A second order modified block copolymer which can be obtained by reacting a first order modified block copolymer with a second order modifier, wherein the first order modified block copolymer comprises a base block copolymer and a functional group-containing first order modifier group bonded to the base block co-polymer, wherein the base block copolymer comprises at least one polymer block comprised mainly of vinyl aromatic hydrocarbon monomer units and at least one polymer block comprised mainly of conjugated diene monomer units, and wherein the second order modifier has a specific functional group which is reactive to the functional group of the first order modifier group of the first order modified block copolymer. A second order modified block copolymer containing polymer composition comprising the second order modified block copolymer as well as a thermoplastic resin and/or a rubbery polymer.