  We establish monoidal model structures on model categories of filtered chain
complexes constructed by Cirici, Egas Santander, Livernet and Whitehouse whose
weak equivalences are the quasi-isomorphisms on the $r$-page of the associated
spectral sequences. In doing so we provide a partial classification of
cofibrant objects and cofibrations of the model structures involving a
boundedness restriction on the filtration. As a consequence we also obtain, by
results of Schwede and Shipley, cofibrantly generated model structures on the
categories of filtered differential graded algebras as well as their modules.
