Have you been to an independent advisor with access to the whole market? Unless you are a bad risk, high LTV, poor credit history etc you should be able to do better than those rates.
Steve
I can, down to 2%......but they all come linked to ridiculous 'administration fees' of up to £3,000. Defeats the purpose of looking for a better deal. 
I think there are two question here, what will the rate be in 3 years and what will they be in 5 years? As these are the points where you will have to renew.
The BOE policy is to keep interest rates low for the foreseeable future until unemployment drops below 7% or inflation goes significantly above
5% 2.5%.
At the current 0.5% base rate the song by Yazz sums up interest rates: "The only way is up".
Now the Government will do all it can to provide an economic feel good factor for the election in 2 years time. After that who knows, what will happen to interest rates, but I think if a recovery has taken hold then with all of the QE money sloshing around in the UK and company profits being hammered since 2007, that they will try to restore them which will mean higher inflation. Add in ever rising energy prices, where UK fracking will have had no impact by then, then I think there is going to be inflationary pressures, but as a borrower, inflation is your friend for the capital, but your enemy for interest rates, as they will be under pressure to rise. If the economic growth does not reach launch velocity so growth continues to be very weak, then ZIRP will continue.
At the moment as a result of the US threatening to taper their QE program, their bond prices have dropped which means yields have risen so they have gone from around 1.7% to 2.7% for 10 years bonds. The same has happened in the UK. How long before real market rates affect what the Government does with official rates, who knows?
Whichever you pick, you could use the savings to accelerate the rate you pay back the capital, so after 3 or 5 years, whatever the market rate, you will have £4000-£7000 less capital to pay interest on which will soften the blow whatever the rates are. Thinking about it, the last sentence actually probably helps provide the answer. I know which I would choose, but that is me and it is your decision.

This is of course assuming where you are one of the richest people
in the UK on OOF that in 3 or 5 years, you won't just pay off the mortgage balance from a bit of loose change (by your standards) you have lying around.
