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Weak approximation of stochastic differential equations and application to derivative pricing

Syoiti Ninomiya, Nicolas Victoir

TL;DR

The paper tackles efficient weak approximation of expectations E[f(Y(1,x))] for SDEs by introducing a second-order weak scheme that uses Bernoulli-Gaussian randomization to drive step updates. It proves the new method achieves an error bound of |E[f(X^{New,n}_1)] − E[f(Y(1,x))]| ≤ C_f/n^2 and demonstrates substantial practical gains when combined with quasi-Monte Carlo for pricing Asian options under the Heston stochastic volatility model. The approach preserves stability and general applicability, avoiding negative volatility issues that can plague Euler-based schemes and enabling fast, high-accuracy derivative pricing. Overall, the combination of a simple, robust 2nd-order weak scheme with QMC yields dramatic speedups over traditional Euler-based methods for high-dimensional, diffusion-driven pricing problems.

Abstract

The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [1] and [2]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast. [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.

Weak approximation of stochastic differential equations and application to derivative pricing

TL;DR

The paper tackles efficient weak approximation of expectations E[f(Y(1,x))] for SDEs by introducing a second-order weak scheme that uses Bernoulli-Gaussian randomization to drive step updates. It proves the new method achieves an error bound of |E[f(X^{New,n}_1)] − E[f(Y(1,x))]| ≤ C_f/n^2 and demonstrates substantial practical gains when combined with quasi-Monte Carlo for pricing Asian options under the Heston stochastic volatility model. The approach preserves stability and general applicability, avoiding negative volatility issues that can plague Euler-based schemes and enabling fast, high-accuracy derivative pricing. Overall, the combination of a simple, robust 2nd-order weak scheme with QMC yields dramatic speedups over traditional Euler-based methods for high-dimensional, diffusion-driven pricing problems.

Abstract

The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [1] and [2]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast. [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.

Paper Structure

This paper contains 20 sections, 1 theorem, 33 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Problem and its Motivation
  3. Notation
  4. Probabilistic Methods
  5. Order 1
  6. Higher order
  7. Romberg Extrapolation
  8. A remark on the Monte Carlo method
  9. A remark on the quasi-Monte Carlo method
  10. Presentation of the new Algorithm
  11. Numerical Example: Application to Finance
  12. Implementation of the algorithm
  13. Solutions of the ODEs
  14. A remark on general implementation
  15. Application of the quasi-Monte Carlo method
  16. ...and 5 more sections

Key Result

theorem 1

Let $\left( \Lambda _{i},Z_{i}\right) _{i\in \left\{ 1,\cdots ,n\right\} }$ be $n$ independent random variables, where each $\Lambda _{i}$ is a Bernoulli random variable independent of $Z_{i}$, which is a standard $d$-dimensional normal random variable. Define $\{X^{({\rm New}),n}_{k/n}\}_{k=0,\dots Then, for all $f\in C^\infty_b(\mathbb{R}^N)$, that is, our new algorithm is of order $2$.

Figures (2)

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

Theorems & Definitions (8)

  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 1
  • proof
  • remark thmcounterremark