You can check your luck value for the day by watching the Fortune Teller on TV. My question is what happens if you do not do this? Are you assigned a default luck value for the day? Or is your luck assigned independently and the Fortune Teller just a means of checking it?
The fortune forecast just returns a phrase reflecting the value of the player's luck for that day. It doesn't change anything about the player's luck whether it's watched or not. The code tests the value of the player's luck and selects appropriate text to reflect it.
Also, the opening phrase (e.g. "Ah... I sense that a new viewer has joined us.") is purely random; it has no effect on gameplay.
(Source: Snooping through the code with a decompiler)
Although the wiki doesn't state much detailed information:
A user in reddit did some code digging to learn more about luck modifiers and variants. As he/she states (emphasis mine):
More information on https://www.reddit.com/r/StardewValley/comments/4c2gx4/a_comprehensive_guide_to_luck/, it's a very interesting thread btw.
So, long story short, luck is assigned when you go to sleep, the Fortune Teller only show you the random luck modifier you got.