[Paper Review] Weak approximation of stochastic differential equations and application to derivative pricing

Syoiti Ninomiya, Nicolas Victoir|arXiv (Cornell University)|May 14, 2006

Stochastic processes and financial applications9 references5 citations

TL;DR

This paper introduces a novel weak approximation algorithm for stochastic differential equations (SDEs) based on cubature on Wiener space, achieving higher accuracy with fewer time steps than Euler-Maruyama. When combined with quasi-Monte Carlo methods, it enables extremely fast and precise pricing of Asian options under the Heston model, outperforming standard methods by up to 80 times in computational speed for 10⁻⁴ accuracy.

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.

Motivation & Objective

• To develop a universal, efficient probabilistic method for weak approximation of SDEs applicable to general diffusion processes.
• To improve computational efficiency in derivative pricing, particularly for path-dependent options like Asian options under stochastic volatility.
• To combine the new algorithm with quasi-Monte Carlo techniques to achieve faster convergence and reduced computational cost.
• To demonstrate superior performance compared to standard Euler-Maruyama and cubature-based methods in terms of accuracy and runtime.

Proposed method

• The method uses a modified cubature on Wiener space approach with a recursive exponential map to approximate the solution of Stratonovich SDEs.
• It applies a time discretization scheme based on the exponential of vector fields, leveraging the Wong-Zakai theorem for weak convergence.
• The algorithm employs a two-level time partitioning strategy with a novel extrapolation technique to enhance convergence order.
• It integrates quasi-Monte Carlo (QMC) sampling to reduce variance and accelerate convergence, especially effective due to the algorithm’s low-discrepancy structure.
• The method is validated through numerical experiments on the Heston model, comparing error, sample size, and CPU time.
• The use of Romberg extrapolation further improves convergence rates without increasing computational complexity unduly.

Experimental results

Research questions

• RQ1 Can a new weak approximation algorithm for SDEs be designed that is both universal and significantly faster than existing methods for derivative pricing?
• RQ2 How does the proposed algorithm perform in combination with quasi-Monte Carlo methods compared to standard Monte Carlo or Euler-Maruyama schemes?
• RQ3 What is the required number of time partitions and sample points to achieve 10⁻⁴ accuracy in Asian option pricing under the Heston model?
• RQ4 Does the proposed method maintain high efficiency across different levels of discretization and sampling strategies?
• RQ5 To what extent does the combination of the new algorithm and quasi-Monte Carlo outperform traditional methods in terms of computational speed and accuracy?

Key findings

• The new algorithm achieves 10⁻⁴ accuracy with only 12 time partitions, compared to 2000 for the standard Euler-Maruyama scheme.
• With Romberg extrapolation, the new method requires just 4+2 partitions to reach 10⁻⁴ accuracy, while Euler-Maruyama needs 16+8 partitions.
• Using quasi-Monte Carlo, the new method requires only 2×10⁵ samples for 10⁻⁴ accuracy, whereas Euler-Maruyama needs 5×10⁶ samples.
• The new algorithm with quasi-Monte Carlo and extrapolation achieves a 1.73-second runtime for 10⁻⁴ accuracy, outperforming Euler-Maruyama by a factor of 80.
• The performance gain is attributed to the algorithm’s structure, which aligns well with the low-discrepancy nature of quasi-Monte Carlo sequences.
• The method demonstrates that combining high-order weak approximation with QMC yields superior convergence and efficiency in derivative pricing.