Mixed Call And Put Expiration Problems

Reading a Mixed Option Position

A portfolio that includes both calls and puts can have several strike prices, position directions, and premiums. The reliable method is to calculate each option leg separately at expiration, add the leg payoffs, and then account for the initial premiums to obtain profit. This avoids treating a portfolio as though it were one option with one strike.

For a call, the holder may buy the underlying asset for the strike price $K$. For a put, the holder may sell it for $K$. At expiration, the underlying price is $S_T$. A long option has a nonnegative payoff to its holder; a short option has the equal-and-opposite payoff to its writer. The option’s premium is paid initially by the long position and received initially by the short position.

Definition: At expiration, the per-unit payoff of a long call with strike $K$ is $\max(S_T-K,0)$, and the payoff of a long put is $\max(K-S_T,0)$. A short position has the negative of the corresponding long payoff. Profit equals the position’s expiration payoff minus its initial net premium cost. These formulas exclude fees, taxes, and interest earned or paid on the premium.

Use a signed position quantity $q$: $q>0$ means long and $q<0$ means short. If $c$ and $p$ are the call and put premiums per unit, the combined payoff of call and put positions is

\text{Payoff at }S_T =\sum_j q_j\max(S_T-K_j,0) +\sum_k q_k\max(K_k-S_T,0).

The signed sum of premiums is the initial net cost: a positive value is paid and a negative value is received. Therefore,

\text{Profit at }S_T =\text{Payoff at }S_T-\text{initial net premium cost}.

If each option contract covers $M$ units of the underlying, multiply the per-unit payoff and profit by $M$ for one contract. When there are multiple contracts, include their signed quantities as well. Keep contract size separate from the per-unit calculation until the final scaling step.

A Reliable Calculation and Sketching Method

A profit diagram plots expiration profit vertically against $S_T$ horizontally. Each option leg changes its formula at its strike, so the strikes are the diagram’s potential kinks. Between consecutive strikes, every leg is either active or inactive, and the total profit is a straight line. A useful sketch therefore comes from identifying the strikes, calculating values at those strikes, and finding the slope in each interval.

List the legs.
Record for each call or put its long or short direction, strike, premium, and number of contracts.

Calculate expiration payoffs.
At each requested $S_T$, apply the call or put maximum formula to each leg, then apply its signed quantity.

Convert payoff to profit.
Subtract the initial net premium cost. A premium received reduces that cost; it is not an extra expiration payoff.

Build the sketch.
Mark the strikes, compute the profit at the strikes and in relevant regions, and connect the points with straight segments having the correct slopes.

The slope gives a quick check. Above a call’s strike, a long call contributes slope $+1$ per unit and a short call contributes $-1$; below that strike, either call contributes zero. Below a put’s strike, a long put contributes slope $-1$ as $S_T$ rises, while a short put contributes $+1$; above the strike, either put contributes zero. Multiply these contributions by the position quantities and add them to obtain the portfolio slope.

This follows directly from the payoff formulas. For example, when $S_T>K$, a long call’s payoff is $S_T-K$, whose change for a one-unit increase in $S_T$ is $+1$. When $S_T<K$, that call’s payoff is zero and its slope is zero. Premiums shift the whole profit diagram vertically but do not change its slopes or kink locations.

Worked Examples

Worked Example: Long Call and Long Put at Different Strikes

A hypothetical investor buys one call with strike $K=50$ for a premium of \$4 per share and one put with strike $K=45$ for a premium of \$2 per share. Both options cover one share for this calculation. Find the combined payoff and profit at $S_T=35,45,50,60$, then describe the profit diagram.

Method. Calculate the two long-option payoffs separately, add them, and subtract the total premium cost of $4+2=\$6$ per share.

At $S_T=35$, the call payoff is $\max(35-50,0)=0$. The put payoff is $\max(45-35,0)=10$. Thus total payoff is $0+10=\$10$, and profit is $10-6=\$4$.

At $S_T=45$, the call payoff is $0$ and the put payoff is $0$. Total payoff is \$0, so profit is $0-6=-\$6$.

At $S_T=50$, both payoffs are still zero: the call is exactly at its strike and the put is above its strike. Profit is therefore $-\$6$.

At $S_T=60$, the call payoff is $60-50=\$10$, and the put payoff is zero. Total payoff is \$10 and profit is $10-6=\$4$.

The kinks occur at \$45 and \$50. Below \$45, only the put is active, so profit is $45-S_T-6=39-S_T$, with slope $-1$. From \$45 through \$50, neither option has a positive payoff, so profit is constant at $-\$6$. Above \$50, only the call is active, so profit is $S_T-50-6=S_T-56$, with slope $+1$. The break-even prices are \$39 and \$56, obtained by setting the applicable profit expressions equal to zero. The computed values at \$35 and \$60 are each \$4, consistent with those break-even points.

Common mistake avoided: Adding both premiums to the payoff would incorrectly report a profit of \$16 at $S_T=35$. The \$6 premium was paid initially, so it must be subtracted from the expiration payoff.

Worked Example: Short Put and Short Call

A hypothetical trader writes one put with strike \$40 and receives a premium of \$3 per share. The trader also writes one call with strike \$60 and receives \$4 per share. Each contract covers 50 shares. Calculate profit at expiration when the underlying is \$30, \$50, and \$75, in dollars for both contracts, and describe the diagram.

Method. First calculate the signed per-share payoff of each short option. The initial premium received is $3+4=\$7$ per share, so the per-share profit equals signed payoff plus \$7.

At $S_T=\$30$, the put’s long-holder payoff would be $40-30=\$10$, so the writer’s payoff is $-\$10$. The call expires without payoff. The combined payoff is $-\$10$ per share, and profit is $-10+7=-\$3$ per share. For 50 shares, profit is $-3(50)=-\$150$.

At $S_T=\$50$, both options expire without payoff: the put is out of the money and the call is out of the money. Profit is the received premium of \$7 per share, or $7(50)=\$350$.

At $S_T=\$75$, the put expires without payoff. The call writer’s payoff is $-(75-60)=-\$15$ per share. Profit is $-15+7=-\$8$ per share, or $-8(50)=-\$400$.

The strikes, \$40 and \$60, are the kinks. Below \$40, the short put’s payoff is $S_T-40$, so profit is $S_T-40+7=S_T-33$, with slope $+1$. Between \$40 and \$60, profit is constant at \$7 per share. Above \$60, profit is $-(S_T-60)+7=67-S_T$, with slope $-1$. The break-even prices are \$33 and \$67. For a sketch, plot these break-even points, the two strike points at profit \$7 per share, and connect each interval with its stated slope.

Common mistake avoided: A short option’s payoff is negative when the option finishes in the money. The premium received is then added to that negative payoff when calculating profit; it does not reverse the sign of the writer’s obligation.

Worked Example: Unequal Quantities and a Contract Multiplier

A hypothetical position consists of two long call contracts, each with strike \$100 and premium \$5 per share, and one short put contract with strike \$90 and premium \$3 per share. Each contract covers 100 shares. Find the position’s payoff and profit at $S_T=\$80,\$90,\$100,\$110,\$120$, both per share-equivalent and for the contracts.

Method. Use signed quantities: two long calls and one short put. The initial net premium cost per share-equivalent is $2(5)-3=\$7$, or \$700 after scaling by 100. For any $S_T$, the combined payoff per share-equivalent is $2\max(S_T-100,0)-\max(90-S_T,0)$.

At \$80, the calls have zero payoff. The long put’s payoff would be $90-80=\$10$, so the short put payoff is $-\$10$. Total payoff is $-\$10$, and profit is $-10-7=-\$17$ per share-equivalent, or $-\$1,700$.

At \$90, all three options have zero payoff, so profit is $-\$7$ per share-equivalent, or $-\$700$. At \$100, the calls are at the strike and the put is above its strike; payoff remains zero and profit remains $-\$700$.

At \$110, each call pays $110-100=\$10$, giving $2(10)=\$20$; the short put has zero payoff. Profit is $20-7=\$13$ per share-equivalent, or \$1,300. At \$120, the two calls pay $2(120-100)=\$40$; the short put again pays zero. Profit is $40-7=\$33$ per share-equivalent, or \$3,300.

The two kinks are \$90 and \$100. Below \$90, the short put payoff is $S_T-90$, so profit is $S_T-97$ per share-equivalent and has slope $+1$. From \$90 to \$100, profit is constant at $-\$7$. Above \$100, the two calls contribute slope $+2$, and profit is $2(S_T-100)-7=2S_T-207$. These expressions give $-\$17$, $-\$7$, and \$13 at \$80, \$90, and \$110 respectively, checking the calculations.

Common mistake avoided: Counting two call premiums but only one call payoff, or forgetting the contract multiplier, produces an inconsistent total. Apply the signed quantity to both premium and payoff, then multiply the net per-share-equivalent result by 100.

Common Payoff and Diagram Errors

A mixed position is easiest to audit when payoff and profit are kept in separate columns. The payoff depends on $S_T$ and the option terms; profit also reflects the initial cash flow. The strike prices determine where formulas change, while premiums determine the vertical level of the profit profile.

Using the wrong option formula. A call is exercised for value only when $S_T>K$; a put is exercised for value only when $S_T<K$. Apply the maximum formula rather than assuming every option pays.
Forgetting the writer’s sign. A short position’s payoff is the negative of the corresponding long payoff. Do not calculate a positive holder payoff and report it as the writer’s payoff.
Mixing payoff and profit. A portfolio may have zero payoff at expiration but still have a loss equal to its initial net premium cost.
Missing a kink or slope change. Put every distinct strike on the horizontal axis. The portfolio slope can change at each strike, even if multiple legs have different effects there.
Applying contract size inconsistently. Scale the complete per-unit result by the contract multiplier and number of contracts, not just the payoff or just the premium.

For a diagram, a few strategically chosen values are more informative than many arbitrary points: the profit at each strike, one value in each interval, and any break-even prices. If a segment’s slope is known, its value at one point determines the line throughout that interval. Check that adjacent segment formulas agree at each strike; option payoffs and profits are continuous at expiration.

Check Your Understanding

For each problem, calculate expiration profit before sketching. Unless specified otherwise, ignore fees, taxes, and interest.

A hypothetical investor buys a put with strike \$55 for \$7 per share and a call with strike \$65 for \$4 per share. Find profit per share at $S_T=\$40,\$55,\$65,\$80$, and give the two break-even prices. (Answer: profits are \$4, $-\$11$, $-\$11$, and \$4; break-even prices are \$44 and \$76.)
A trader writes a put with strike \$30 for \$2.50 per share and a call with strike \$45 for \$3.50 per share. Each contract covers 20 shares. Find total profit at $S_T=\$25$ and $S_T=\$52$, and state the break-even prices. (Answer: \$20 at \$25 and $-\$20$ at \$52; break-even prices are \$24 and \$51.)
A position is long one call with strike \$40 and premium \$3, and short two puts, each with strike \$35 and premium \$1. Find profit per share-equivalent at $S_T=\$20,\$35,\$40,\$50$. (Answer: $-\$31,-\$1,-\$1,\$9$.)
A trader writes a call with strike \$100 for \$8 and buys a call with strike \$110 for \$3, with one share per position. Find profit at $S_T=\$100,\$107,\$110,\$120$, and give the maximum and minimum profit. (Answer: \$5, $-\$2$, $-\$5$, and $-\$5$; maximum profit is \$5 and minimum profit is $-\$5$.)
A position is long a put with strike \$75 for \$6 and short a call with strike \$90 for \$4. State the kinks and slopes of the profit profile, and calculate profit at $S_T=\$60$ and $S_T=\$100$. (Answer: kinks at \$75 and \$90; slopes are $-1,0,-1$ from left to right; profits are \$13 and $-\$12$.)

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