Convolution-FFT for option pricing in the Heston model

ArXiv ID: 2512.05326 “View on arXiv”

Authors: Xiang Gao, Cody Hyndman

Abstract

We propose a convolution-FFT method for pricing European options under the Heston model that leverages a continuously differentiable representation of the joint characteristic function. Unlike existing Fourier-based methods that rely on branch-cut adjustments or empirically tuned damping parameters, our approach yields a stable integrand even under large frequency oscillations. Crucially, we derive fully analytical error bounds that quantify both truncation error and discretization error in terms of model parameters and grid settings. To the best of our knowledge, this is the first work to provide such explicit, closed-form error estimates for an FFT-based convolution method specialized to the Heston model. Numerical experiments confirm the theoretical rates and illustrate robust, high-accuracy option pricing at modest computational cost.

Keywords: Heston Model, Fast Fourier Transform (FFT), European Option Pricing, Analytical Error Bounds, Convolution Method, Equity Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper involves advanced mathematical concepts including stochastic differential equations, characteristic functions, and deriving explicit analytical error bounds, indicating high math complexity. The empirical rigor is high due to detailed numerical experiments that confirm theoretical rates and compare with established methods like the Carr-Madan approach, though it lacks raw code or datasets, focusing more on algorithmic implementation.

  flowchart TD
    A["Research Goal: Develop a stable<br/>FFT method for Heston model<br/>with analytical error bounds"] --> B["Methodology: Derive continuously<br/>differentiable characteristic function<br/>for convolution-FFT method"]
    
    B --> C{"Inputs & Setup"}
    C --> C1["Heston Model Parameters<br/>Variance, Volatility of Volatility, Correlation"]
    C --> C2["Numerical Grid Settings<br/>Discretization steps, Truncation limit"]
    
    C1 & C2 --> D["Computational Process:<br/>Fast Fourier Transform<br/>on discretized characteristic function"]
    
    D --> E["Key Findings:<br/>• Fully analytical error bounds<br/>• Stable integrand without damping<br/>• High accuracy at modest cost<br/>• First explicit error estimates for Heston FFT"]
    
    style A fill:#e1f5fe
    style B fill:#fff3e0
    style C fill:#f3e5f5
    style C1 fill:#e8f5e8
    style C2 fill:#e8f5e8
    style D fill:#fce4ec
    style E fill:#e8f5e8