Is is true that a callable option is most beneficial for Bank2 when Bank1 calls it after the first payment at end of 6m?A 10year non-call 6m ---Bank1 (Issuer): Issues a rangeswap (say 6%*n/d) indexed to LIBORBank2: Pay LIBOR Bank1 has right to cancel this notethanks

In your case, the callable option is written by Bank 2 hence there is an option premium received by Bank 2. In a non-standard case, the contract may have callable/cancellable conditions. For example, If bank 1 calls the swap within the first two years of the deal, bank 1 may incurr some penalty charge for early termination.

Thanks.Let me quickly rephrase the question: Why would Bank2 ever do a *callable* range-note?Of course, for a non-callable range-note, Bank2 is negative gamma in the range (it wants the LIBOR to stay in the range.) and then makes money.In the callable case however, it will quickly be called by Bank1. If out of the range, Bank2 keeps paying LIBOR.So then what is the incentive for Bank2 to ever do such a *callable* trade at all?

When NPV is zero, you expect much higher coupon on the range leg compared to LIBOR. However, as issuer of the range, you don't want to end up in a situation where you are always in the range (e.g. if the trade was done 1 years ago with range defined as pay 6% when USD 3M LIBOR > 4.5%, you would like to buy some insurance such that you don't end up paying 6% when LIBOR is 5.25% .... To bank 2 (In fact, most of the end user are corporate clients): 1) This is off-balance sheet (If they use it for hedging purpose)2) The callable feasure is yield enhancing 3) Bank 2 has an opposite view Everything has a price.

xdlyn is correct.To expand on this. Main reason is increased coupon.Bank 2 generally wants the highest coupon level possible and is willing to sell some optionallity back to bank 1 to get this.