price, and \(C\) represents the call option premium received by the seller.

The payoff of a short call at expiration, before considering the premium received, is:

\[
\text{Payoff}_{\text{Short Call}} = -\max(S_T-K,0)
\]

To calculate the actual profit or loss at expiration, the premium \(C\) received when the option was sold must also be included:

\[
\text{Profit}_{\text{Short Call}} = C-\max(S_T-K,0)
\]

If \(S_T\) is equal to or below \(K\), the option expires worthless and the seller keeps the entire premium. Therefore, the maximum profit of a short call is limited to the premium received:

\[
\text{Maximum Profit} = C
\]

If \(S_T\) rises above \(K\), the seller begins to incur losses as the value of the call payoff increases. Because the underlying price has no fixed upper limit, an uncovered short call has theoretically unlimited loss potential.

The breakeven price at expiration is the strike price plus the premium received:

\[
S_T = K + C
\]

For example, if a call with a strike price of $100 is sold for a premium of $5, the breakeven price is $105. If the underlying finishes at or below $100, the seller keeps the full $5 premium. Above $105, the position generates a net loss that increases as the underlying price rises.

Long Put

A long put gives the holder the right, but not the obligation, to sell the underlying asset at the strike price. In the formulas below, \(S_T\) represents the price of the underlying asset at expiration, \(K\) represents the strike price, and \(P_0\) represents the put option premium paid by the buyer.

The payoff of a long put at expiration is:

\[
\text{Payoff}_{\text{Long Put}} = \max(K-S_T,0)
\]

The payoff does not include the premium paid to purchase the option. To calculate the actual profit or loss at expiration, the put premium \(P_0\) must also be taken into account:

\[
\text{Profit}_{\text{Long Put}} = \max(K-S_T,0)-P_0
\]

If \(S_T\) is equal to or above \(K\), the option expires worthless and the holder loses the premium paid. Therefore, the maximum loss of a long put is limited to \(P_0\).

The position benefits as the price of the underlying asset falls below the strike price. Because the lowest possible price of the underlying is zero, the maximum payoff occurs when \(S_T=0\):

\[
\text{Maximum Payoff} = K
\]

After accounting for the premium paid, the maximum profit is:

\[
\text{Maximum Profit} = K-P_0
\]

The breakeven price at expiration is the strike price minus the premium paid:

\[
S_T = K-P_0
\]

For example, if a put with a strike price of $100 is purchased for a premium of $5, the breakeven price is $95. Below $95, the position generates a net profit. If the underlying finishes at or above $100, the option expires worthless and the maximum loss is limited to the $5 premium.

Short Put

A short put is created by selling a put option. The seller receives the option premium but assumes the obligation to buy the underlying asset at the strike price if the option is exercised. In the formulas below, \(S_T\) represents the price of the underlying asset at expiration, \(K\) represents the strike price, and \(P_0\) represents the put option premium received by the seller.

The payoff of a short put at expiration, before considering the premium received, is:

\[
\text{Payoff}_{\text{Short Put}} = -\max(K-S_T,0)
\]

To calculate the actual profit or loss at expiration, the premium \(P_0\) received when the option was sold must also be included:

\[
\text{Profit}_{\text{Short Put}} = P_0-\max(K-S_T,0)
\]

If \(S_T\) is equal to or above \(K\), the option expires worthless and the seller keeps the entire premium. Therefore, the maximum profit of a short put is limited to the premium received:

\[
\text{Maximum Profit} = P_0
\]

If \(S_T\) falls below \(K\), the seller begins to incur losses because the underlying may have to be purchased at the strike price \(K\), even though its market value is lower.

The maximum loss occurs if the underlying price falls to zero. In that case, the seller must effectively buy an asset worth zero for the strike price, partially offset by the premium received:

\[
\text{Maximum Loss} = K-P_0
\]

The breakeven price at expiration is the strike price minus the premium received:

\[
S_T = K-P_0
\]

For example, if a put with a strike price of $100 is sold for a premium of $5, the breakeven price is $95. If the underlying finishes at or above $100, the seller keeps the full $5 premium. Below $95, the position generates a net loss, with the maximum loss occurring if the underlying price falls to zero.

Numerical Example

Assume a stock is currently $100 and a one-month call with strike $105 costs $3. If the stock finishes at $115, the option payoff is $10 and profit is $7. If the stock finishes at $103, the call expires worthless and the loss is the $3 premium. The breakeven is $108.

Why Payoff Diagrams Matter

Payoff diagrams make nonlinear risk intuitive. They show where profits begin, where losses are capped and where risk can expand rapidly. For multi-leg option strategies, combining the individual payoff functions provides a direct view of the strategy’s expiration profile.

Key Takeaways

The four basic option positions form the building blocks of more complex strategies. Long options exchange a premium for asymmetric upside or protection. Short options collect premium but accept potentially large obligations. Always distinguish payoff from profit and include the premium when calculating breakeven.

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