Before revision:  Although there appeared a large amount of literature to deal with a so-called multi-curve framework, a complete picture for a multi-currency setup with currency funding spreads, which are necessary to explain non-zero cross currency basis , can be rarely found since our initial attempts 9, 10, 11 . This note gives an extension of these works regarding a general framework of interest rates for a fully collateralized market.
After revision:  Although there appeared a large amount of literature to deal with a so-called multi-curve framework, a complete picture for a multi-currency setup with currency funding spreads, which are necessary to explain non-zero cross currency basis , can be rarely found since our initial attempts 9, 10, 11 . This work gives its extension regarding a general framework of interest rates for a fully collateralized market.