Valuation of Option

Can anyone explain why the value of an options is determined by the underlying. As well as the logic that as the price of the underlying increases the value of a call option increases. Whilst an increase in the price will also decrease a put option. thanks

An option is nothing else than a contract that gives the owner the to right buy or sell an underlying at a certain price (at a certain date in the case of European options), thus the value must be somehow connected to the underlying.

Let’s look at an example:

Imagine I sold you a contract right now for $1, that gives you the right to buy all of my flashcards for the Level 2 CFA exam at a price of $10 anytime you want to. Depending on whether you pass the Level 1 or not, you might exercise that option. Let’s say, this June 4th you actually pass the exam, but you realize that you are actually much better off writing your own flashcards so you decide not to exercise that option. Instead you want to sell that option here in the forum. Now what is that option worth?

It depends on what my flashcards are worth currently. Say I advertised a lot, and the flash cards are selling for $15. Your option gives the owner the right to buy the flash cards for $10 instead. Thus, the option is worth the difference, $5 (put differently, someone would be willing to pay up to $5). On the other hand, imagine people realize that my flash cards are not worth the paper they are written on, and the current price is $2. Then your option is useless, since it gives the owner the right to buy the flash cards for $10 while they can buy them without the option for $2.

So the call option is worth the current value of the underlying minus the exercise price. And If the exercise price is bigger than the price of the underlying, the option is worth 0.