The options can be classified as European options and American options. The difference between those is that the American options can be exercised anytime whereas the European options can be exercised only at the expiration.
The value of European options at the expiration is straightforward.
The value of a European call option at expiration is the greater of zero or the difference between the underlying and the exercise price. The value is calculated for the option buyer and the value can never be negative. Suppose a trader buys a European call option to get the right to buy the stock at a strike (exercise) price of $35, and at the expiration, the price is $37. Then he will have a profit of $2 at the expiration as he can buy the stock at $35 and sell the stock at $37 in the market. If the spot price is $33 at the expiration, then the option buyer will not exercise his right to buy the stock at $35 because he can get the stock at a lower price from the spot market.
If cT is the European call option price at expiration, ST is the spot price of the underlying asset at expiration, and X is the exercise price, then
cT = Max(0, ST - X)
Similarly, the value of a European put option at the expiration is equal to the greater of zero or the difference between the exercise price and the underlying price. If the underlying price ends below the exercise price at expiration, then the put option buyer will exercise the option as he can sell the underlying at a higher exercise price. If the underlying price ends above the exercise price, then the put option buyer will not exercise his right to sell the underlying at the exercise price because he can sell the underlying at a higher price in the spot market.
If pT is the European put option price at expiration, ST is the spot price of the underlying asset at expiration, and X is the exercise price, then
pT = Max(0, X - ST)
Check your concepts:
(58.14) What will the value of a call option at expiration if the stock price settles at $39? The exercise price of the option is $40.
(a) -$1
(b) $0
(c) $1
(58.15) If both the call option and put option are taken at the same exercise price on the same underlying, then what can be said about their values at expiration? Assume that the underlying price at expiration differs from the exercise price.
(a) Equal but opposite in sign
(b) One will have a positive value and the other will have a zero value
(c) Unequal and opposite in sign
Solutions:
(58.14) Correct Answer is B: As the stock price closed lower than the exercise price, the option will expire worthlessly and will have zero value.
(58.15) Correct Answer is B: The call options and put options cannot have a negative value. When the exercise price is different than the underlying price, then one position will have a positive value and the other will have a zero value depending on whether the underlying price is greater/lower than the exercise price.
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