NUFFT for the Fast COS Method

ArXiv ID: 2507.13186 “View on arXiv”

Authors: Fabien LeFloc’h

Abstract

The COS method is a very efficient way to compute European option prices under Lévy models or affine stochastic volatility models, based on a Fourier Cosine expansion of the density, involving the characteristic function. This note shows how to compute the COS method formula with a non-uniform fast Fourier transform, thus allowing to price many options of the same maturity but different strikes at an unprecedented speed.

Keywords: COS Method, Fast Fourier Transform (FFT), Option Pricing, Lévy Models, Affine Stochastic Volatility, Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 7.8/10
• Quadrant: Holy Grail
• Why: The paper is deeply mathematical, featuring advanced Fourier analysis (NUFFT type-2/3), complex characteristic function derivations, and detailed truncation range formulas, warranting a high math score. It also provides strong empirical backing with concrete speed benchmarks, error metrics (RMSE/MAE) for two major models (Variance Gamma and Heston), and a direct comparison to established methods, justifying a high rigor score.

  flowchart TD
    A["Research Goal<br>Speed up COS method for<br>multi-strike option pricing"] --> B{"Key Methodology"}
    B --> C["Leverage COS Method Fourier expansion"]
    B --> D["Apply NUFFT<br>Non-Uniform Fast Fourier Transform"]
    
    C & D --> E{"Data & Inputs"}
    E --> F["Lévy / Affine SV Models<br>Characteristic Function"]
    E --> G["Option Strikes<br>Non-uniform grid"]
    
    F & G --> H["Computational Process<br>NUFFT-based COS Formula"]
    
    H --> I{"Key Outcomes"}
    I --> J["Unprecedented Speed<br>for batch pricing"]
    I --> K["High Accuracy<br>preserved"]