A new weak approximation scheme of stochastic differential equations and the Runge-Kutta method

Abstract

In this paper, authors successfully construct a new algorithm for the new higher order scheme of weak approximation of SDEs. The algorithm presented here is based on [1][2]. Although this algorithm shares some features with the algorithm presented by [3], algorithms themselves are completely different and the diversity is not trivial. They apply this new algorithm to the problem of pricing Asian options under the Heston stochastic volatility model and obtain encouraging results. [1] Shigeo Kusuoka, "Approximation of Expectation of Diffusion Process and Mathematical Finance," Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, "Cubature on Wiener Space," Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198. [3] Syoiti Ninomiya, Nicolas Victoir, "Weak approximation of stochastic differential equations and application to derivative pricing," Applied Mathematical Finance, Volume 15, Issue 2 April 2008, pages 107--121