Semi-Closed Form Cubature and Applications to Financial Diffusion Models

TL;DR

This paper develops a semi-closed form cubature method for financial diffusion models, enabling explicit solutions to involved ODEs, reducing computational effort, and extending applicability beyond existing algorithms.

Contribution

It introduces a semi-closed form approach for cubature methods, expanding the class of models where explicit solutions are feasible, and proposes a variation of the Ninomiya-Victoir algorithm.

Findings

Explicit solutions for ODEs in certain models

Significant reduction in computation time

Extended applicability of cubature methods

Abstract

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.