When you compare the market price to the theoretical price, your difference is model dependent.

Let's say you are using Black Scholes and you're using some other vol that you think is more appropriate (you're not using the IV because that would give you the market price)

Under the Black Scholes assumptions of continuous trading and no execution costs your P/L would be the intrinsic value of the position when you delta hedged. Since you are long, if you delta hedge when the options are OTM, your P/L will probably be close to zero because the optionaliy will eventually expire and its as if the settlement price is the initial hedge price.
If the options are ITM you would guarantee the intrinsic value.

This is obviously very theoretical, since these BS assumption are not observed in the real world. Also, don't forget from the start that your delta depends on the vol you considered.