  High order discretization schemes of SDEs by using free Lie algebra valued
random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and
Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving
the flows of vector fields associated with SDEs and it is usually done by
numerical methods. The authors found a special Lie algebraic structure on the
vector fields in the major financial diffusion models. Using this structure, we
can solve the flows associated with vector fields analytically and efficiently.
Numerical examples show that our method saves the computation time drastically.
