In this paper, we study a semiparametric multiplicative volatility model, which splits up into a nonparametric part and a parametric GARCH component. The nonparametric part is modelled as a product of a deterministic time trend com- ponent and of further components that depend on stochastic regressors. We propose a two-step procedure to estimate the model. To estimate the nonpara- metric components, we transform the model in order to apply the backfitting procedure used in Vogt and Walsh (2019). The GARCH parameters are esti- mated in a second step via quasi maximum likelihood. We show consistency and asymptotic normality of our estimators. Our results are obtained using mixing properties and local stationarity. We illustrate our method using finan- cial data. Finally, a small simulation study illustrates a substantial bias in the GARCH parameter estimates when omitting the stochastic regressors.