The theory of valued difference fields (K, σ, υ) depends on how the valuation υ interacts with the automorphism σ. Two special cases have already been worked out - the isometric case, where υ(σ(x)) = υ(x) for all x Î G has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where υ(σ(x)) > nv(x) for all x Î K x with υ(x) > 0 and n Î N, has been worked out by Salih (...) Azgin. In this paper we deal with a more general version, the multiplicative case, where υ(σ(x)) = p · υ(x), where p (> 0) is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for this theory. (shrink)