  Cubature methods, a powerful alternative to Monte Carlo due to
Kusuoka~[Adv.~Math.~Econ.~6, 69--83, 2004] and
Lyons--Victoir~[Proc.~R.~Soc.\\Lond.~Ser.~A 460, 169--198, 2004], involve the
solution to numerous auxiliary ordinary differential equations. With focus on
the Ninomiya-Victoir algorithm~[Appl.~Math.~Fin.~15, 107--121, 2008], which
corresponds to a concrete level $5$ cubature method, we study some parametric
diffusion models motivated from financial applications, and exhibit structural
conditions under which all involved ODEs can be solved explicitly and
efficiently. We then enlarge the class of models for which this technique
applies, by introducing a (model-dependent) variation of the Ninomiya-Victoir
method. Our method remains easy to implement; numerical examples illustrate the
savings in computation time.
