Where you are. Every instrument so far has obliged both sides. A future’s two payoffs are mirror images, straight lines, symmetric in every direction. This lesson introduces the first instrument in the course where the two sides are not mirror images at all, and that asymmetry is the reason the next two lessons need a pricing model.
The right to change your mind
A future obliges you to buy at 100 in December. If the price is 70 you buy at 100 anyway, and you lose thirty.
Now buy the right to buy at 100 instead of the obligation. Price is 130 in December, you exercise, you gain thirty. Price is 70, you simply do not exercise - you walk away and buy at 70 like everyone else.
Read that again, because it is obviously too good: unlimited upside, no downside. It is too good, and the correction is the premium. The right is not free. Somebody sold it to you, and what you pay them is the entire cost of never having to take the bad outcome.
The idea in one paragraph
An option is the right, without the obligation, to trade at an agreed price. A call is the right to buy, a put the right to sell, and the agreed price is the strike. The buyer pays a premium up front and can lose no more than that premium, because a right that has become worthless is simply declined. The writer receives the premium and takes the other side of the buyer’s choice, which means they are obliged whenever it hurts and released whenever it does not. That is the asymmetry: the buyer’s loss is capped and the writer’s is not. Everything else about options - the premium’s size, the pricing model, the sensitivities - follows from having to put a number on an asymmetric payoff.
Four payoffs, one strike
The exercise computes all four - long call, short call, long put, short put - across a grid of settlement prices, premium included.
Two properties come out. Each pair sums to zero, because the premium simply changes hands between buyer and writer: options are zero sum exactly as futures are. And the long positions are bounded below at minus the premium, while the short positions are bounded above at plus the premium, with the other tail open.
Why anyone writes them
If the writer’s loss is unbounded, why write at all?
Because the premium is received with certainty and the loss is contingent. Most options expire worthless, so most premiums are kept, and a writer who prices correctly earns a steady income from a rare, large obligation. It is a business, and it is the insurer’s business.
It also has the insurer’s failure mode. A writer who sells cheaply for years looks profitable for years, because the years look identical right up until the one that is not. Nothing in the income statement distinguishes correct pricing from cheap pricing until the tail arrives.
Check yourself
1. If a call gives unlimited upside and no downside, why is it not free money?
Because it costs the premium. The buyer pays for the right up front and loses that amount whenever the option expires worthless, which is most of the time. The payoff is asymmetric, and the premium is the price of that asymmetry, which is exactly what the next lesson computes.
2. Where does the asymmetry between buyer and writer live?
In who chooses. The buyer decides whether to exercise, so they exercise only when it helps them, which means the writer is obliged only when it hurts. The buyer’s loss is therefore capped at the premium and the writer’s is capped by nothing.
3. Most options expire worthless. Why is writing them still a risky business?
Because the income is small, steady and certain while the loss is large, rare and contingent. Years of kept premiums look identical whether the pricing was correct or far too cheap, and the difference only shows up when the rare outcome arrives - which is the insurer’s problem in its purest form.
4. Someone says “I trade options, so my risk is limited to what I put in.” When is that false?
When they write options rather than buy them. A bought option’s loss is capped at the premium; a written option’s is not, and the premium received is only a fraction of the exposure taken on. The sentence is true for exactly one of the two sides of the same trade.
Do this
Compute all four payoffs and check the symmetry that remains.
python3 code/option_payoff.pyThe starter leaves you payoff, which returns the profit at expiry for a given kind and side, premium included: a long call earns its intrinsic value less the premium, a long put the mirror of that, and a short is the exact negative of the matching long. The assertions check that a declined right costs only the premium, that a thirty-point in-the-money move pays thirty less the premium, and that every buyer and writer pair sums to zero. Success prints the grid and the line:
the buyer's loss is capped at the premium and the seller's is not: that asymmetry is the whole instrumentThe completed version is in solutions/option_payoff.py.
What you can now do. You can compute all four basic payoffs and say precisely where the asymmetry sits. What you cannot yet do is say what the premium should be, which is a harder question than any pricing this course has asked so far, because the answer depends on something that has not happened. The next lesson builds a small machine that answers it.