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

Mariko Ninomiya, Syoiti Ninomiya

TL;DR

The paper tackles the problem of efficiently computing weak approximations of SDEs, which is critical for fast derivative pricing in finance. It proposes a new high-order weak approximation scheme built on the Kusuoka–Ninomiya–Lyons–Victoir framework, recasting the SDE into an ODE-valued random variable via a Lie algebra representation and realizing the exponential flows with Runge–Kutta methods. Theoretical contributions include error bounds showing an $O(s^{(m+1)/2})$ convergence and a constructive procedure for the Lie-algebra valued random variables $Z_1,\dots,Z_M$, along with a Romberg extrapolation path to higher order. The method is demonstrated on Asian option pricing under the Heston model, achieving significant speedups compared to Euler–Maruyama and Ninomiya–Victoir schemes, aided by quasi-Monte Carlo sampling and reduced integration dimensionality.

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

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

TL;DR

The paper tackles the problem of efficiently computing weak approximations of SDEs, which is critical for fast derivative pricing in finance. It proposes a new high-order weak approximation scheme built on the Kusuoka–Ninomiya–Lyons–Victoir framework, recasting the SDE into an ODE-valued random variable via a Lie algebra representation and realizing the exponential flows with Runge–Kutta methods. Theoretical contributions include error bounds showing an convergence and a constructive procedure for the Lie-algebra valued random variables , along with a Romberg extrapolation path to higher order. The method is demonstrated on Asian option pricing under the Heston model, achieving significant speedups compared to Euler–Maruyama and Ninomiya–Victoir schemes, aided by quasi-Monte Carlo sampling and reduced integration dimensionality.

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

Paper Structure

This paper contains 22 sections, 22 theorems, 88 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The problem and background
  3. The problem
  4. Background
  5. Our results
  6. Notation
  7. Main results
  8. Proof of Theorem \ref{['contp_krk_main_thm1']}
  9. Construction of the $\mathcal{L}_\mathbb{R}((A))$-valued random variables $Z_1,\dots, Z_M$
  10. The Runge--Kutta method
  11. The new simulation scheme and Corollary \ref{['contp_krk_main_cor']}
  12. Application
  13. Simulation
  14. The algorithm and competitors
  15. The algorithm of the new method
  16. ...and 7 more sections

Key Result

theorem 1.3

Let $m\geq 1$, $M\geq 2$, and $Z_1,\dots, Z_M$ be $\mathcal{L}_{\mathbb R}((A))$-valued random variables. Assume that $Z_1,\dots, Z_M$ satisfy the followings: Then for $p\in [1, \infty)$ and arbitrary $g_1,\dots,g_M\in\mathcal{IS}(m)$, there exists a positive constant $C_{m,M}$ such that for $s\in(0,1]$ where $C_{m,M}$ depends only on $m$ and $M$. Here for functions $f$ and $g$, $f\circ g(x)$ de

Figures (2)

  • Figure 1: Error coming from the discretization
  • Figure 2: Convergence Error from quasi-Monte Carlo and Monte Carlo

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • proof
  • ...and 34 more