This thesis presents a new method to value the hedge ratio based on the COS method of Fang and Oosterlee (2008), which determines a European option value by using Fourier-cosine expansion of the density of log stock price. The key insight is in the close relation of the characteristic function with series coefficients of the Fouriercosine expansion of the density. This finding allows us to value the hedge ratio, the first derivative of option price with respect to the underlying asset, the distribution and the integration of the density of log stock price. We prove that the derivative of the option is equal to the derivative of the coefficients of its Fourier-cosine expansion under the exponential L´evy models, thus the hedge ratio can be approximated by the Fourier-cosine expansion that is similar to price the option using COS method. Furthermore, we analysis whether the error can be made arbitrarily small or not to consider if the COS method is suitable for the hedge ratio.