Semi-closed form cubature and applications to financial diffusion models

Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [<italic>Adv. Math. Econ</italic>., 2004, <bold>6</bold>, 69--83] and Lyons--Victoir [<italic>Proc. R. Soc. Lond. Ser. A</italic>, 2004, <bold>460</bold>, 169--198], involve the solution to numerous auxiliary ordinary differential equations (ODEs). With focus on the Ninomiya--Victoir algorithm [<italic>Appl. Math. Finance</italic>, 2008, <bold>15</bold>, 107--121], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and show the 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.