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Boundary error control for numerical solution of BSDEs by the convolution-FFT method

Xiang Gao, Cody Hyndman

TL;DR

This work tackles boundary truncation errors in convolution-FFT solutions to BSDEs used for option pricing. It introduces a damping-and-shifting strategy, forming a damped target $\tilde{u}(x)=e^{\alpha x}(u(x)-h(x))$ with $\alpha<0$ and $h(x)=Ae^{x}+B$, and fixes the damping while updating shifts to achieve periodicity, enabling stable Fourier-domain updates for $Y$ and $Z$. The authors derive explicit multipliers in the Fourier domain, provide local and global error bounds, and demonstrate through European option pricing that boundary artifacts are markedly reduced and delta surfaces become more reliable. The method enhances the practical applicability of convolution-FFT BSDE solvers and is poised for extension to higher-dimensional and more general BSDE settings.

Abstract

We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method.

Boundary error control for numerical solution of BSDEs by the convolution-FFT method

TL;DR

This work tackles boundary truncation errors in convolution-FFT solutions to BSDEs used for option pricing. It introduces a damping-and-shifting strategy, forming a damped target with and , and fixes the damping while updating shifts to achieve periodicity, enabling stable Fourier-domain updates for and . The authors derive explicit multipliers in the Fourier domain, provide local and global error bounds, and demonstrate through European option pricing that boundary artifacts are markedly reduced and delta surfaces become more reliable. The method enhances the practical applicability of convolution-FFT BSDE solvers and is poised for extension to higher-dimensional and more general BSDE settings.

Abstract

We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method.
Paper Structure (11 sections, 4 theorems, 54 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 4 theorems, 54 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Assumptions and the convolution method
  3. Boundary control schemes
  4. Boundary effects and damping--shifting strategy
  5. Fourier-domain representation of the damped--shifted scheme
  6. Discrete Fourier implementation
  7. Error analysis
  8. Numerical result of option pricing
  9. Conclusion
  10. Appendix
  11. Proof of Theorem \ref{['thm_conv_bsde']}

Key Result

Lemma 3.1

Assuming the integrable function $f(x)$ is bounded by $\mathaccentV{bar}0\symboldoperators16 f$ on $[-\frac{L}{2},\frac{L}{2}]$ and admits the Fourier expansion with coefficients defined by Suppose the discrete Fourier coefficient has bounded error $\left|F_j - \mathaccentV{bar}0\symboldoperators16 F_j\right|\leq \epsilon_L N^{-m}$ for $m\geq 2$ and some constant $\epsilon_L>0$. Then the convol

Figures (3)

  • Figure 1: Call option price and delta errors - convolution method of hyndman2017convolution
  • Figure 2: Call option price and delta errors - convolution method with boundary error control.
  • Figure 3: Call option delta surface - convolution method with boundary error control

Theorems & Definitions (7)

  • Lemma 3.1: Error of the convolution method
  • Remark 3.1: Boundary problem
  • Remark 3.2: Error transfer with damping parameter
  • Lemma 3.2: Local error of the convolution method with damping and shifting
  • Proposition 3.1: Stability and convergence
  • Theorem 3.1: Global error bounds
  • proof