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9. Options

An option is a contractual agreement that gives the buyer the right but not the obligation to buy or sell a financial asset on or before a specified date. However, the seller of the option is obliged to follow the buyer's decision.

Call option

Right to buy an financial asset at a specified exercise price (strike price) on or before the exercise date Put option

Right to sell an financial asset at a specified exercise price on or before the exercise date

Exercise price (Striking price)

The price at which you buy or sell the security

Expiration date

The last date on which the option can be exercised The rights and obligations of the buyer and seller of call and put options are summarized below.

Buyer

Seller

Call option

Right to buy asset

Obligation to sell asset if option is exercised

Put option

Right to sell asset

Obligation to buy asset if option is exercised

The decision to buy a call option is referred to as taking a long position, whereas the decision to sell a call option is a short position.

If the exercise price of a option is equal to the current price on the asset the option is said to be at the money. A call (put) option is in the money when the current price on the asset is above (below) the exercise price. Similarly, a call (put) option is out of the money if the current price is below (above) the exercise price.

With respect to the right to exercise the option there exist two general types of options:

- American call which can be exercised on or before the exercise date

- European call which can only be exercised at the exercise date

9.1. Option value

The value of an option at expiration is a function of the stock price and the exercise price. To see this consider the option value to the buyer of a call and put option with an exercise price of €18 on the Nokia stock.

Stock price

€15

€16

€17

€18

€19

€20

€21

Call value

0

0

0

0

1

2

3

Put value

3

2

1

0

0

0

0

If the stock price is 18, both the call and the put option are worth 0 as the exercise price is equal to the market value of the Nokia stock. When the stock price raises above €18 the buyer of the call option will exercise the option and gain the difference between the stock price and the exercise price. Thus, the value of the call option is €1, €2, and €3 if the stock price rises to €19, €20, and €21, respectively. When the stock price is lower than the exercise price the buyer will not exercise and, hence, the value is equal to 0. Vice versa with the put option.

The value to the buyer of a call and a put options can be graphically illustrated in a position diagram:

As the seller of a call and a put option takes the opposite position of the buyer, the value of a call and put option can be illustrated as:

The total payoff of a option is the sum of the initial price and the value of the option when exercised. The following diagram illustrates the profits to buying a call option with an exercise price of €18 priced at €2 and a put option with an exercise price of €18 priced at €1.5.

Note that although the profits to the call option buyer is negative when the difference between the share price and exercise price is between 0 and €2 it is still optimal to exercise the option as the value of the option is positive. The same holds for the buyer of the put option: its optimal to exercise the put whenever the share price is below the exercise price.

9.2. What determines option value?

The following table summarizes the effect on the expected value of call and put option of an increase in the underlying stock price, exercise price, volatility of the stock price, time to maturity and discount rate.

The impact on the ... option price of an increase in...

Call

Put

1. Underlying stock price (P)

Positive

Negative

2. Exercise price (EX)

Negative

Positive

3. Volatility of the stock price (a)

Positive

Positive

4. Time to option expiration (t)

Positive

Positive

5. Discount rate (r)

Positive

Negative

1. Underlying stock price

The effect on the option price of an increase in the underlying stock price follows intuitively from the position diagram. If the underlying stock price increases the value of the call (put) option for a given exercise price increases (decreases).

2. Exercise price

This follows directly from the position diagram as the value of the call (put) option is the difference between the underlying stock price and the exercise price (the exercise price and underlying stock price). For a given underlying stock price the value of the call decreases (put increases) when the exercise price increases

3. Volatility of the underlying stock price

Consider call options on two stocks. The only difference between the two call options is the volatility in the underlying stock price: One stock has low stock price volatility, whereas the other has high. This difference is illustrated in the position diagrams where the bell-shaped line depicts the probability distribution of future stock prices.

For both stocks there is a 50% probability that the stock price exceeds the exercise price, which implies that the option value is positive. However, for the option to the right the probability of observing large positive option values is significantly higher compared to the option to the left. Thus, it follows that the expected option value is increasing in the underlying stock price volatility.

4. Time to option expiration

If volatility in the underlying stock price is positively related to option value and volatility, 02, is measured per period, it follows that the cumulative volatility over t sub periods is t-02. Thus, option value is positively related to the time to expiration.

5. Discount rate

If the discount rate increases the present value of the exercise price decreases. Everything else equal, the option value increases when the present value of the exercise price decreases.

 
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