Contingent Claims
A contingent claim is a derivative in which the outcome or payoff is dependent on the outcome or payoff of an underlying asset. Although this characteristic is also associated with forward commitments, a contingent claim has come to be associated with a right, but not an obligation, to make a final payment contingent on the performance of the underlying. Given that the holder of the contingent claim has a choice, the term contingent claim has become synonymous with the term option. The holder has a choice of whether or not to exercise the option. This choice creates a payoff that transforms the underlying payoff in a more pronounced manner than does a forward, futures, or swap. Those instruments provide linear payoffs: As the underlying goes up (down), the derivative gains (loses). The further up (down) the underlying goes, the more the derivative gains (loses). Options are different in that they limit losses in one direction. In addition, options can pay off as the underlying goes down. Hence, they transform the payoffs of the underlying into something quite different.
We might say that an option, as a contingent claim, grants the right but not the obligation to buy an asset at a later date and at a price agreed on when the option is initiated. But there are so many variations of options that we cannot settle on this statement as a good formal definition. For one thing, options can also grant the right to sell instead of the right to buy. Moreover, they can grant the right to buy or sell earlier than at expiration. So, let us see whether we can combine these points into an all-encompassing definition of an option.
An option is a derivative contract in which one party, the buyer, pays a sum of money to the other party, the seller or writer, and receives the right to either buy or sell an underlying asset at a fixed price either on a specific expiration date or at any time prior to the expiration date.
Unfortunately, even that definition does not cover every unique aspect of options. For example, options can be created in the OTC market and customized to the terms of each party, or they can be created and traded on options exchanges and standardized. As with forward contracts and swaps, customized options are subject to default but are less regulated and relatively transparent. Exchange-traded options are protected against default by the clearinghouse of the options exchange and are relatively transparent and regulated at the national level. As noted in the definition above, options can be terminated early or at their expirations. When an option is terminated, either early or at expiration, the holder of the option chooses whether to exercise it. If he exercises it, he either buys or sells the underlying asset, but he does not have both rights. The right to buy is one type of option, referred to as a call or call option, whereas the right to sell is another type of option, referred to as a put or put option. With one very unusual and advanced exception that we do not cover, an option is either a call or a put, and that point is made clear in the contract.
An option is also designated as exercisable early (before expiration) or only at expiration. Options that can be exercised early are referred to as American-style. Options that can be exercised only at expiration are referred to as European-style. It is extremely important that you do not associate these terms with where these options are traded. Both types of options trade on all continents.13
As with forwards and futures, an option can be exercised by physical delivery or cash settlement, as written in the contract. For a call option with physical delivery, upon exercise the underlying asset is delivered to the call buyer, who pays the call seller the exercise price. For a put option with physical delivery, upon exercise the put buyer delivers the underlying asset to the put seller and receives the strike price. For a cash settlement option, exercise results in the seller paying the buyer the cash equivalent value as if the asset were delivered and paid for.
The fixed price at which the underlying asset can be purchased is called the exercise price (also called the “strike price,” the “strike,” or the “striking price”). This price is somewhat analogous to the forward price because it represents the price at which the underlying will be purchased or sold if the option is exercised. The forward price, however, is set in the pricing of the contract such that the contract value at the start is zero. The strike price of the option is chosen by the participants. The actual price or value of the option is an altogether different concept.
As noted, the buyer pays the writer a sum of money called the option premium, or just the “premium.” It represents a fair price of the option, and in a well-functioning market, it would be the value of the option. Consistent with everything we know about finance, it is the present value of the cash flows that are expected to be received by the holder of the option during the life of the option. At this point, we will not get into how this price is determined, but you will learn that later. For now, there are some fundamental concepts you need to understand, which form a basis for understanding how options are priced and why anyone would use an option.
Because the option buyer (the long) does not have to exercise the option, beyond the initial payment of the premium, there is no obligation of the long to the short. Thus, only the short can default, which would occur if the long exercises the option and the short fails to do what it is supposed to do. Thus, in contrast to forwards and swaps, in which either party could default to the other, default in options is possible only from the short to the long.
Ruling out the possibility of default for now, let us examine what happens when an option expires. Using the same notation used previously, let ST be the price of the underlying at the expiration date, T, and X be the exercise price of the option. Remember that a call option allows the holder, or long, to pay X and receive the underlying. It should be obvious that the long would exercise the option at expiration if ST is greater than X, meaning that the underlying value is greater than what he would pay to obtain the underlying. Otherwise, he would simply let the option expire. Thus, on the expiration date, the option is described as having a payoff of Max(0,ST – X). Because the holder of the option would be entitled to exercise it and claim this amount, it also represents the value of the option at expiration. Let us denote that value as cT. Thus,
which is read as “take the maximum of either zero or ST – X.” Thus, if the underlying value exceeds the exercise price (ST > X), then the option value is positive and equal to ST – X. The call option is then said to be in the money. If the underlying value is less than the exercise price (ST < X), then ST – X is negative; zero is greater than a negative number, so the option value would be zero. When the underlying value is less than the exercise price, the call option is said to be out of the money. When ST = X, the call option is said to be at the money, although at the money is, for all practical purposes, out of the money because the value is still zero.
This payoff amount is also the value of the option at expiration. It represents value because it is what the option is worth at that point. If the holder of the option sells it to someone else an instant before expiration, it should sell for that amount because the new owner would exercise it and capture that amount. To the seller, the value of the option at that point is ‒Max(0,ST – X), which is negative to the seller if the option is in the money and zero otherwise.
Using the payoff value and the price paid for the option, we can determine the profit from the strategy, which is denoted with the Greek symbol Π. Let us say the buyer paid c0 for the option at time 0. Then the profit is
To the seller, who received the premium at the start, the payoff is
The profit is
