Options

In this lesson, we learned about put and call options, both American and European types.  In actuality, most of this lesson was a review since I learned about American call and put options in a class I had last semester, ECON 355 - Financial Markets.

European call option - right (but not the obligation) to buy 1 unit of underlying asset at price K at specific time T.
American call option - right (but not the obligation) to buy 1 unit of underlying asset at price K at any time until specific time T.
European put option - right (but not the obligation) to sell 1 unit of underlying asset at price K at specific time T.
American put option - right (but not the obligation) to sell 1 unit of underlying asset at price K at any time until specific time T.

The difference between the two is that in the American calls/puts, you can exercise the option any time between purchase of the option and the expiration, while with the European call/put, you can only exercise it at the specific time T.

The payoff for a european call option is: max{S(T) - K,0}.
The payoff for an american call option is: max{S(t) - K,0}.
The payoff for a european put option is: max{K - S(T),0}.
The payoff for an american put option is: max{K - S(T),0}.

A call option is considered in the money if S(t/T) > K
A call option is considered at the money if S(t/T) = K
A call option is considered out of the money if S(t/T) < K

A put option is considered in the money if S(t/T) < K
A put option is considered at the money if S(t/T) = K
A put option is considered out of the money if S(t/T) > K

The following symbols are used for call and put options:

The following is known as the put-call parity:

Bounds on prices of European options

The price of the American call option will be greater than or equal to the price of the European call option since the American option gives more choices on when to exercise, so therefore you pay a higher price for more freedom.

Bounds on prices of American options

The price of an American call option is always greater than the exercise value of the call option.  Thus, it is never optimal to exercise a call option immediately.

In the lesson online, it is asserted that it is never optimal to exercise the call early and thus:

However, I disagree with this conclusion, since it may be optimal to exercise the call option after purchasing it, but before time T.