If ABC stock is trading at $5 and I buy the Jan 15 strike put for $8.25, what happens to the value of this put at expiration if ABC only meandered between $5 to $7 during the entire period?
The value at expiration does not depend on the price path for a plain vanilla European or American option. At expiration, the value would simply be:
max[K - S_T, 0],
where: K is the strike price,
and S_T is the underlying price at expiration.
$15 - $5 = $10
How did you possibly buy a put for less than the intrinsic value of the option, at $8.25
So we can infer that you would have had to get this put when the stock price was AT LEAST $6.75, but given the 3 months of theta left, it was likely above $7
The value of the put if the price of the underlying asset (the stock ABC) meandered between $5 - $7 would be somewhere between $10 - $8 at expiration.
So you don't really stand to lose much in this scenario, and can make a decent gain in this scenario. I mean decent if you were trading stocks and were trying to beat the S&P or keep up with Warren Buffett, but a pretty poor gain since you are trading options!
If the stock moves above $7 this is where the put starts to substantially lose value.