Lloyds Banking Group sold 35 per cent per cent of TSB in an initial public offering compared with the planned 25 per cent as investors signalled their faith in the prospects for the self-styled challenger bank.
The offer was ten times over subscribed by investors.
Some 30 per cent of the shares sold went to 60,000 retail investors.
Those that applied for up to £2,000 worth of shares received their full amount.
António Horta-Osório, chief executive of Lloyds Banking Group, said: "TSB has a national network of branches, a strong capital base, robust liquidity and significant economic protection against legacy issues.
"It is already operating on the UK high street and is proving to be a strong and effective challenger, further enhancing competition in the UK banking sector."
TSB has a conduct indemnity which means responsibility for past mis-selling of products like Payment Protection Insurance rests with Lloyds.
Some investors may have been encouraged to apply for shares in the belief that expected increases in interest rates will be good news for TSB, which has a big mortgage business.
Banks may try to increase the margins they make on loans if the Bank of England raises the base rate from 0.5 per cent.
"One element of investor interest has centred on the rate sensitivity of the name but you've also got a general recovery in the UK economically," said Joe Dickerson, an analyst at Jefferies investment bank.
However TSB has said it does not expect to pay a dividend until the 2017 financial year.
The bank set the offer price at 260p giving TSB a market capitalisation of £1.3 billion, with a book value of around £1.5bn.
After shares closed up 30p, at 290p, yesterday TSB had a stock market value of around £1.45 billionn.
The initial 260p valuation was just above the mid-point of the 220p-290p expected price range announced earlier this month
The float raised £455m for Lloyds and saw TSB return to the market for the first time since 1995 when it merged with Lloyds.
The business is Britain's seventh largest lender. It has 631 branches, including 189 in Scotland.
The bank can trace its roots to the formation of the trustee savings bank movement in 1810 by the Reverend Henry Duncan in Dumfriesshire.
While the executive team led by chief executive Paul Pester is based in London, TSB Group's registered office remains in Edinburgh.
Lloyds is required to sell the business as a condition of the £20bn bail-out of the bank following its rescue takeover of Edinburgh-based Halifax Bank of Scotland during the financial crisis.
The TSB share sale is another step on the recovery path for Lloyds and will help clear the way for the government to sell its remaining 25 per cent stake.
The flotation comes a year after the collapse of the planned sale of the business, which has 4.5 million customers, to Co-operative Bank.
It was revealed shortly afterwards that Co-op Bank had a large capital shortfall .