Option Payoffs 611 Basic Options

Options are instruments that give their holder the right to buy or sell an asset at a specified price until a specified expiration date. The specified delivery price is known as the delivery price, exercise price, or strike price, and is denoted by K.

Options to buy are call options; options to sell are put options. As options confer a right to the purchaser of the option, but not an obligation, they will be exercised only if they generate profits. In contrast, forwards involve an obligation to either buy or sell and can generate profits or losses. Like forward contracts, options can be either purchased or sold. In the latter case, the seller is said to write the option.

Depending on the timing of exercise, options can be classified into European or American options. European options can be exercised at maturity only. American options can be exercised at any time, before or at maturity. Because American options include the right to exercise at maturity, they must be at least as valuable as European options. In practice, however, the value of this early exercise feature is small, as an investor can generally receive better value by reselling the option on the open market instead of exercising it.

We use these notations, in addition to those in the previous chapter:

K = exercise price c = value of European call option C = value of American call option p = value of European put option P = value of American put option

To illustrate, take an option on an asset that currently trades at $85 with a delivery price of $100 in one year. If the spot price stays at $85, the holder of the call will not exercise the option, because the option is not profitable with a stock price less than $100. In contrast, if the price goes to $120, the holder will exercise the right to buy at $100, will acquire the stock now worth $120, and will enjoy a "paper" profit of $20. This profit can be realized by selling the stock. For put options, a profit accrues if the spot price falls below the exercise price K = $100.

Thus the payoff profile of a long position in the call option at expiration is

The payoff profile of a long position in a put option is

If the current asset price St is close to the strike price K, the option is said to be at-the-money. If the current asset price St is such that the option could be exercised at a profit, the option is said to be in-the-money. If the remaining situation, the option is said to be out-of-the-money. A call will be in-the-money if St > K; a put will be in-the-money if St < K;

As in the case of forward contracts, the payoff at expiration can be cash settled. Instead of actually buying the asset, the contract could simply pay $20 if the price of the asset is $120.

Because buying options can generate only profits (at worst zero) at expiration, an option contract must be a valuable asset (or at worst have zero value). This means that a payment is needed to acquire the contract. This up-front payment, which is much like an insurance premium, is called the option "premium." This premium cannot be negative. An option becomes more expensive as it moves in-the-money.

Thus the payoffs on options must take into account this cost (for long positions) or benefit (for short positions). To be complete, we should translate all option payoffs by the future value of the premium, that is, cerT for European call options.

Figure 6-1 compares the payoff patterns on long and short positions in a call and a put contract. Unlike those of forwards, these payoffs are nonlinear in the underlying spot price. Sometimes they are referred to as the "hockey stick" diagrams. This is because forwards are obligations, whereas options are rights. Note that the positions are symmetrical around the horizontal axis. For a given spot price, the sum of the profit or loss for the long and for the short is zero.

So far, we have covered options on cash instruments. Options can also be struck on futures. When exercising a call, the investor becomes long the futures at a price set to the strike price. Conversely, exercising a put creates a short position in the futures contract.

FIGURE 6-1 Profit Payoffs on Long and Short Calls and Puts

Buy call Buy put

FIGURE 6-1 Profit Payoffs on Long and Short Calls and Puts

Buy call Buy put

Because positions in futures are equivalent to leveraged positions in the underlying cash instrument, options on cash instruments and on futures are also equivalent. The only conceptual difference lies in the income payment to the underlying instrument. With an option on cash, the income is the dividend or interest on the cash instrument. In contrast, with a futures contract, the economically equivalent stream of income is the riskless interest rate. The intuition is that a futures can be viewed as equivalent to a position in the underlying asset with the investor setting aside an amount of cash equivalent to the present value of F.

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