How Strike Changes the Payoff Diagram

Why the Strike Changes the Payoff Diagram

The strike determines the price at which an option holder may exercise. Changing the strike therefore changes both where the payoff diagram bends and, at some terminal prices, the amount paid at expiration. To compare two diagrams cleanly, hold the option type and position fixed and change only the strike. This isolates the effect of the strike from other features of the contract.

The comparisons here concern expiration payoff per option, not profit. Premiums are not included. Assume a European option on one underlying asset, with strike $K>0$, expiration time $T$, and terminal asset price $S_T\geq 0$. Ignore transaction costs and any settlement complications. The payoff formulas describe the cash flow at expiration; they do not require assumptions about the asset’s price path before expiration.

Definition: For one option at expiration, a long call has payoff $\max(S_T-K,0)$, and a written call has payoff $-\max(S_T-K,0)$. A long put has payoff $\max(K-S_T,0)$, and a written put has payoff $-\max(K-S_T,0)$. The strike is the kink: the point where the payoff’s slope changes.

A call holder benefits from a higher terminal asset price, but receives no call payoff when $S_T\leq K$. A put holder benefits from a lower terminal asset price, but receives no put payoff when $S_T\geq K$. Thus, raising the strike moves the kink to the right on the horizontal price axis for either option type. It does not move the kink in the same way as a change in terminal price: the strike changes the contract, while $S_T$ identifies the point at which that contract is evaluated.

Comparing Two Strikes

Let $K_L<K_H$, so $K_L$ is the lower strike and $K_H$ is the higher strike. Compare payoffs in three regions: below both strikes, between the strikes, and above both strikes. This partition matters because the option’s exercise status can differ only in the middle region.

Mark both strikes.
Place $K_L$ and $K_H$ on the terminal-price axis, with $K_L$ to the left of $K_H$.

Identify each option’s exercise region.
A call has positive holder payoff above its own strike; a put has positive holder payoff below its own strike.

Evaluate each payoff separately.
Use the same $S_T$ in both formulas. Do not assume both options are either exercised or unexercised when $S_T$ lies between the strikes.

Reverse signs for the writer.
A written option’s payoff is the negative of the corresponding long option’s payoff, so every comparison reverses.

For long calls, the lower strike never gives a smaller payoff than the higher strike. For long puts, the higher strike never gives a smaller payoff than the lower strike. The written positions reverse those inequalities. The size of the difference depends on the region, as the following result shows.

Result: If $K_L<K_H$, then the difference between the long-call payoff at $K_L$ and the long-call payoff at $K_H$ is $$ \max(S_T-K_L,0)-\max(S_T-K_H,0) = \begin{cases} 0, & S_T\leq K_L,\\ S_T-K_L, & K_L<S_T\leq K_H,\\ K_H-K_L, & S_T>K_H. \end{cases} $$ The difference between the long-put payoff at $K_H$ and the long-put payoff at $K_L$ is $$ \max(K_H-S_T,0)-\max(K_L-S_T,0) = \begin{cases} K_H-K_L, & S_T\leq K_L,\\ K_H-S_T, & K_L<S_T<K_H,\\ 0, & S_T\geq K_H. \end{cases} $$ For written options, the corresponding differences have the opposite sign.

Why the Payoff Differences Have These Shapes

The formulas follow directly by evaluating each option in each price region. For calls, if $S_T\leq K_L$, then $S_T$ is below both strikes, so both call payoffs are zero. If $K_L<S_T\leq K_H$, the lower-strike call pays $S_T-K_L$, while the higher-strike call pays zero. If $S_T>K_H$, both calls pay a positive amount, and subtracting the higher-strike payoff from the lower-strike payoff gives a constant:

$$ (S_T-K_L)-(S_T-K_H)=K_H-K_L. $$

For puts, when $S_T\leq K_L$, both options have positive payoff. Subtracting the lower-strike payoff from the higher-strike payoff gives $K_H-K_L$. Between the strikes, the higher-strike put pays $K_H-S_T$, while the lower-strike put pays zero. Above both strikes, both put payoffs are zero. These cases establish the stated piecewise formulas. A written option has the negative of the long payoff by definition, so its payoff difference is the negative of the corresponding long-position difference.

The result also explains the diagram. A long-call payoff with the lower strike begins rising earlier; once the terminal price is above both strikes, its vertical advantage is the fixed strike gap $K_H-K_L$. A long-put payoff with the higher strike starts declining later as $S_T$ rises, and its advantage is largest below both strikes. For written positions, the vertical ordering reverses, but the strike locations of the kinks do not change.

Worked Example: Comparing Long Calls Below, Between, and Above the Strikes

A hypothetical investor holds one long call with strike $K_L=\$40$ and compares it with one otherwise identical long call with strike $K_H=\$50$. Calculate both expiration payoffs at terminal prices $S_T=\$35$, $\$45$, and $\$60$. Identify the kink locations and check the strike comparison.

Method. Apply the long-call payoff $\max(S_T-K,0)$ separately for each strike. The strikes are $\$40$ and $\$50$, so the selected terminal prices fall below both strikes, between them, and above both.

At $S_T=\$35$, the strike-$\$40$ call pays $\max(35-40,0)=\max(-5,0)=\$0$. The strike-$\$50$ call pays $\max(35-50,0)=\max(-15,0)=\$0$. The difference is $\$0-\$0=\$0$.

At $S_T=\$45$, the strike-$\$40$ call pays $\max(45-40,0)=\$5$. The strike-$\$50$ call pays $\max(45-50,0)=\max(-5,0)=\$0$. The lower-strike call pays $\$5$ more.

At $S_T=\$60$, the strike-$\$40$ call pays $\max(60-40,0)=\$20$. The strike-$\$50$ call pays $\max(60-50,0)=\$10$. The difference is $\$20-\$10=\$10$, equal to the strike gap $\$50-\$40=\$10$.

The payoff kinks are at $\$40$ and $\$50$, respectively. The results agree with the comparison formula: the lower-strike call has no advantage below both strikes, a growing advantage between them, and a constant $\$10$ advantage above both. This check avoids treating the strike gap as the payoff difference at every terminal price.

Worked Example: Comparing Written Puts at a Price Between the Strikes

A hypothetical trader has written one put with strike $K_H=\$72$. Compare its payoff with the payoff from writing an otherwise identical put with strike $K_L=\$64$, when $S_T=\$60$, $\$68$, and $\$75$. State which written put has the more negative payoff in each case.

Method. A written put’s payoff is $-\max(K-S_T,0)$. Calculate the long-put amount first and then apply the negative sign. This is especially important between the strikes, where only the higher-strike put has a positive holder payoff.

At $S_T=\$60$, the written $\$72$ put pays $-\max(72-60,0)=-\$12$. The written $\$64$ put pays $-\max(64-60,0)=-\$4$. Thus the higher-strike written put has the more negative payoff by $\$12-\$4=\$8$, which equals the strike gap.

At $S_T=\$68$, the written $\$72$ put pays $-\max(72-68,0)=-\$4$. The written $\$64$ put pays $-\max(64-68,0)=-\$0=\$0$. The higher-strike written put is again more negative, now by $\$4$.

At $S_T=\$75$, both strikes are below the terminal price. The written $\$72$ put pays $-\max(72-75,0)=\$0$, and the written $\$64$ put pays $-\max(64-75,0)=\$0$. Neither has a negative expiration payoff.

The kinks are at $\$64$ and $\$72$. Below both strikes, the difference between the written put payoffs is $-\$8$; between the strikes it increases from $-\$8$ to zero; above both strikes it is zero. The sign check confirms that the higher-strike written put, not the lower-strike one, has the more negative payoff in the first two cases.

Worked Example: Comparing Written Calls Above Both Strikes

Compare one written call with strike $K_L=\$95$ and one written call with strike $K_H=\$105$, evaluated separately at $S_T=\$90$, $\$100$, and $\$115$. Find each payoff and explain how the lower strike changes the writer’s payoff.

Method. Use $-\max(S_T-K,0)$ for each written call. The lower strike is exercised first as the terminal price rises, so the writer’s payoff becomes negative sooner.

At $S_T=\$90$, the written $\$95$ call pays $-\max(90-95,0)=\$0$, and the written $\$105$ call pays $-\max(90-105,0)=\$0$.

At $S_T=\$100$, the written $\$95$ call pays $-\max(100-95,0)=-\$5$. The written $\$105$ call pays $-\max(100-105,0)=\$0$. The lower-strike writer has a payoff $\$5$ lower.

At $S_T=\$115$, the written $\$95$ call pays $-\max(115-95,0)=-\$20$, while the written $\$105$ call pays $-\max(115-105,0)=-\$10$. The difference is $-\$20-(-\$10)=-\$10$, the negative of the strike gap.

The kinks remain at $\$95$ and $\$105$. Below both, both writer payoffs are zero; between them, only the lower-strike call has a negative payoff; above both, the lower-strike writer’s payoff is lower by the fixed $\$10$ strike gap. These results check the reversal from the long-call comparison.

Common Mistake: Applying the Strike Gap Everywhere

A frequent error is to say that two options with strikes $\$40$ and $\$50$ always differ in payoff by $\$10$. That is true for long calls only when $S_T$ is above both strikes; it is false below the strikes and generally false between them. For example, at $S_T=\$45$, the long-call payoffs are $\$5$ and $\$0$, a difference of $\$5$, not $\$10$. At $S_T=\$35$, both payoffs are zero, so the difference is zero.

Another trap is to keep the long-position ordering for a written option. With a written call, a lower strike produces an equal or more negative payoff, not an equal or more positive one. Calculate the long payoff and explicitly reverse its sign, or use the written payoff formula directly. Finally, do not include a premium in these calculations: premium affects profit, whereas this comparison concerns expiration payoff only.

Key takeaway: Changing the strike moves the kink to the new strike and can change payoff at a fixed terminal price. For long calls, a lower strike gives an equal or greater payoff; for long puts, a higher strike gives an equal or greater payoff. The ordering reverses for written options. These payoff comparisons are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each question, compare expiration payoffs per option. Do not include premiums.

Two long calls have strikes $\$36$ and $\$44$. At $S_T=\$40$, calculate both payoffs and their difference. (Answer: $\$4$ for the $\$36$ call; $\$0$ for the $\$44$ call; the lower-strike call pays $\$4$ more.)
Two long puts have strikes $\$58$ and $\$66$. At $S_T=\$62$, calculate both payoffs. (Answer: $\$0$ for the $\$58$ put; $\$4$ for the $\$66$ put.)
Two written puts have strikes $\$80$ and $\$74$. At $S_T=\$70$, calculate both payoffs and identify the more negative payoff. (Answer: $-\$10$ for the $\$80$ put and $-\$4$ for the $\$74$ put; the $\$80$ written put is more negative.)
Two written calls have strikes $\$51$ and $\$59$. At $S_T=\$55$, calculate both payoffs and identify the kink locations. (Answer: $-\$4$ for the $\$51$ call and $\$0$ for the $\$59$ call; the kinks are at $\$51$ and $\$59$.)
Two long calls have strikes $\$25$ and $\$30$. Calculate both payoffs at $S_T=\$20$, $\$28$, and $\$35$, and describe how the difference changes across the regions. (Answer: at $\$20$, $\$0$ and $\$0$; at $\$28$, $\$3$ and $\$0$; at $\$35$, $\$10$ and $\$5$. The lower-strike call’s advantage is $\$0$, then $\$3$, then the constant strike gap of $\$5$.)

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