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.

傅里叶余弦展开（COS）方法用于以非常有效的方式对欧式期权进行数字定价。要应用 COS 方法，必须指定两个参数：对数返回密度的截断范围和用余弦级数近似截断密度的项数N。如何选择截断范围是已知的。在这里，我们能够为N以及定价和敏感性找到一个明确且有用的界限，即希腊 Delta 和 Gamma，前提是对数回报的密度是平滑的。我们进一步证明，当密度平滑并呈指数衰减时，COS 方法具有指数级收敛。然而，当密度平滑且具有重尾时，如在有限矩对数稳定模型中，COS 方法不具有指数级收敛。数值实验证实了理论结果。