[Paper Review] A weak approximation with asymptotic expansion and multidimensional Malliavin weights

Takahashi, Akihiko, Yamada, Toshihiro|arXiv (Cornell University)|May 4, 2016

Stochastic processes and financial applications37 citations

TL;DR

This paper introduces a novel weak approximation scheme for expectations of diffusion processes by combining asymptotic expansion with multidimensional Malliavin weights, achieving high accuracy—especially for deep out-of-the-money options—using only a few time partitions. The method leverages Watanabe and Kusuoka theories in Malliavin calculus to rigorously justify the approximation, significantly improving convergence rates over standard asymptotic expansions.

ABSTRACT

This paper develops a new efficient scheme for approximations of expectations of the solutions to stochastic differential equations (SDEs). In particular, we present a method for connecting approximate operators based on an asymptotic expansion with multidimensional Malliavin weights to compute a target expectation value precisely. The mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin calculus. Moreover, numerical experiments for option pricing under local and stochastic volatility models confirm the effectiveness of our scheme. Especially, our weak approximation substantially improves the accuracy at deep Out-of-The-Moneys (OTMs).

Motivation & Objective

• To develop a high-accuracy weak approximation scheme for expectations of solutions to stochastic differential equations (SDEs), particularly for financial derivatives pricing.
• To address the well-known limitation of asymptotic expansions—poor tail accuracy—especially in deep out-of-the-money (OTM) regions.
• To integrate asymptotic expansion-based operators with multidimensional Malliavin weights to enhance precision in Monte Carlo simulations.
• To provide a mathematically rigorous foundation using Watanabe and Kusuoka theories in Malliavin calculus.
• To demonstrate through numerical experiments that only a small number of time partitions (e.g., n=2) can yield substantial accuracy improvements over standard asymptotic expansions.

Proposed method

• The method constructs a sequence of approximate operators Qε,m_s based on asymptotic expansion up to order m for short time intervals s.
• It composes these operators sequentially over a time partition π = {t₀ < t₁ < ... < tₙ = T} to approximate the full-time transition operator Pε_T.
• Multidimensional Malliavin weights are used to represent the derivative of the solution with respect to the perturbation parameter ε, enabling precise integration by parts.
• The approach relies on Kusuoka–Stroock function spaces Kr and Malliavin calculus to ensure mathematical validity and regularity of the approximations.
• Theoretical justification is grounded in Watanabe’s integration by parts formula and Kusuoka’s high-order weak approximation framework.
• Numerical implementation uses the Kusuoka scheme for high-order weak approximation, combined with asymptotic expansion for the density correction.

Experimental results

Research questions

• RQ1 Can asymptotic expansion be systematically combined with Malliavin calculus to improve weak approximation accuracy in SDE expectations?
• RQ2 How does the proposed scheme perform in the deep out-of-the-money (OTM) region, where standard asymptotic expansions fail?
• RQ3 What is the minimal number of time partitions required to achieve high accuracy with the new scheme?
• RQ4 Can the use of multidimensional Malliavin weights significantly reduce approximation error in Monte Carlo simulations?
• RQ5 How does the convergence rate of the proposed scheme compare to standard asymptotic expansions and Euler–Maruyama schemes?

Key findings

• The proposed scheme achieves substantially improved accuracy for deep out-of-the-money (OTM) options, where standard asymptotic expansions fail due to tail deviation.
• With only n=2 time partitions, the method reduces errors significantly compared to first- and second-order asymptotic expansions.
• Theoretical error bounds show convergence rates of O(ε^{m+1}) for smooth functions, O(ε^{m+2}) for Lipschitz functions, and O(ε^{m+1}) for bounded continuous functions.
• For γ > (l−2)/l, the error decays as O(n^{-(l−2)/2}) when using n partitions, indicating favorable convergence with increasing partition size.
• Numerical experiments confirm the effectiveness of the scheme under both local and stochastic volatility models, validating the theoretical claims.
• The method maintains high computational efficiency while achieving high accuracy, making it suitable for real-time risk management and trading systems.