The inventive composition includes a block copolymer which includes four blocks and has the following block arrangement: B-S-B-S. The two polymer blocks B comprise conjugated diene monomer constitutional units; and the two polymer blocks S comprise vinyl aromatic hydrocarbon monomer constitutional units. The invention also includes methods of making such inventive compositions as well as asphalt compositions and/or thermoplastic elastomers which include such block copolymer.