While the Blues have enjoyed domestic success since Roman Abramovich's takeover in 2003, European club football's top prize has eluded them.
In that time the club have had six managers with many believed to have been sacked by the Russian billionaire due to their failure to deliver the trophy.
"Three years from now I think we'll have a 36-year-old manager who has been with us for four years," Buck told reporters, alluding to Villas-Boas.
"We'll have won a couple of trophies including one with the word 'European' in its title, and we will have a contented fan base and a contented owner."
The closest Chelsea have come to winning the Champions League was in 2008 when they lost to Manchester United on penalties in the final in Moscow.