On the number of terms in the COS method for European option pricing

Paper: arXiv 2303.16012

Authors: Gero Junike

Abstract

The Fourier-cosine expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the sensitivities, i.e., the Greeks Delta and Gamma, provided the density of the log-returns is smooth. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper provides valuable theoretical contributions by deriving explicit truncation bounds and convergence rates for the COS method in option pricing, including heavy-tailed distributions. While heavily mathematical with thorough analytical derivations, empirical verification is limited to standard numerical experiments rather than empirical market data backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Establish explicit bound for N & convergence rate of COS method for options & Greeks] --> B[Inputs: Log-return density, Truncation range, Accuracy tolerance]
    B --> C[Theoretical Analysis: Error bounds for smooth densities with exponential vs. heavy tails]
    C --> D[Derive Explicit Bounds for N for Option Prices, Delta, and Gamma]
    C --> E[Numerical Experiments: Test standard vs. Heavy-Tailed models e.g., FMLS]
    D --> F[Findings: COS achieves exponential convergence for exponential tails, but loses it for heavy tails]
    E --> F