n  option is a contract between two parties:  a buyer and a seller (or option writer).   The buyer pays the seller a price called the premium, and in exchange, the option writer gives the buyer the right (but not the obligation) to buy or sell some underlying security at some pre-specified price for some pre-specified period of time.

An option to buy is a call option, and an option to sell is a put option.  The pre-specified price is called the strike or exercise price, and the option's life is called the time to maturity or time to expiration.

If the right to exercise is "any time until maturity" then the option is called an American option.  If the right to exercise is only "at the time of maturity" then the option is referred to as a European option.  The majority of options traded are American options.  All stock options, for example, are American, as are most options on stock indices, on futures, and on currencies.  Commonly traded European options include those on the S&P 500 index and on some currencies.

The specified rights and obligations in an option contract determine the cash flows from the contract.  Consider the rights and obligations specified by a European call option.  This option can be exercised only at its time of maturity, and the decision whether to exercise depends upon whether the option finishes in-the-money or out-of-the-money.  An option is said to be in-the-money or out-of-the-money based upon its current exercise value.

For example, assume the possible stock values at the time a European strike 30 call option expires are 20 and 40.  If you hold one call option, you have the right, but not the obligation, to buy the stock for 30.

Suppose the stock value is 20.  If you exercise your option, you are buying the stock for 30, when you could acquire it in the market for 20.  That is, you would realize a loss of 10 by exercising.  In this case the option is  out-of-the-money.

If, on the other hand, the stock price is 40, it pays you to exercise the option.  By exercising, you get to buy the stock for 30 when its current price is 40.  Here your option is in-the-money.

In summary, your exercise decision at the end of the life of a European option is determined by whether the option is in-the-money or out-of-the-money.  The exercise decision for an American option is not this straightforward and is discussed in greater detail in Chapter 3 (see topics 3.3-3.5, American Call Option, American Put Option and Dividends and American Options.

In general, if X is the strike price and S is the value of the stock at the time the option expires, then the payoff C from a call option is

C = max{0, S - X}.

This terminal payoff for a call option is shown in Figure 1.1:

Figure 1.1

Terminal Payoff for a Call Option

 The call is worthless if S < X, but its value increases linearly with the stock price as long as S > X.  That is, if you own a call option, you receive cash from exercising if the stock price is greater than the exercise price.  Otherwise, you do not exercise the option.

Of course, owning a call requires buying it, so you pay something for it.  Whether you gain from owning a European option at the time of maturity depends upon two things.  First, it must finish in-the-money and second, the amount by which it finishes in-the-money must exceed what you paid for it.  Of course, if significant time has elapsed between the time at which you purchase the option and the time you exercise it, you must also account for interest lost on the money you used to buy the option.

The terminal payoff for a put option is:

P = max{0, X - S}, shown in Figure 1.2 where "terminal" means at maturity.

Figure 1.2

Terminal Payoff for a Put Option

In this case, you receive cash from exercising if the price of the stock falls below the  strike price.  Otherwise, you do not exercise the option.