Error analysis of the COS method for options pricing
This thesis investigates the use of the Fourier cosine (COS) method for pricing European options under a variety of Lévy process models. The COS method is a Fourier-based technique that leverages the characteristic function of an asset’s returns to efficiently compute option prices. We apply it to five representative Lévy models: the Variance Gamma (VG), CGMY, Normal Inverse Gaussian (NIG), Merton jump-diffusion, and Kou double-exponential jump model, which encompass a range of jump and heavy-tailed behaviors. Through extensive numerical experiments, we demonstrate that the COS method achieves high accuracy and rapid convergence across all tested models. For instance, under the classical Black–Scholes (GBM) model, the COS expansion attains near machine-precision accuracy with only around \( N = 32 \) terms. Even in jump-heavy scenarios like the CGMY and VG models, careful choice of the integration range and number of expansion terms is shown to yield an exponential decay of pricing errors. We find that selecting the truncation interval based on distribution cumulants helps provide a robust, model-consistent choice for the integration range. The method remains effective for jump-diffusion cases such as Merton’s and Kou’s models, highlighting its robustness despite the presence of asymmetry and heavy tails. A convergence analysis confirms that with appropriately chosen truncation limits and sufficient terms, pricing errors decrease exponentially while computation time grows only linearly. These results underscore the practical performance of the COS approach, establishing it as an accurate and efficient alternative for option pricing in a broad range of Lévy-driven markets.
Meistaraprófsritgerðir, Verkfræði, Fjármálaverkfræði, Fourier-based methods, COS method
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