The purpose of this paper is to incorporate both skewness and kurtosis explicitly through extending Zhang (1994) to provide bounds for the prices of and expected payoffs for options, given the first two moments of skewness and kurtosis. The rest of this paper is organized as follows. Section II distributions of terminal stock prices with given expected prices, standard deviation, skewness, and kurtosis under the assumption that the underlying asset price is continuously distributed. Similar to the results given in (1), the bounds derived in this paper depend on the information of the cumulative distribution of the underlying asset, However, it is shown in Section II that for each set of moments, there always exists one semi-parametric upper bound which is independent of the information of any distribution of the underlying asset. This semi-parametric upper bound has the same property as that in the case of the first two moments in Zhang (1994), that is, it is always greater than or equal to the distribution dependent bounds.
