Selling an option: the seller’s profit
1. The idea
Every option has a seller, also called its writer. The seller receives the premium when the option is sold, and at expiry pays the buyer’s payoff. So the seller’s profit is the premium minus the buyer’s payoff:
\[ \begin{aligned} \text{sold call:}\quad \text{profit} &= c - \max(S_T - K,\ 0),\\ \text{sold put:}\quad \text{profit} &= p - \max(K - S_T,\ 0), \end{aligned} \]
where \(S_T\) is the share price at expiry, \(K\) the strike price, and \(c\) and \(p\) the premiums received for the call and the put. The seller’s profit is the buyer’s profit with the opposite sign: what one side gains, the other loses. So the break-even share price is the same for both sides: \(K + c\) for a call and \(K - p\) for a put.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The seller’s side on one chart
In the question, we sold a call with a strike price of $100 for a premium of $8. The share ends at $103, so the buyer exercises and buys the share from us at $100. The buyer’s payoff is $3, and that is our cost. Our profit is the premium minus the buyer’s payoff: $8 − $3 = $5.
Now move the share price at expiry, and switch between a sold call and a sold put. Our payoff as the seller is never above $0, so the most we can make is the premium.
For the sold call, the chart has three cases:
- At or below the $100 strike price. The buyer does not exercise, and we keep the whole $8 premium.
- Between $100 and the $108 break-even. The buyer exercises, but the buyer’s payoff is less than the $8 premium, so we still make a profit. This is the case most often missed.
- At or above $108. The buyer’s payoff takes the whole premium or more. For a sold call the loss has no limit: the higher the share price, the larger the loss.
4. One worked example, then your turn
The worked example is a sold put in the middle case: the buyer exercises, and we still keep part of the premium.
Now a sold call with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers, for a sold call or a sold put. You master the skill when you get three right in a row, with at least one right answer in each of the three cases. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: