Failure of Fourier pricing techniques to approximate the Greeks

ArXiv ID: 2306.08421 “View on arXiv”

Authors: Unknown

Abstract

The Greeks Delta and Gamma of plain vanilla options play a fundamental role in finance, e.g., in hedging or risk management. These Greeks are approximated in many models such as the widely used Variance Gamma model by Fourier techniques such as the Carr-Madan formula, the COS method or the Lewis formula. However, for some realistic market parameters, we show empirically that these three Fourier methods completely fail to approximate the Greeks. As an application we show that the Delta-Gamma VaR is severely underestimated in realistic market environments. As a solution, we propose to use finite differences instead to obtain the Greeks.

Keywords: Greeks (Delta/Gamma), Variance Gamma model, Fourier techniques, Value at Risk (VaR), Carr-Madan formula, Options / Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper exhibits high mathematical density through advanced Fourier analysis, characteristic functions, and rigorous proofs of conditions for method validity, warranting a high math score. It demonstrates strong empirical rigor with detailed numerical experiments using realistic market parameters (VG and ME models), Monte Carlo simulations for ground truth, and quantitative error analysis of VaR underestimation, placing it in the Holy Grail quadrant.

  flowchart TD
    A["Research Goal:<br>Can Fourier methods<br>reliably compute<br>option Greeks?"] --> B["Data/Inputs:<br>Variance Gamma model<br>parameters, market data"]
    B --> C["Methodology:<br>Compare Fourier methods<br>Carr-Madan, COS, Lewis<br>vs. Finite Differences"]
    C --> D{"Computational Process:<br>Evaluate Greeks"}
    D --> E["Outcome 1:<br>Fourier methods fail<br>for specific parameters"]
    D --> F["Outcome 2:<br>Delta-Gamma VaR<br>severely underestimated"]
    E --> G["Recommendation:<br>Use Finite Differences<br>for accurate Greeks"]
    F --> G