thanks. Just for the GARCH gamma, in the book it says it is difference between present delta and future delta. it's just so confusing here what it means by saying future delta and 'future' historical volatility and 'future' implied volatility??? what the 'future' means??
I think you're trying to run before you can walk. Take your time, go through each concept. You're not really explaining the context very well, and asking too many things, so it is hard to help.
What you seem to be talking about is models for a volatility that's changing in time, hence present and future.
Future historical volatility is probably confusing as a term, so if it does confuse you, switch the word historical with 'realised' for example or with 'statistical'.
We have info now about realised (historical) volatility, we also have implied volatility. That's now. As time goes by we'll have new data that has been realised and we could determine a new realised vol, and also the implied vol might change. So the question then is, what can we tell or estimate about these two things - the future realised vol, and the future implied vol - given that we're stuck in the present. Something? Nothing?
So we may for example pick a mathematical model, like GARCH, which models the vol and from this model derive something useful. We may consider that deep out of the money options tell us something about future vol.
Then again, I could be way off because I don't really have the context. In any case, this is reasonably advanced, and only a day or two ago you were struggling with greeks in the simple constant vol black scholes model. Slow down...
EDIT:- Why are you listed as a trading arcade or prop trading company?