## Definitions

The buyer of an option has the right, *but not the obligation*, to buy or sell
a given quantity of an underlying asset at a specified price on or before a
predetermined date.

- An option giving the right to buy the underlying asset is called a
**call**. - An option giving the right to sell the underlying
asset is called a
**put**. - An option that can be exercised at any time before the expiration date is
called an
**American**option. - An option that can only be exercised on the expiration date is called a
**European**option. - The amount paid by the option buyer to the seller is called the
**premium**. - The price determined in advance is the
**strike price**. - The expiration date is also called the
**exercise date**. After this date, the option ceases to exist. **Exercising**an option means that the buyer exercises his right to buy or sell.

## The underlying assets

There are two main types of underlying assets.

They can be real assets, i.e. those that are traded on the spot market and that can also be used as the basis for futures contracts: securities, stock market indices, interest rates, currencies, precious metals, commodities etc. This is known as a spot option. The exercise of the option is for the buyer to actually receive the underlying asset in return for payment in the case of a call option, or to actually deliver the asset in return for payment by the seller.

Options can also be based on a futures contract. In this case, exercising an option means that the call buyer buys the corresponding futures contract or the put buyer sells the underlying futures contract on the futures market.

## Use of options

The option thus allows the buyer not only to fix in advance a price for the asset he is interested in, but also to settle or receive this amount only if the market conditions are such that it becomes interesting to do so. If at maturity it is attractive for the buyer to exercise his option, it is said to be "in the money", i.e.:

- for a call, the price of the asset on the spot market is higher than the strike price
- for a put, the price of the asset on the spot market is lower than the strike price

Conversely, if the asset price makes it unattractive to exercise the option, it is said to be "out of the money". And finally, if the exercise price is strictly equal to the asset price, the option is "at par".

Whatever the buyer's decision to exercise or not, the premium remains with the seller. This premium actually compensates for the risk he takes by selling the option. Indeed, the risk taken by the seller of a call option is potentially unlimited, since there is in principle no upper limit to the price that the underlying asset can reach. If the buyer decides to exercise his option, the seller will have to buy the asset at the market price in order to sell it back to the buyer at the strike price, or, if he already holds the asset in his portfolio, lose the opportunity to resell it on the market at a much better price than the strike price of the option! The cash flow generated by the exercise of an option is called the pay-off and to be attractive it must exceed the level of the premium paid to the option writer.

The risk taken by a put writer is limited by the zero value of the underlying asset. Instead of buying the asset for nothing, the seller will have to buy it at the strike price.

As for the option buyer, his maximum risk is simply the value of the premium paid to the seller, since he can, if market conditions are not attractive, choose not to exercise his option.

## Options trading

Unlike futures, which are traded only on organised markets, options are traded either over-the-counter or on regulated markets.

OTC options are often traded between a bank and its customers, with the customer usually the buyer and the bank the seller. There is no intermediary or clearing house. Once the two counterparties have committed to the transaction, it is very difficult for them to reverse this decision. Moreover, the counterparties are committed to each other without an intermediary, which creates a risk that one or the other will not fulfil its obligations (counterparty risk).

Exchange-traded options are standardised products traded on stock exchanges that centralise offers to buy or sell. Traders can unwind their position at any time, which allows them to revise their strategy at low cost. Finally, the presence of a clearing house, which acts as a buyer to all sellers and a seller to all buyers, and guarantees the successful completion of all transactions, eliminates counterparty risk.

## The value of an option

Unlike an asset traded on the spot market, whose value depends mainly on supply and demand, the value of an option (i.e. ultimately the premium the seller charges) depends on multiple factors. Let us first look at how we define the value of an option.

First of all, an option is said to have an **
intrinsic value** if it is "in the money". For example, if the underlying
asset is quoted at 20 on the spot market, the intrinsic value of the right to
buy this same asset (call) at 15 is: 20 - 15 = 5.
On the other hand, the option to sell the same asset at 15 has no intrinsic
value. Indeed, it is more interesting to sell on the spot market at 20!

However, this same put option could gain value over time, if the prices of the underlying asset fall. The longer the maturity of the option, the more this time value increases, since in fact the higher the probability that prices will move in the direction favourable to the buyer. This probability obviously determines the risk taken by the seller of the option, which he passes on in the premium he charges! Therefore, the time value of an option is defined as the difference between the premium and the intrinsic value. In our previous example, if the asset is quoted at 20 and the call option with a strike price of 15 is sold 7, the time value of the option is 2 (= 7 - (20 - 15)).

## Determinants of the value of an option

The main factors involved in calculating the price of an option are:

- the price of the underlying asset
- the exercise price of the option
- the time remaining before the option expires
- the volatility of the underlying asset

These factors may work in opposite directions depending on whether they apply to the call or put option, or they may work in the same direction:

- The call price is an increasing function of the price of the underlying asset. The put price is a decreasing function of the price of the underlying asset.
- The call price is a decreasing function of the strike price. The put price is an increasing function of the strike price of the option.
- The price of an option, call or put, is an increasing function of the expiration time and also an increasing function of the volatility of the underlying asset.

## On the Web

- Read this article from the Barbican Consulting Web site