Option

An option is a right, not an obligation, to buy or sell something at an agreed price by a certain date. You pay for that right up front with a fee called the premium. If the deal does not pay off, you simply do not use the option and lose only the premium.

The basic terms

• Underlying asset: what the option is on, most often a share or a stock index.
• Strike price: the agreed price at which you can buy or sell.
• Expiry: the date until which the option is valid.
• Premium: the price of the option, paid by the buyer to the seller.
• Exercise: using the right, i.e. the actual purchase or sale at the strike price.

Call and put options

A call option gives the right to buy. It makes money when the price of the underlying rises above the strike.

A put option gives the right to sell. It makes money when the price of the underlying falls below the strike. It often serves as insurance against a fall in shares you already own.

The value of an option at expiry, before deducting the premium, is:

$$\text{call: } \max(S_T - K,\ 0) \qquad \text{put: } \max(K - S_T,\ 0)$$

• $S_T$: the price of the underlying on the expiry date
• $K$: the strike price

The $\max(\cdot,\ 0)$ expresses exactly that an option is a right: if exercising would lose money, you do not exercise, and its value is zero.

A worked example

A share costs €100 today. You buy a call option with a strike of €100 and one year to expiry for a premium of €8.

• The share rises to €125. You exercise: you buy for €100 a share worth €125. The option is worth €25; after the premium you make €17.
• The share falls to €90. Paying €100 for something worth €90 makes no sense. You do not exercise and lose only the €8 premium.
• The break-even point is €108: only from there do you also recover the premium paid.

Had you bought the share itself instead, the fall to €90 would have cost you €10. The option limited the loss to the premium, but on the way up you made less from it.

American and European options

Options are also divided by when you can exercise them:

• A European option can be exercised only on the expiry date.
• An American option can be exercised at any time up to and including expiry.

The names have nothing to do with geography. An American option is not one traded in the USA, and a European one is not from Europe: both are traded on exchanges worldwide. Most options on individual shares are American-style, most options on stock indices European-style.

Because an American option gives everything a European one does, plus the chance to decide earlier, it is never worth less than the equivalent European option. When early exercise actually pays off, and how an American option is priced, is covered in detail in the chapter American and exotic options of the financial derivatives course.

What to watch out for

The buyer and the seller carry different risks. The buyer of an option can lose at most the premium. The seller (writer) receives the premium, but selling a call without owning the share carries a loss that is, in theory, unlimited.

An option usually loses value over time. If the price of the underlying does not move, the value of a bought option usually falls as expiry approaches, because there is less time left for a favourable move.

Volatility has a large effect on the price. Besides the distance between the underlying's price and the strike, and the time to expiry: the more the underlying's price swings, the more expensive the option, because a large move in its favour is more likely.

Where to go next

How a fair option price is worked out step by step is shown by the binomial model calculator, which prices European options. The whole route from first principles to the Black–Scholes formula is covered by the course, starting with the chapter What a derivative is and what it is for.