The payoff graphs of the options show the proceeds, gross and net from the premium paid, from the future buy or sell transactions depending on the value of the underlying stock (Figures 6.3-6.5). The put option is said to be "in-the-money" when it provides a positive gross payoff, or "out-of-the money" if the gross payoff is zero, and "at-the-money" when the strike price is equal to the current stock price.

The call option is said to be "in-the-money" when it provides a positive gross payoff, "out-of-the money" if the gross payoff is zero, and "at-the-money" when the strike price is equal to the current stock price. The net payoff translates downward the gross payoff by the amount of the premium paid.

FIGURE 6.3 Payoff of a put option at maturity

FIGURE 6.4 Net payoff of a put option at maturity

The payoff of the buyer of a put option is:

payoff buyer of put — maximum (strike price - stock price; 0) The payoff of the buyer of a call option is:

payoff buyer of call — maximum (stock price - strike price; 0)

FIGURE 6.5 Payoff of a call option on stock

Sellers of Options

Sellers of options, unlike buyers of options, have the obligation to comply with their commitment to the buyer. The seller of a call option on stock will have the obligation to sell the stock to the buyer at the strike price if the buyer exercises. The seller of a put option on stock has the obligation to buy that stock at the strike price if the seller exercises.

The seller, or writer, of the call option has the obligation to pay any positive difference between stock price and strike price when the call option is "in the money," which might be potentially very high. The payoff is the mirror image of the gain of the buyer:

The seller, or writer, of the put option has the obligation to pay to the buyer any positive difference between the strike price and the stock price, when the put is "in the money," which might be potentially as high as the strike price. The payoff is the mirror image of the gain of the buyer:

payoff seller of put — -maximum (strike price - stock price; 0)

Sellers of options are banks. They gain the premium but they have to hedge themselves against the losses that they would incur when the buyer makes a gain. In order to do so, they use a technique called delta-hedging. The delta of an option is the change of value of the option when there is a change in the price of the underlying asset.

The delta ("8") changes when the underlying asset price changes. This results from the "kink" observed in the payoff. The value of the option is always above the payoff because, at any point in time before maturity, there is still a chance that the option gets in the money. Accordingly the price of the option is a curve, such as the one for a call. The difference between the price and the exercise price, which is the payoff from immediate exercise, is called the time value of the option. The delta is the slope of the curve, which is always between 0 (extremely out-of-the-money) and 1 when the call option is well in-the-money. It changes because the curve shows convexity. Because of convexity, options are called non-linear instruments.

The underlying principle for delta-hedging is relatively simple. The seller of a call loses when the option is in the money. If the option has a delta of 0.8, for example, any small variation of the stock price of 1 is matched by a variation of 0.8 of the call option. If the stock price increases further the value of the call also increases, and this makes up an additional loss for the seller of the call. The seller can offset this loss by taking a long position in the stock. Assume that the stock price is S— 120, and the strike is 100, the call delta is 0.8, and a call (C) provides the right to buy a single stock. Then being short the call and long by 0.8 the stock, the variation of value of the trader's portfolio is zero. Note that delta-hedging is dynamic, since the delta changes when the underlying changes. This is an application of the replicating principle, whereby the traders replicates a long position in the call by mixing the call with a long position in the stock.

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