Get a read on your appointment letter. To be safe, just follow the notice period. Once u received your offer letter and all the conditions stated are acceptable, just tender your resignation. If u are released earlier then u can inform the future company to amend the offer letter. Key point is to get the offer letter ready.

Nope it is not ok to reject your 1st employer since u have signed the offer letter. However u can try to talk to your 1st employer on the conditions to let u go off without any payment in lieu. If that didn't work out and u still persist to join the bank, then just pay the salary in lieu and tender your resignation on the 1st day of work at the 1st employer. U'll get blacklisted though.

If u fulfill the conditions stated in your offer letter in regard to termination/resignation then unlikely the company will sue u. Suggest u read carefully and talk to the HR of that company. Try to achieve mutual content on your intention to leave on the 1st day of work or earlier.