Option vocabulary and call payoffs

The vocabulary, from one contract

A Tamarack Foods call: strike $50\ast \ast ,\ast \ast premium$3.20, expiring in three months. Every option term is in that line.

• A call is the right to buy the underlying at the strike; a put (Lesson 12.5) is the right to sell.
• The holder (buyer) pays the premium today and receives the right. The writer (seller) receives the premium and takes on the obligation — to deliver the shares at $50 if the holder exercises, however high the market has gone.
• The strike (exercise price) is the fixed price in the contract; the expiry is the last day the right exists.
• Contract size: one Montréal Exchange equity option covers 100 shares. A premium of 3.20 a share is $320 a contract.
• American options can be exercised any time up to expiry; European only on the expiry date. MX equity options are American style and physically settled, cleared by CDCC; they expire monthly (and weekly), with the last trading day the third Friday of the expiry month.

The asymmetry is the whole idea: the holder has a right it will use only when it pays; the writer has an obligation it will be held to only when it hurts. The premium is the price of that asymmetry.

Payoff, then profit

At expiry the call is worth its payoff:

$$\text{call payoff}=\max (S_T-K,0)$$

If Tamarack is at 58, the holder buys at 50 what is worth 58: payoff =MAX(58-50,0) = 8.00. If Tamarack is at 46, the right to pay 50 for a 46 share is worth nothing — payoff 0; the holder lets it expire. The payoff can never be negative; that is what the MAX does.

Profit subtracts what the right cost:

$$\text{holder profit}=\max (S_T-K,0)-\text{premium}$$

At 58: 8.00 − 3.20 = 4.80 a share, =4.80*100 = $480 a contract. At 46: 0 − 3.20 = −3.20, −$320 a contract — the premium, gone. The holder's loss is capped at the premium whatever happens; the gain is unbounded as the share rises.

Breakeven: the share must rise far enough above the strike to earn back the premium — $K+\text{premium}$ = 50 + 3.20 = 53.20. Between 50 and 53.20 the holder exercises (the payoff is positive) but still loses money overall.

The writer is the mirror

The writer received 3.20 and owes the payoff:

$$\text{writer profit}=\text{premium}-\max (S_T-K,0)$$

At 58: 3.20 − 8.00 = −4.80. At 46: 3.20 − 0 = +3.20 — the writer keeps the premium. Every dollar the holder makes the writer loses, and vice versa: the two profit lines are mirror images through the horizontal axis. So the writer's maximum gain is the premium and the writer's loss is unlimited — a naked call writer has sold a claim on a share that can go to any price.

Moneyness

A call is in the money when $S>K$ (exercising pays), at the money when $S=K$, out of the money when $S<K$. In the money does not mean profitable — at 52 the Tamarack call is in the money by 2.00 and still 1.20 short of breakeven.

Recompute

At 62: payoff 12.00, holder profit 8.80 (880 a contract), writer −8.80. At 50: payoff 0, holder −3.20, writer +3.20. Cold, both formulas: holder = MAX(S_T − K, 0) − premium; writer = premium − MAX(S_T − K, 0).