Paper: arXiv 2512.24714

Authors: Xiang Gao, Cody Hyndman

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper involves advanced stochastic calculus, partial differential equations, and Fourier analysis with detailed error bounds and convergence proofs. The empirical component is limited to numerical illustrations and theoretical error analysis without real-world data, backtests, or implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Improve boundary error control in<br>CFFT method for BSDEs"] --> B["Methodology<br>Modify damping & shifting schemes<br>Time-dependent shifting for boundary error"]
  B --> C["Computational Process<br>Implement modified CFFT with<br>numerical experiments"]
  C --> D["Key Findings<br>Significantly reduced boundary errors<br>Faster convergence & higher accuracy<br>Validated with error analysis"]

Cited References (17)

  1. Convolution-FFT for option pricing in the Heston model — → 2512.05326
  2. Efficient numerical Fourier methods for coupled forward-backward SDEs (2016) — link
  3. A Fourier Interpolation Method for Numerical Solution of FBSDEs: Global Convergence, Stability, and Higher Order Discretizations (2014) — link
  4. A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations (2013) — link
  5. A forward scheme for backward SDEs (2007) — link
  6. Optimal Fourier Inversion in Semi-Analytical Option Pricing (2007) — link
  7. Forward-Backward Stochastic Differential Equations and their Applications (2007) — link
  8. Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations (2006) — link
and 9 more
  1. Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations (2004) — link
  2. Backward stochastic differential equations and partial differential equations with quadratic growth (2000) — link
  3. Numerical methods for forward-backward stochastic differential equations (1996) — link
  4. Backward-Forward Stochastic Differential Equations (1993) — link
  5. Adapted solution of a backward stochastic differential equation (1990) — link
  6. Stochastic control, numerical methods, and machine learning in finance and insurance (2021)
  7. Discrete-Time Approximation of BSDEs and Probabilistic Schemes for Fully Nonlinear PDEs (2009)
  8. Option valuation using the fast Fourier transform (1999) — link
  9. The Fokker-Planck Equation (2nd ed.) (1996)

1 of 17 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.