Does time premium include the Saturday at expiration or it should technically be 0 (OTM) on Friday at 4pm?
Since time premium is the part of the option price above its intrinsic value. Since you can't trade a option after market close on the expiration Friday, I believe that the time premium is zero.
From the end of trading to the exercise deadline, there should be a "small" amount of time value remaining to reflect the possibility that something can happen to bring the option in-the-money during that final interval.
In theory Naz is right but in reality time premium is zero at the close friday since you cant trade em. TV of time premium is pretty meaningless on Friday of expiration. I doubt there are any firms left who price to Saturday.
Well the SPX options, which cease trading at 4 PM on Thursday and expire at the Friday open, still have a fair amount of time value priced into them right up to 3:59:59 on Thursday. Today the SPX closed at 1165.76. The 1165 calls, with 0.76 of intrinsic value, have a closing price of 3.80. The 1170 calls, wholly OTM, went for 0.95 at the very end of the trading day. So obviously somebody is pricing in a significant SPX overnight move. Of course this is quite different from an option that expires Saturday, as markets don't open on Saturday. So whether time value is significant after the last minute of trading depends on the kind of option you're talking about.
Since he asked about saturday the SPX reference is moot. More than that the SPX is a whole different animal since there is so much volatility in the settlement price.