Clarification on exercise and assignment

[CXMelga]

 

Can someone kindly clarify the following couple of questions for me please, thank you

Looking at America style options first
I understand they can be exercised at any time. 

So if sell a call option (write) for a strike price of $55 when the stock (spot) price is $45 with an expiry of 30 days (just to give some numbers) 

Question 1: 

If the stock never reaches its strike price during this 30 days the holder of the option (buyer) 'can not' exercise this option (as the contract terms have not been met). Therefore the contract cannot be assigned, is this correct ?

Even thought the option holder (buyer) cannot exercise (until strike price is reached) they could sell their option contract (with less days left on it), at what ever market premium they can get for the option at that time. (which is not the same as an option being exercises/assigned) ?

Question 2: 

When it comes to European style options which can only be exercised at at expiration (small time window) again using the numbers above (but on an Index) 

If the index goes goes well above is strike price 15 days from expatriation, but then goes back down below the strike price at expiration, I assume the option expires worthless. As the buyer of the option was 'unable to exercise it when it went past the strike price' is the above also correct ? 

However as above even though the holder of the option could be exercise this European style option exactly when they wanted, they could sell the option on (with the remaining time until expiry and same strike price), is that also correct ?

Thanks very much in advance

 

[Kim]

If the stock is below the strike price, the option owner can still exercise American option - it just doesn't make sense to do it. And yes, they can sell it at any time, as we mentioned in the previous posts.

[Rebeker]

Old thread but I just needed it.