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

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