NUFFT for the Fast COS Method

Paper: arXiv 2507.13186

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.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 7.8/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: 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.

Research Flowchart

  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"]

Cited References (11)

1. High-Performance Applications of the Non-Uniform Fast Fourier Transform to Option Pricing (2022)
2. A novel pricing method for European options based on Fourier -cosine series expansions (2009)
3. More robust pricing of European options based on Fourier cosine series expansions (2020) — link
4. Fourier integration and stochastic volatility calibration (2014)
5. Option valuation using the fast Fourier transform (1999)
6. Speed and biases of Fourier-based pricing choices: a numerical analysis (2018)
7. Optimal Fourier inversion in semi-analytical option pricing (2006)
8. On the number of terms in the COS method for European option pricing — → 2303.16012

1 of 11 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.