A Note on the Conditions for COS Convergence

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The COS method is a fundamental tool in quantitative finance for option pricing and density approximation. This paper strengthens the theoretical guarantees of the COS method, particularly for heavy-tailed distributions (Student-t with small degrees of freedom) commonly used in financial modeling. While the paper is purely theoretical, its results directly impact the reliability of COS-based pricing engines used in practice. The extension to multivariate settings is relevant for basket options and multi-asset derivatives. However, the paper does not propose new algorithms or trading strategies.

The paper is purely theoretical with no implementation required. The COS method itself (Fang and Oosterlee, 2009) is well-established and straightforward to implement. The contribution here is a theoretical condition (checking whether a density has a finite weighted L2 moment of order p > 1) that is simple to verify analytically for common distributions. No code or computational infrastructure is needed.

The paper is purely theoretical with complete proofs provided. All mathematical arguments are self-contained and verifiable. No computational experiments or code are needed to validate the results. The conditions stated (f ∈ L1 ∩ L2, finite weighted L2 moment of order p > 1) are simple and verifiable for any given density.