Chelsea have won seven major competitions since Abramovich bought the west London club eight years ago, but the one trophy that has eluded the Russian billionaire is the Champions League.
The closest the Blues have come to lifting the biggest club competition in Europe was in 2008 when they lost to Manchester United in the final in Moscow on penalties.
Abramovich has gone through six managers trying to win the competition but there were indications from the Chelsea hierarchy they firmly believe Villas-Boas, 33, is the man to succeed where his predecessors have failed.
Buck said: "In three or four years' time, we expect to have a 36-year-old manager who's been with us for four years, we expect to have won a couple of trophies - preferably a big one in Europe - and of course we hope and expect to have a contented fan base, a contented board and a contented Mr Abramovich."
As well as his desire to win the Champions League, Abramovich is thought to be keen to own a team devoted to playing attractive football.
Sometimes winning trophies and playing good football do not go hand in hand, but Buck thinks it is possible.
"It's other people who say you cannot win trophies while you are developing your creativity," Buck add.
"We don't agree with that. It might take time, it might not. Why are we here but to have a successful football club?"
Buck spoke openly at the Leaders in Football conference about the possibility Chelsea may leave Stamford Bridge.
The Blues have offered to buy back the 12,000 Stamford Bridge shares owned by Chelsea Pitch Owners (CPO) so they can negotiate a potential deal for a new stadium in London.
The CPO shareholders will vote on whether to sell their shares on October 27, but the plan has already drawn strong criticism from some fans, who say they should not agree to sell their shares without knowing exactly where Chelsea could relocate.
Buck moved to address those concerns by saying: "Plenty of developers have come to us about various sites around London but they won't negotiate with us if they don't know for certain whether we can come to a deal.
"Therefore we have come to the view that if someone comes along with a proposal, and it makes sense to us, and that we want to negotiate a deal with them, we want to be able to negotiate with them and the current structure does not permit us to do that."
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