How Premium Changes the Profit Diagram

Why Premium Changes Profit but Not Payoff

An option premium is paid or received when a position is initiated. The option’s payoff, by contrast, is determined by the option contract and the underlying asset’s value at expiration. Keeping those two amounts separate is essential when comparing profit diagrams: changing the premium changes the position’s profit, but it does not change what the option contract pays at expiration.

This tutorial uses the standard expiration profit-diagram convention: profit is the expiration payoff minus the initial premium for a long position, or the initial premium minus the expiration payoff for a written position. The convention does not accumulate or discount the premium for interest, and excludes transaction costs, taxes, and other cash flows. The comparisons assume the option type, underlying, strike, and expiration are the same; only the premium changes. If a problem includes financing or other cash flows, those must be incorporated separately.

Definition: Let $S_T$ be the underlying asset price at expiration, $K$ the strike price, $c$ the premium of a call, and $p$ the premium of a put. For one option unit, a call’s expiration payoff is $\max(S_T-K,0)$, and a put’s expiration payoff is $\max(K-S_T,0)$. A long position pays the premium and has profit equal to payoff minus premium. A written position receives the premium and has profit equal to premium minus payoff.

A “higher-premium option” in the comparisons below means an otherwise identical contract offered at a higher initial premium. That change affects the initial cash flow, not the strike or the exercise decision at expiration.

The Vertical Shift in a Profit Diagram

Suppose the premium for a call increases from $c_1$ to $c_2$, with $c_2=c_1+\Delta$ and $\Delta>0$. For a long call, the payoff is $\max(S_T-K,0)$ under either premium. Subtracting the two profit expressions gives a constant difference at every expiration price:

\begin{aligned} \Pi_{\text{long call},2}(S_T)-\Pi_{\text{long call},1}(S_T) &=\bigl[\max(S_T-K,0)-c_2\bigr] -\bigl[\max(S_T-K,0)-c_1\bigr]\\ &=-(c_2-c_1)=-\Delta. \end{aligned}

Thus the higher-premium long-call profit diagram is shifted down by $\Delta$. A written call has profit $c-\max(S_T-K,0)$, so increasing its premium by $\Delta$ shifts its profit diagram up by $\Delta$. The same directions apply to puts: a higher premium shifts a long-put diagram down and a written-put diagram up. In every case, the option payoff itself is unchanged.

Result: For otherwise identical options, changing the premium by $\Delta$ changes profit by a constant amount at every $S_T$. A long position’s profit changes by $-\Delta$; a written position’s profit changes by $+\Delta$. The expiration payoff is unchanged because it depends on $S_T$ and $K$, not on the premium.

This conclusion is pointwise: choose any expiration price, and the profit difference is the same. The kink at the strike stays at $K$, and the slope on each side of the strike stays the same. Only the vertical position of the profit graph changes.

Break-Even Moves When Premium Changes

Break-even is the expiration price at which profit equals zero. Solving the profit equation shows how a premium change moves that price. For a long call, profit is zero when the call is in the money and $S_T-K-c=0$, giving $S_T=K+c$. A higher premium therefore moves the long-call break-even price upward by the premium increase.

For a long put, the in-the-money zero-profit equation is $K-S_T-p=0$, so $S_T=K-p$. Increasing the premium moves this break-even price downward. For written calls and puts, the profit equations are the negatives of the corresponding long-option profits, so they have the same break-even prices: $K+c$ for a written call and $K-p$ for a written put. These statements presume a break-even price is feasible. In particular, $K-p$ is not a nonnegative stock price if $p>K$.

\begin{array}{c|c|c} \text{Position} & \text{Profit at expiration} & \text{Break-even }S_T\\ \hline \text{Long call} & \max(S_T-K,0)-c & K+c\\ \text{Written call} & c-\max(S_T-K,0) & K+c\\ \text{Long put} & \max(K-S_T,0)-p & K-p\\ \text{Written put} & p-\max(K-S_T,0) & K-p \end{array}

For a long call, a higher premium lowers profit at every price and raises the zero-profit threshold. For a long put, it also lowers profit at every price, but the break-even price moves down because a lower $S_T$ is needed to produce a larger put payoff. For written positions, the profit moves upward, but the break-even shifts in the same direction as for the corresponding long position: the zero-profit equation is unchanged when both sides of a long position’s profit equation are multiplied by $-1$.

Worked Examples: Comparing Premiums

Worked Example: A Higher Premium on a Long Call

A hypothetical call on an asset has strike $K=\$50$. Compare a long position with premium $c_1=\$6$ to an otherwise identical long position with premium $c_2=\$9$. Find both profits at $S_T=\$44,\$50,\$56,\$65$, and compare the break-even prices. Use the expiration profit convention defined above.

Method. Compute the common call payoff $\max(S_T-50,0)$ at each price, then subtract each premium. The premium difference is $\Delta=\$9-\$6=\$3$.

At $S_T=\$44$, the payoff is $\max(44-50,0)=\$0$. Profits are $0-6=-\$6$ and $0-9=-\$9$.

At $S_T=\$50$, the payoff is $\max(50-50,0)=\$0$. Profits are $0-6=-\$6$ and $0-9=-\$9$.

At $S_T=\$56$, the payoff is $\max(56-50,0)=\$6$. Profits are $6-6=\$0$ and $6-9=-\$3$.

At $S_T=\$65$, the payoff is $\max(65-50,0)=\$15$. Profits are $15-6=\$9$ and $15-9=\$6$.

$S_T$	Call payoff	Profit with $c_1=\$6$	Profit with $c_2=\$9$
\$44	\$0	-\$6	-\$9
\$50	\$0	-\$6	-\$9
\$56	\$6	\$0	-\$3
\$65	\$15	\$9	\$6

The break-even prices are $K+c_1=50+6=\$56$ and $K+c_2=50+9=\$59$. At each listed price, the higher-premium profit is exactly $\$3$ lower, while the call payoff is identical. At $S_T=\$56$, the first position breaks even and the second loses $\$3$, which checks the calculation. This avoids the common mistake of treating the higher premium as a change to the strike.

Worked Example: A Higher Premium on a Long Put

A hypothetical put has strike $K=\$80$. Compare long positions costing $p_1=\$7$ and $p_2=\$11$. Calculate their profits at $S_T=\$60,\$69,\$80,\$92$, and determine how the break-even changes.

Method. Find the common put payoff $\max(80-S_T,0)$, then subtract each premium. The premium increase is $\Delta=11-7=\$4$.

At $S_T=\$60$, payoff $=\max(80-60,0)=\$20$. Profits are $20-7=\$13$ and $20-11=\$9$.

At $S_T=\$69$, payoff $=\max(80-69,0)=\$11$. Profits are $11-7=\$4$ and $11-11=\$0$.

At $S_T=\$80$, payoff $=\max(80-80,0)=\$0$. Profits are $0-7=-\$7$ and $0-11=-\$11$.

At $S_T=\$92$, payoff $=\max(80-92,0)=\$0$. Profits remain $-\$7$ and $-\$11$.

The break-even prices are $80-7=\$73$ and $80-11=\$69$. The higher-premium profit is $\$4$ lower at every price, and its break-even is $\$4$ lower. At $S_T=\$69$, the higher-premium put’s payoff is $\$11$, exactly its premium, confirming zero profit. The strike remains $\$80$; only the profit level and break-even change.

Worked Example: A Higher Premium on a Written Call

A hypothetical call has strike $K=\$40$. Compare written positions that receive $c_1=\$3$ and $c_2=\$5$. Calculate profit at $S_T=\$35,\$43,\$45,\$52$, and compare the profit shift and break-even prices.

Method. For each price, calculate the call payoff and subtract it from the premium received. The premium increase is $\Delta=5-3=\$2$.

At $S_T=\$35$, payoff $=\max(35-40,0)=\$0$. Written profits are $3-0=\$3$ and $5-0=\$5$.

At $S_T=\$43$, payoff $=\max(43-40,0)=\$3$. Written profits are $3-3=\$0$ and $5-3=\$2$.

At $S_T=\$45$, payoff $=\max(45-40,0)=\$5$. Written profits are $3-5=-\$2$ and $5-5=\$0$.

At $S_T=\$52$, payoff $=\max(52-40,0)=\$12$. Written profits are $3-12=-\$9$ and $5-12=-\$7$.

The break-even prices are $40+3=\$43$ and $40+5=\$45$. The higher premium raises the writer’s profit by $\$2$ at every price, and moves the break-even upward by $\$2$. At $S_T=\$45$, the call payoff and higher premium are both $\$5$, so that written position breaks even. The same payoff values apply to both contracts because only the premium differs.

Common Trap: Moving the Strike Instead of the Profit

A premium is not part of the option’s expiration payoff formula. For example, with $K=\$50$, $c=\$6$, and $S_T=\$55$, the call payoff is $\max(55-50,0)=\$5$, while the long-call profit is $5-6=-\$1$. A tempting but incorrect calculation is to replace the strike by $K+c=\$56$, giving the supposed payoff $\max(55-56,0)=\$0$. That expression is not the contractual payoff; it confuses a break-even threshold with the strike. The correction is to keep the payoff calculation at $K=\$50$, and then subtract the premium to obtain profit.

When comparing two premiums, calculate the payoff once, using the unchanged strike, and then apply each premium separately. This makes both the invariant payoff and the constant profit shift visible. It also prevents a related error: assuming that a higher premium moves the kink in the graph. The kink remains at $S_T=K$; the graph shifts vertically.

Check Your Understanding

For each problem, distinguish the expiration payoff from profit and use the stated premium convention.

A long call has strike $\$30$. Compare premiums of $\$4$ and $\$7$. At $S_T=\$36$, find the common payoff and both profits, then give both break-even prices. (Answer: Payoff $\$6$; profits $\$2$ and $-\$1$; break-even prices $\$34$ and $\$37$.)
A long put has strike $\$65$. Its premium increases from $\$5$ to $\$9$. Find the profit shift at every expiration price and both break-even prices. (Answer: Profit shifts down by $\$4$; break-even prices are $\$60$ and $\$56$.)
A written put has strike $\$42$. Compare premiums of $\$3$ and $\$6$. At $S_T=\$38$, compute the payoff and each writer’s profit, then find both break-even prices. (Answer: Payoff $\$4$; profits $-\$1$ and $\$2$; break-even prices $\$39$ and $\$36$.)
A call with strike $\$75$ has premiums of $\$8$ and $\$12$. For each premium, state the payoff and profit at $S_T=\$70$ and $S_T=\$90$, and explain what happens to the kink. (Answer: Payoffs are $\$0$ and $\$15$; profits are $-\$8,\$7$ for the $\$8$ premium and $-\$12,\$3$ for the $\$12$ premium; the kink remains at $\$75$ and the higher-premium long-call diagram is $\$4$ lower.)
A written call has strike $\$55$, and its premium rises from $\$2$ to $\$5$. At $S_T=\$60$, calculate both profits and compare the two break-even prices. (Answer: The payoff is $\$5$; profits are $-\$3$ and $\$0$; break-even prices are $\$57$ and $\$60$.)

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