Options and volatility · O1 · 8 min
Option payoffs
Say what calls and puts are, work out what they pay at expiry, and find the price where a trade breaks even.
In the news: “Traders bought calls before the results”: who makes money, who loses, and from what price.
The essentials
- Calls and puts
- A is the right, not the duty, to buy at a fixed price, the , until expiry; a is the right to sell. The buyer pays the seller a up front.
- Payoff and breakeven
- At expiry a call pays the amount the price is above the strike and a put the amount below, or nothing. A buyer breaks even at the strike plus the premium for a call, minus it for a put: a 100 call bought for 5 needs 105.
- Buyer and seller
- The buyer can lose only the premium. The seller keeps at most the premium and can lose far more: a short call’s loss has no limit, and a short put can lose the strike less the premium.
- Moneyness
- A call is in the money when the price is above the strike and out of the money when it is below; a put is the reverse. With a share at 100, the 95 call is in the money and the 105 call out of it.
- Intrinsic and time value
- is what the option would pay if it expired now; is the rest, and falls to zero by expiry. A 95 call at 7.50, share at 100, is 5 intrinsic and 2.50 time value.
Payoff and breakeven at expiry
Call = max(S − K, 0); put = max(K − S, 0); breakeven = K ± premium
- S
- price of the underlying at expiry
- K
- the strike
- ±
- + for a call, − for a put; the same for buyer and seller
How it works
1. An option is a reserved price
An option is the right, but not the duty, to trade at a fixed price later, bought for a small amount now. It works like paying a shop £5 to hold a jacket for you at £100 for a month: if its price rises to £130, you buy at £100; if it falls to £80, you walk away and lose only the £5.
2. Calls and puts
The right to buy at a fixed price is a ; the right to sell at a fixed price is a . The fixed price is the , and the right lasts until the expiry date. A call gains as the price rises. A put gains as it falls, which is why investors buy puts as insurance.
3. The premium
The buyer pays the seller up front for the right. That price is the . The seller keeps it whatever happens, and in return must buy or sell at the strike if the buyer asks. Share options are priced per share, and a listed US contract covers 100 shares: a premium of 2.50 costs 250 a contract.
4. What a call pays at expiry
At expiry a call pays the amount the price is above the strike, or nothing. With a 100 strike and the share at 112, it pays 12; with the share at 95, it pays 0. As a rule: call payoff = max(S − K, 0), where S is the share price and K the strike. Take off the premium you paid to get your profit or loss.
5. What a put pays at expiry
A put pays the amount the price is below the strike, or nothing: put payoff = max(K − S, 0). A 100-strike put pays 15 if the share ends at 85 and nothing if it ends at 105. The most it can ever pay is the strike itself, if the share falls to zero.
6. Breakeven
You make money only once the payoff covers the premium. A 100-strike call bought for 5 needs the share above 105 at expiry; a 100-strike put bought for 5 needs it below 95. So the breakeven is the strike plus the premium for a call, and the strike minus the premium for a put. The seller has the same breakeven.
7. The seller’s side
Every option has a seller, whose result is the buyer’s turned upside down. A buyer can lose only the premium. A seller can make only the premium and can lose much more: a call seller’s loss has no limit, because a price has no ceiling. People sell options to be paid for a risk they can carry or hedge.
8. In, at or out of the money
says where the strike sits against today’s price. A call is in the money when the price is above the strike, out of the money when it is below, and at the money when they are about equal; for a put it is the reverse. So a call’s strike can sit below the price: with a share at 100, the 95 call is in the money. Currency options work the same way: a EUR call / USD put (the right to buy euros with dollars) struck at 1.05 is in the money with EUR/USD at 1.08.
9. Intrinsic value and time value
An option’s price has two parts. is what it would pay if it expired now: 5 for the 95 call with the share at 100. The rest is , paid for the chance of more before expiry. If the 95 call costs 7.50, its time value is 2.50, and that part shrinks to zero by expiry.
Try it
Choose long or short, call or put. The dark line shows your profit or loss at expiry; the ticks on the zero line are the breakevens.
P&L per share against the stock priceAt expiryToday, 1 year to expiry
- Premium paid
- 7.97
- time value 7.97
- Breakeven
- 107.97
- Delta
- +0.54
- Gamma
- +0.020
- per 1 point
- Theta
- −0.011
- per day
- Vega
- +0.40
- per vol point
| Stock price | P&L at expiry | P&L today |
|---|---|---|
| 70 | −7.97 | −7.72 |
| 80 | −7.97 | −6.78 |
| 90 | −7.97 | −4.38 |
| 100 | −7.97 | 0.00 |
| 110 | 2.03 | 6.33 |
| 120 | 12.03 | 14.18 |
| 130 | 22.03 | 23.04 |
Worked example
A share trades at 100. You buy a 105-strike call for 2.50 a share. At expiry the share is at 112.
- The strike is above today’s price, so the call starts out of the money: all 2.50 is time value.
- At expiry it pays 112 − 105 = 7 a share.
- Take off the premium: 7 − 2.50 = 4.50 profit a share.
- One contract covers 100 shares: 4.50 × 100 = 450.
- Breakeven was 105 + 2.50 = 107.50. Anywhere below 105 you would have lost the whole 2.50.
+4.50 a share (+450 a contract), with a breakeven of 107.50; the most you could have lost was the 2.50 premium.
Check yourself
9 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
What’s the most you can lose selling a call?
Break an option’s price into its two parts.
Why would anyone sell options?
Remember
Breakeven = strike ± premium
Buyers can lose only the premium; sellers keep at most the premium, and a short call’s loss has no limit.