Maximum Profit of a Covered Call

Let $S$ be the share price at expiry. The premiums received are $100(10.50+9.50)=\$2000$. The position’s profit is

Work through the three regions of $S$:

• $S\le 240$: both calls expire worthless and you keep all $200$ shares, so

  $$\Pi(S)=200S-44000 .$$

  This is increasing in $S$, and $\Pi(240)=4000$.

• $240\le S\le 250$: the $240$-calls are exercised, so $100$ shares are called away at $\$240$ (a gain of $\$10$ each). The other $100$ shares are still held. Then

  $$\Pi(S)=200S-46000+2000-100(S-240)=100S-20000 .$$

  This is still increasing, and $\Pi(250)=5000$.

• $S\ge 250$: all $200$ shares are called away, $100$ at $\$240$ and $100$ at $\$250$. The profit no longer depends on $S$:

  $$\Pi(S)=100(240-230)+100(250-230)+2000=1000+2000+2000=5000 .$$

So $\Pi$ is non-decreasing and constant from $S=250$ onwards. Its maximum is reached once both calls are in the money.

The reference figure of $4000 is wrong. At $S=250$ it counts only the shares still held and drops the $1000 already earned on the $100$ shares sold at $\$240$.

Key idea. A covered call’s profit is capped by the strikes. Once every short call is exercised, the profit is the premiums plus each share’s gain up to the strike at which it is sold.