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
Exhibit 3 illustrates the payoffs and profits to the call buyer and seller as graphical representations of these equations, with the payoff or value at expiration indicated by the dark line and the profit indicated by the light line. Note in Panel A that the buyer has no upper limit on the profit and has a fixed downside loss limit equal to the premium paid for the option. Such a condition, with limited loss and unlimited gain, is a temptation to many unsuspecting investors, but keep in mind that the graph does not indicate the frequency with which gains and losses will occur. Panel B is the mirror image of Panel A and shows that the seller has unlimited losses and limited gains. One might suspect that selling a call is, therefore, the worst investment strategy possible. Indeed, it is a risky strategy, but at this point these are only simple strategies. Other strategies can be added to mitigate the seller’s risk to a substantial degree.
EXHIBIT 3 Payoff and Profit from a Call Option
Now let us consider put options. Recall that a put option allows its holder to sell the underlying asset at the exercise price. Thus, the holder should exercise the put at expiration if the underlying asset is worth less than the exercise price (ST < X). In that case, the put is said to be in the money. If the underlying asset is worth the same as the exercise price (ST = X), meaning the put is at the money, or more than the exercise price (ST > X), meaning the put is out of the money, the option holder would not exercise it and it would expire with zero value. Thus, the payoff to the put holder is
If the put buyer paid p0 for the put at time 0, the profit is
And for the seller, the payoff is
And the profit is
Exhibit 4 illustrates the payoffs and profits to the buyer and seller of a put.
EXHIBIT 4 Payoff and Profit from a Put Option
The put buyer has a limited loss, and although the gain is limited by the fact that the underlying value cannot go below zero, the put buyer does gain more the lower the value of the underlying. In this manner, we see how a put option is like insurance. Bad outcomes for the underlying trigger a payoff for both the insurance policy and the put, whereas good outcomes result only in loss of the premium. The put seller, like the insurer, has a limited gain and a loss that is larger the lower the value of the underlying. As with call options, these graphs must be considered carefully because they do not indicate the frequency with which gains and losses will occur. At this point, it should be apparent that buying a call option is consistent with a bullish point of view and buying a put option is consistent with a bearish
