Higher order K-scheme and application to derivative pricing

Abstract

The author presents a new discretization method of stochastic differential equations (SDEs) without proof. The method is a kind of K-scheme (Kusuoka approximation, Kusuoka-Lyons-Ninomiya-Victoir method). K-scheme is a new framework of the higher order discretization methods for approximating SDEs weakly, proposed by Kusuoka. Ninomiya-Victoir and Ninomiya-Ninomiya methods are known as practical realizations of K-scheme, and some extrapolations of these methods are proposed. The new method is higher order than these methods without the use of extrapolations, and the author applies it to a financial problem of derivative pricing.

Higher-order Discretization Methods of Forward-backward SDEs Using KLNV-scheme and Their Applications to XVA Pricing

“…To overcome this problem, the TBBA (Crisan and Lyons 2002) was applied to these methods in (Ninomiya 2003b;Ninomiya and Mitsuzono 2004;Ninomiya 2010;Crisan and Ortiz-Latorre 2013). On the other hand, using the framework of KLNVscheme and the Gaussian random variables, the Ninomiya-Victoir (N-V) (Ninomiya and Victoir 2008) and Ninomiya-Ninomiya (N-N) (Ninomiya and Ninomiya 2009) methods have been proposed as practically feasible second-order methods, and the Q ð7;2Þ ðsÞ method (Shinozaki 2016(Shinozaki , 2017 as a third order method. Note that the N-V and N-N methods are implemented by using only substitutions and linear combinations in the same way as explicit Runge-Kutta methods are and are free from symbolic calculations.…”