## Payoff Profile for Calls and Puts

The following abbreviations will be used in this section for the diagrams and formulas concerning options and their uses.

 S = Current stock price St = Stock price at time T X = Exercise price C = Price of call option P = Price of put option

Call Options

A call option gives the holder the right to buy an asset for a specified price on or before a given expiration (maturity) date. The payoff profile (gain or loss) for the holder and writer of a call option is shown in Figure 10.1.

Option Clearing Corporation (OCC)

Call Holder Call Writer

Figure 10.1: Payoff Profile for Call Holder and Call Writer

Payoff for call The diagram on the left side of Figure 10.1 is the payoff for the option holder holder of a call option. The solid line represents the payoff for the call option. The dashed line is the payoff of the option, net the cost of the option. If the price of the stock never rises above the exercise price, the holder of the option is only liable for the premium paid to buy the call option.

This payoff of the option is also expressed using the following mathematical relationship:

When written out, this means that the payoff to the holder of a call option is the maximum of 0 (zero) and the difference between the price of the underlying stock (S) and the exercise price (X) of the option.

Example Here is an example of how this works. LDM Company has a call option with an exercise price of \$30. If the stock price (X) of LDM Company is less than \$30, the option is out of the money, and the payoff of the option is \$0. If the stock price rises above \$30, the option is in the money, and the payoff for the call option is the price of the stock minus the exercise price (S - X). For instance, if the stock rises to \$42, the option payoff is \$42 - \$30 = \$12 per share.

An option buyer is able to participate in any price change in the underlying asset without having to buy the asset itself, which would require a substantially larger investment. The option's cost (premium) is usually only a small fraction of the underlying asset's market price. Most options are written for a block of shares, such as 100.

With an American option, the payoff profile is valid anytime during the life of the option. In other words, the holder of the option can exercise the option and claim the payout. In a European option, the only stock price we can consider is the price at the time the option expires.

### Payoff for a call option writer

The diagram on the right side of Figure 10.1 is the payoff profile for the call option writer. In this case, the writer receives the price of the call option up front. Essentially, the writer is making a bet that the price of the underlying stock will not rise above the exercise price. If the bet is correct, the holder of the option will never exercise the option and the writer has the premium as profit. Options trading is a zero-sum transaction — any profits gained by one counterparty are exactly matched by losses incurred by the other counterparty.

If the price of the stock is greater than the exercise price on the option plus the premium, the holder of the option will most likely exercise the option. In this case, the writer is obligated to sell the shares of stock to the holder at the exercise price. If the writer did not own the underlying stock when the option was written (called writing a naked option), then the writer has to buy the stock at the current market price and sell it to the holder of the option at the exercise price. The writer's loss will be the difference in the two prices (net of the price received for the call option).

### Put Options

A put option gives the holder the right to sell an asset for a specified price on or before a given expiration (or maturity) date. The payoff profiles for the holder and writer of a put option are given in Figure 10.2.

Put Holder Put Writer

Figure 10.2: Payoff Profile for Put Holder and Put Writer holder

Maximum The figure on the left is the payoff profile for the holder of a put pay°ff for put option. The maximum profit is the exercise price of the put option less the price paid for the option. This would occur if the stock price fell to \$0.

The mathematical relationship is:

With a put option, the option is in the money when the underlying stock price is less than the exercise price. The payoff to the holder is the difference between the exercise price and the underlying stock price less the amount paid for the put option. Once again, the solid lines represent the payoff of the put option; the dashed lines are payoff, net the price paid for the option.

Example As an example, LDM Company put options have an exercise price of

\$45. As long as the stock price of LDM remains below \$45, these put options are in the money. If the stock price is at \$32, the payoff for the holder of a put option would be \$45 - \$32 = \$13 per share. If the investor paid \$90 for the put option contract, and it was for 100 shares, the net payoff would be (\$13 x 100) - \$90 = \$1,210.

### Payoff for put writer

The profits for the writer of a put option are exactly the opposite of the holder of the put. Essentially, the writer of the put is betting that the price of the stock will rise above the exercise price. If that occurs, the writer's payoff is the put option premium. If the stock price is not above the exercise price, the writer of the put is obligated to purchase the underlying stock for the exercise price. This position (called writing a naked put) is somewhat risky, and most investors combine writing puts with other strategies to limit their potential losses.

The transactions for put options work in the same way as the transaction for call options. Investors' accounts with brokers are credited and debited for the net amount of each option transaction.

In the world of finance, very few option contracts are completed. Usually, investors will close out their option positions by taking the opposite side of the transaction before the exercise date; that is, the holder of a call option will become the writer of an identical option shortly before the exercise date.

processing

Option transactions take place electronically through the OCC and its member brokers. The clearinghouse processes all transactions and acts as the counterparty on both sides of an option contract to ensure performance. If an option holder exercises an option, the OCC randomly assigns an exercise notice to a broker's account that reflects the writing of the same option. The broker then assigns the notice to one of its clients (option investors) on either a random or a "first in-first out" basis.

### Margin money and margin calls

Investors are required to post margin money with their brokers to assure performance of their obligations. The broker can then deposit or withdraw the funds from the investor's account to correspond with the profit or loss on the option transactions. If an investor's account balance becomes too low (a point where the broker no longer feels that the investor can meet the possible obligations), the investor will receive a margin call. A margin call requires the investor to deposit more funds into the margin account.

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