Options: calls and puts · RiskAtoms Academy

By the end you will be able to

1Define calls, puts, strike, expiry and moneyness.
2Say whether an option is in, at or out of the money, and find its intrinsic value.
3Calculate the payoff and profit of a call or a put at expiry.
4Explain the difference between buying and writing an option, and what time value is.

Why it matters

A forward locks you in, for better or worse. What if you could lock in only the good side: keep the right to buy at today's price, but walk away if the price falls? You can. It costs a fee, and the contract is called an option.

Core explanation

A right, not an obligation

Picture a house you like. You pay the owner a deposit of 5,000 for the right to buy it at 500,000 at any time in the next three months. If house prices jump, you buy it at 500,000 and are ahead. If they fall, you walk away and lose only the deposit. That deposit has bought you an option.

Every option has the same few parts:

Type

Call: the right to buy. Put: the right to sell.

Strike, K

The fixed price in the contract: the strike price.

Expiry, T

The expiry date, after which the right is gone.

Premium

The premium the buyer pays the seller upfront.

The buyer, or holder, is long the option and has the right. The seller, called the writer, is short. The writer keeps the premium but must deliver if the holder chooses to exercise. So the holder's loss is limited to the premium, while a call writer's loss has no limit if the price keeps rising.

A European option can only be exercised at expiry; an American option can be exercised on any day up to expiry. The names have nothing to do with where they trade. Options on shares and indices trade on exchanges around the world; listed US equity options usually cover 100 shares per contract. Banks also write tailored options over the counter, on currencies, interest rates and commodities.

Call or put?

4 cards

You own shares in a Singapore bank and want protection against a fall over the next three months.
You think a share will rise sharply after a product launch, but you only want to risk a small sum.
An airline wants to cap the price it will pay for jet fuel next winter.
A gold miner wants a minimum price for next year's output.

Moneyness

Traders describe an option by comparing the share price today with the strike. A call is in the money when the share is above the strike, because exercising now would pay. It is out of the money when the share is below the strike, and at the money when the two are about equal. A put is the mirror image. What an option would pay if exercised now is its intrinsic value.

In, at or out of the money?

Share price today (strike is 100)110

call, strike 100in the money

Intrinsic value: 10.00

Exercise now: buy at 100 a share worth 110. That is worth doing.

put, strike 100out of the money

Intrinsic value: 0.00

Exercise now: sell at 100 a share worth 110. Not worth doing, so the value is zero.

An option usually costs more than its intrinsic value. Even an out-of-the-money option has a chance of ending in the money before expiry. That extra is its time value: premium = intrinsic value + time value. L1.7.09 looks at what drives it.

The formal version

At expiry the holder makes one simple choice: exercise, or walk away. Writing that choice down needs one new piece of notation, $\max(a, b)$ , which means “the larger of a and b”. As before, $S_T$ is the share price at expiry and $K$ the strike.

The formulas, for reference

\text{Call payoff} = \max(S_T - K,\ 0) \qquad \text{Put payoff} = \max(K - S_T,\ 0)

\text{Profit (holder)} = \text{payoff} - \text{premium} \qquad \text{Profit (writer)} = \text{premium} - \text{payoff}

Setting the profit to zero gives the breakeven: $K + \text{premium}$ for a call and $K - \text{premium}$ for a put.

Try it

The payoff diagram plots what an option pays at expiry against the share price then. Switch between the four basic positions, change the strike and premium, and drag across the chart. Notice that the writer's line is the holder's turned upside down.

Payoff diagramOne option

Price an option today →

Legs

callKpremium

Tap or drag the chart to move the line

Price of the underlying at expiry$110

At an expiry price of $110

Payoff10.00

Profit+5.00

Net cost today$5.00

Most you can makeUnlimited

Most you can lose$5.00

Breakeven105.00

Worked example

A call and a put on the same share

A US share trades at $100. Three-month European options with strike $100 cost $5.00 for the call and $4.00 for the put, per share. Both are at the money today.

Three possible prices at expiry

Step 1 of 5

1
Payoffs
Share at expiry Call payoff Put payoff
$85 0.00 15.00
$100 0.00 0.00
$112 12.00 0.00

Share at expiry	Call payoff	Put payoff
$85	0.00	15.00
$100	0.00	0.00
$112	12.00	0.00

Regulatory lens · looking ahead

An option's value does not move in a straight line with the underlying, so simple risk measures can miss its risk. Bank capital rules for market risk (FRTB, L4.3) add a separate curvature charge for this. Retail investors buying options are also protected by conduct rules, such as the appropriateness tests for complex products under MiFID II in the EU.

Sparks

SparkThales and the olive presses

In his Politics, Aristotle tells how the philosopher Thales of Miletus, sure of a good olive harvest, paid small deposits in winter to secure the use of the local olive presses. When the harvest came in large, he rented them out at high prices. He had bought call options on press time.

Spark1973, a big year for options

The Chicago Board Options Exchange opened in April 1973, the first exchange to list standard share options. The same year, Fischer Black and Myron Scholes published their option pricing formula. Both helped turn options from a niche product into a huge market.

Key takeaways

A call is the right to buy, and a put the right to sell, at the strike K until expiry T. The buyer pays a premium.
Call payoff = max(S_T − K, 0); put payoff = max(K − S_T, 0). Profit = payoff − premium.
The holder can lose at most the premium; the writer keeps the premium and takes the risk.
Moneyness compares the share price with the strike: in, at or out of the money.
Premium = intrinsic value + time value.

Reaction

Check what stuck

Question 1 · objective 1

Which option gives you the right to sell a share at a fixed price?

AA callBA putCA forwardDA future

Question 2 · objective 3

You buy a call with strike 50 for a premium of 3. At expiry the share is at 58. What is your profit per share?

Question 3 · objective 2

A share trades at 40. A put on it has a strike of 45. The put is:

AIn the moneyBAt the moneyCOut of the moneyDWorthless

Question 4 · objective 3

You buy a put with strike 80 for a premium of 2.50. At expiry the share is at 70. What is your profit per share?

Question 5 · objective 4

A call with strike 110 costs 14 when the share is at 120. What is its time value?

Glossary

Option: A contract giving its holder the right, but not the obligation, to buy or sell an asset at a fixed price.
Call: An option to buy the underlying at the strike price.
Put: An option to sell the underlying at the strike price.
Strike price: The fixed price at which the option lets you buy or sell. Written K.
Expiry: The last date on which the option can be used.
Premium: The price the buyer pays the seller for the option, upfront.
Writer: The seller of an option. They receive the premium and must deliver if the holder exercises.
Exercise: Using the option: buying (call) or selling (put) at the strike.
European option: An option that can only be exercised on its expiry date.
American option: An option that can be exercised on any day up to and including expiry.
In the money: An option that would pay something if exercised now: a call with the share above the strike, or a put with the share below it.
At the money: An option whose strike equals, or is very close to, the current share price.
Out of the money: An option that would pay nothing if exercised now.
Intrinsic value: What an option would pay if exercised right now: max(S − K, 0) for a call, max(K − S, 0) for a put.
Time value: The part of an option's price above its intrinsic value: the value of the chance that things improve before expiry.

Deep Dives & Origins

Deep Dives

John C. Hull, Options, Futures, and Other Derivatives, 11th edition (Pearson, 2021), the chapter “Mechanics of Options Markets”. Beginner.

Origins

John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021.
Aristotle, Politics, Book I, chapter 11 (the story of Thales).
Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81(3), 1973.

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