Value of an Option at Expiration

A holder of a call option will choose to exercise, at expiration, only when the value of the underlying asset, $$S_{T}$$, is _greater than_ the exercise price $$X$$. In this case, the option is worth $$S_{T}-X$$, and if it is not greater than the exercise price, the option will expire worthless. So, at expiration, the __exercise value__ is the greater of zero or the price of the underlying asset less the exercise price, or: $$\displaystyle c_{T}=Max\left ( 0,S_{T}-X \right )$$

Suppose your father expresses an interest in trading options, but since he is a beginner, he comes to you for a tutorial on the basics of valuation. If he recently purchased a European call option with an exercise price of USD 55, what is the value of his option if the price of the underlying stock at expiration is USD 50?

Exactly! Since the price of the underlying stock at expiration of USD 50 is below the exercise price, your father's call option will expire worthless: $$\displaystyle c_{T}=Max\left ( 0,S_{T}-X \right )$$ $$\displaystyle c_{T}=Max\left ( 0,50-55 \right ) = 0 $$

Incorrect. Your father's call option allows him to purchase the underlying stock for USD 55, but since the stock is trading for only USD 50, there is no advantage for him to exercise the option. He is better served buying the stock in the market, so his option cannot be worth USD 5.

Incorrect. USD 50 is the price that the underlying stock is trading at upon expiration of your father's call option. It is not the value of his option, which is the value of the stock relative to the option's exercise price.

The holder of a put option will choose to exercise, at expiration, only when the value of the underlying asset, $$S_{T}$$, is _less than_ the exercise price $$X$$, in which case the option is worth $$X-S_{T}$$. And if it is not less than, the option will expire worthless. So, at expiration, the exercise value is the greater of zero or the exercise price less the price of the underlying asset, or: $$\displaystyle p_{T}=Max\left ( 0,X-S_{T}\right )$$

Among the European options your father is considering for his growing portfolio is a put option with an exercise price of USD 45. If the price of the underlying stock at the time of his option purchase is USD 60, but drops to USD 35 prior to expiration, and then to USD 25 at expiration, what is the value of his option if he exercises it?

Incorrect. If your father's put option is purchased, the immediate exercise value is USD 0. However, his European option only allows him to exercise at expiration, so his option would have a value greater than USD 0.

Incorrect. Your father is considering purchasing a _European_ put option, which gives him the right to sell the underlying stock _only at expiration_, but not at any point before that. So even if the price of the stock initially falls to USD 35, he is unable to exercise at that point. But that is a good thing for him, since the price would have fallen even further, to USD 25, at expiration.

Correct! Your father's European put option allows him to sell the underlying stock at USD 45 at expiration, but not before it. So, the exercise value is: $$\displaystyle p_{T}=Max\left ( 0,X-S_{T}\right )$$ $$\displaystyle p_{T}=Max\left ( 0,45-25\right ) = 20 $$

In summary: [[summary]]

USD 0

USD 5

USD 50

USD 0

USD 10

USD 20

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