Confusion over option payoff and value, and price

CFA Level I Derivatives
#1

(1) I read payoff means the value at expiration. So, if the exercise price (X) = 55, and spot price (S) = 50, then pay off for put holder is 55-50 = 5. Clearly given in Schweser.

(2) The price of an option is the premium paid for option.

(3) Value is option value is Intrinsic value + time value. Time value as per the definition is the premium over & above the intrinsic value. => option premium = time value + intrinsic value? and by this theory, option premium = option value. And through this we conclude option price (premium) = option value? which is also called payoff (value at expiration)?

-----What I understand is these terms are all same at different times and circumstances. Price/Premium means that the value to the option writer. After the buyer buys option, he takes over the option value, which is both intrinsic and time. And finally, upon expiration, this value is only intrinsic, and is called payoff.

(4) And what is profit? I think profit is after subtratcing the cost of premium as well.

#2

The pay off incorporates the premium paid for the option. So if the premium = $3, and the intrinsic value of your option is $3 (I disregard the time value here), you would exercice your option right away (assuming its an american one and you are before expiration) and break-even. The intrinsic value is simply the difference between the strike price and the stock price (assuming we are on equity instruments). So it’s basically the in-at-out the money characteristic.

The value of the option is the intrinsic value of the option in addition to the time value of the option (the further the expiration date, the higher the time value).

#3

The payoff DOES NOT incorporate the premium paid for the option.

The profit DOES, however.

#4

This is very confusing though. We use “gross pay off” usually for the intrinsic value, and “net pay off” for the profit/loss. I have re-checked the cfa terminology and I believe you’re right, I apologize for the confusion.