Long Call Profit and Break-Even

From Call Payoff to Profit

A purchased call gives its holder the right, but not the obligation, to buy an underlying asset for a specified strike price. At expiration, the holder’s payoff depends on the terminal asset price and the strike. But a positive payoff does not necessarily mean a profit: the buyer also paid a premium to acquire the option. The break-even terminal price is the price at which the expiration payoff exactly offsets that premium.

This tutorial uses the standard Exam FM convention that profit is the expiration payoff minus the premium, without accumulating the premium at an interest rate. The distinction matters: the premium is paid at the outset, whereas the payoff is determined at expiration. If a problem instead asks for terminal wealth and specifies an interest rate for financing the premium, that accumulated cost must be used.

Definition: Let $K$ be the call’s strike price, $S_T$ the underlying asset’s price at expiration $T$, and $c$ the premium paid per unit of the underlying. For one long call, the expiration payoff per unit is $\max(S_T-K,0)$. Under the standard convention of subtracting the unaccumulated premium, expiration profit per unit is $$ \Pi_T=\max(S_T-K,0)-c. $$

Assume $K>0$, $c>0$, and that the option and underlying are quoted in matching units. The call’s payoff is zero when $S_T\leq K$; when $S_T>K$, the call is exercised and its payoff is $S_T-K$. Profit subtracts the premium in either case. Therefore, the option can have a positive payoff and still make a net loss.

Deriving the Break-Even Price

To break even, profit must equal zero. Below or at the strike, the payoff is zero, so profit is $-c$, which is negative when the premium is positive. Above the strike, the payoff is $S_T-K$. Setting that payoff equal to the premium gives the break-even condition:

S_T-K-c=0 \quad\Longrightarrow\quad S_T=K+c.

Thus the break-even terminal price is the strike plus the premium per unit. This derivation also identifies the profit and loss regions: below $K+c$, profit is negative; at $K+c$, profit is zero; and above $K+c$, profit is positive. In the interval from $K$ to $K+c$, the call is exercised, but its payoff is not yet large enough to recover the premium.

Theorem: For a long call with positive premium $c$, the unique break-even terminal price under the standard profit convention is $K+c$. If $S_T<K+c$, the position has a loss; if $S_T=K+c$, it breaks even; and if $S_T>K+c$, it earns a profit.

The result follows directly from the payoff, not from whether the option is exercised. Exercise begins above $K$, while profit becomes positive only above $K+c$. Keeping these two thresholds separate is central to solving long-call questions correctly.

Worked Example: A Call With Three Possible Expiration Prices

An investor buys one call with strike $K=\$48$ for a premium of $c=\$3.25$ per share. Ignore fees and taxes. Find the break-even price and the profit or loss if the share price at expiration is (a) $\$46$, (b) $\$50$, or (c) $\$53.25$.

Method. First find the break-even price, then calculate payoff minus premium for each stated terminal price. This avoids treating exercise as equivalent to profit.

The break-even price is $$ K+c=\$48+\$3.25=\$51.25. $$

(a) At $S_T=\$46$, the price is below the strike, so the call is not exercised. Its payoff is $\max(\$46-\$48,0)=\$0$. Profit is $\$0-\$3.25=-\$3.25$ per share.

(b) At $S_T=\$50$, the call is exercised. Its payoff is $\$50-\$48=\$2.00$. Profit is $\$2.00-\$3.25=-\$1.25$ per share. Exercise produces a positive payoff, but the position still has a loss.

(c) At $S_T=\$53.25$, the payoff is $\$53.25-\$48=\$5.25$. Profit is $\$5.25-\$3.25=\$2.00$ per share. This price is above the break-even price, so a profit is consistent with the threshold.

Check. At the break-even price of $\$51.25$, payoff is $\$51.25-\$48=\$3.25$, exactly equal to the premium. Net profit is zero.

Using the Result With Contract Multipliers

Option premiums are commonly quoted per unit of the underlying, while a contract may represent multiple units. A multiplier converts a per-unit payoff or profit into a total dollar amount. If a position contains $N$ identical long-call contracts and each contract covers $M$ units, then the total profit is $NM$ times the per-unit profit. The break-even terminal price per unit remains $K+c$; the multiplier changes the dollar amount at risk or earned, not the price threshold.

Before calculating, check whether a premium is quoted per unit or stated as the total amount paid for a contract or group of contracts. Use matching units when subtracting premium from payoff. If the premium is a total dollar amount, divide it by the number of underlying units to find the per-unit premium before using $K+c$.

Worked Example: Converting Per-Share Profit to Contract Profit

A hypothetical exchange-traded call has strike $K=\$72$, premium $\$4.60$ per share, and a contract multiplier of 100 shares. An investor buys two contracts. At expiration, the share price is $\$78.20$. Find the break-even share price and the total profit on the two contracts. Ignore fees and taxes.

Method. Calculate the break-even price and profit per share first, then multiply by the total number of shares represented by the contracts.

The break-even share price is $$ \$72+\$4.60=\$76.60. $$

The terminal price exceeds the strike, so the payoff per share is $$ \$78.20-\$72=\$6.20. $$

Subtract the premium to find profit per share: $$ \$6.20-\$4.60=\$1.60. $$

The two contracts represent $2\times100=200$ shares. Total profit is therefore $$ 200\times\$1.60=\$320.00. $$

Check. The terminal price $\$78.20$ is $\$1.60$ above the break-even price $\$76.60$, matching the calculated profit of $\$1.60$ per share. The total is $\$320.00$.

Worked Example: Recovering the Per-Unit Premium From a Total Cost

An investor pays $\$540$ in total for three identical call contracts. Each contract represents 100 shares, and each call has strike $K=\$125$. At expiration, the share price is $\$130.20$. Find the premium per share, the break-even share price, and the investor’s total profit. Ignore fees and taxes.

Method. Convert the total premium to a per-share amount before applying the break-even formula or calculating per-share profit.

The contracts cover $3\times100=300$ shares. The premium per share is $$ \frac{\$540}{300}=\$1.80. $$

The break-even share price is $$ K+c=\$125+\$1.80=\$126.80. $$

At expiration, the call payoff per share is $$ \$130.20-\$125=\$5.20. $$

Profit per share is $\$5.20-\$1.80=\$3.40$. Multiplying by the 300 shares gives total profit $$ 300\times\$3.40=\$1{,}020.00. $$

Check. The terminal price is $\$130.20-\$126.80=\$3.40$ above break-even. Across 300 shares, this gives $300\times\$3.40=\$1{,}020.00$, as calculated.

Common Exam Trap: Confusing Payoff With Profit

A frequent error is to see that a call finishes in the money and conclude that the buyer made a profit. The option’s exercise payoff is not its net profit: the premium must still be subtracted. Another common error is to add the premium to the payoff rather than subtract it.

For example, suppose $K=\$60$, $c=\$2.50$, and $S_T=\$61$. The incorrect calculation “the call pays $\$61-\$60=\$1$, so profit is $\$1$” reports the payoff as if it were profit. The correct calculation is $$ \text{profit}=\$1-\$2.50=-\$1.50. $$ The call is exercised, but the buyer loses $\$1.50$ per share. Its break-even price is $\$60+\$2.50=\$62.50$, not the strike price.

A second unit trap occurs when the premium is given as a total cost but the strike and terminal price are per share. A total premium cannot be subtracted directly from a per-share payoff. Convert the premium to a per-share amount, calculate per-share profit, and then scale by the number of shares.

Key takeaway: Under the standard convention, long-call profit per unit equals expiration payoff minus premium. For a positive premium, the break-even terminal price is $K+c$: exercise may begin above $K$, but profit begins only above $K+c$.

Check Your Understanding

For each problem, use the standard profit convention and ignore fees and taxes unless stated otherwise.

A call has strike $\$40$ and premium $\$2.75$ per share. Find its break-even price and the profit per share if the terminal share price is $\$45$. (Answer: Break-even $\$42.75$; profit $\$2.25$ per share.)
A call with strike $\$96$ costs $\$5.40$ per share. At expiration the underlying is $\$100.25$. Is the call exercised, and what is the profit or loss per share? (Answer: Exercised; loss of $\$1.15$ per share.)
An investor buys two call contracts, each covering 50 shares. The strike is $\$210$, the premium is $\$3.20$ per share, and the terminal share price is $\$216.50$. Find the break-even share price and total profit. (Answer: Break-even $\$213.20$; total profit $\$330.00$.)
Four call contracts cover 100 shares each. The investor pays $\$720$ total for the options. Each call has strike $\$35$, and the terminal share price is $\$38.10$. Find the premium per share, break-even price, and total profit. (Answer: Premium $\$1.80$ per share; break-even $\$36.80$; total profit $\$520.00$.)
An investor buys five identical call contracts, each covering 100 shares. The strike is $\$44$, and the premium is $\$2.50$ per share. What terminal share price produces a total profit of $\$875$? Verify by calculating the per-share profit. (Answer: $\$48.25$; per-share profit $\$1.75$.)

These payoff-and-profit calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

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